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#matrix-inversion
A common use of the pseudoinverse is to compute a 'best fit' (least squares) solution to a system of linear equations that lacks a unique solution
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Moore–Penrose inverse - Wikipedia
tegral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse. <span>A common use of the pseudoinverse is to compute a 'best fit' (least squares) solution to a system of linear equations that lacks a unique solution (see below under § Applications). Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the




#matrix-inversion

The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition.

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Moore–Penrose inverse - Wikipedia
tion (see below under § Applications). Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra. <span>The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition. Contents [hide] 1 Notation 2 Definition 3 Properties 3.1 Existence and uniqueness 3.2 Basic properties 3.2.1 Identities 3.3 Reduction to Hermitian case 3.4 Products 3.5




#matrix-inversion

For , a pseudoinverse of is defined as a matrix satisfying all of the following four criteria:

  1. ( AA+ need not be the general identity matrix, but it maps all column vectors of A to themselves);
  2. ( A+ is a weak inverse for the multiplicative semigroup);
  3. ( AA+ is Hermitian); and
  4. ( A+A is also Hermitian).

Moore-Penrose Pseudo-inverse

exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula.

In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as:

...
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Moore–Penrose inverse - Wikipedia
; K ) {\displaystyle I_{n}\in \mathrm {M} (n,n;K)} denotes the n × n {\displaystyle n\times n} identity matrix. Definition[edit source] <span>For A ∈ M ( m , n ; K ) {\displaystyle A\in \mathrm {M} (m,n;K)} , a pseudoinverse of A {\displaystyle A} is defined as a matrix A + ∈ M ( n , m ; K ) {\displaystyle A^{+}\in \mathrm {M} (n,m;K)} satisfying all of the following four criteria: [8] [9] A A + A = A {\displaystyle AA^{+}A=A\,\!} (AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); A + A A + = A + {\displaystyle A^{+}AA^{+}=A^{+}\,\!} (A + is a weak inverse for the multiplicative semigroup); ( A A + ) ∗ = A A + {\displaystyle (AA^{+})^{*}=AA^{+}\,\!} (AA + is Hermitian); and ( A + A ) ∗ = A + A {\displaystyle (A^{+}A)^{*}=A^{+}A\,\!} (A + A is also Hermitian). A + {\displaystyle A^{+}} exists for any matrix A {\displaystyle A} , but when the latter has full rank, A + {\displaystyle A^{+}} can be expressed as a simple algebraic formula. In particular, when A {\displaystyle A} has linearly independent columns (and thus matrix A ∗ A {\displaystyle A^{*}A} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = ( A ∗ A ) − 1 A ∗ . {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}\,.} This particular pseudoinverse constitutes a left inverse, since, in this case, A + A = I {\displaystyle A^{+}A=I} . When A {\displaystyle A} has linearly independent rows (matrix A A ∗ {\displaystyle AA^{*}} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = A ∗ ( A A ∗ ) − 1 . {\displaystyle A^{+}=A^{*}(AA^{*})^{-1}\,.} This is a right inverse, as A A + = I {\displaystyle AA^{+}=I} . Properties[edit source] Proofs for some of these facts may be found on a separate page Proofs involving the Moore–Penrose inverse. Existence and uniqueness[edit source] The pseu




#linear-algebra #matrix-decomposition

In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way.

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Eigendecomposition of a matrix - Wikipedia
| ocultar ahora Eigendecomposition of a matrix From Wikipedia, the free encyclopedia (Redirected from Eigendecomposition) Jump to: navigation, search <span>In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. Contents [hide] 1 Fundamental theory of matrix eigenvectors and eigenvalues 2 Eigendecomposition of a matrix 2.1 Example 2.2 Matrix inverse via eigendecomposition 2.2.1 Pr




#linear-algebra #matrix-decomposition
The eigendecomposition can be derived from the fundamental property of eigenvectors: and thus which yields .
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Eigendecomposition of a matrix - Wikipedia
, {\displaystyle v_{i}\,\,(i=1,\dots ,N),} can also be used as the columns of Q. That can be understood by noting that the magnitude of the eigenvectors in Q gets canceled in the decomposition by the presence of Q −1 . <span>The decomposition can be derived from the fundamental property of eigenvectors: A v = λ v {\displaystyle \mathbf {A} \mathbf {v} =\lambda \mathbf {v} } and thus A Q = Q Λ {\displaystyle \mathbf {A} \mathbf {Q} =\mathbf {Q} \mathbf {\Lambda } } which yields A = Q Λ Q − 1 {\displaystyle \mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{-1}} . Example[edit source] Taking a 2 × 2 real matrix A = [




#gaussian-process
Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data.
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Gaussian process - Wikipedia
f them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. <span>Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data. The prediction is not just an estimate for that point, but also has uncertainty information—it is a one-dimensional Gaussian distribution (which is the marginal distribution at that poi




#gaussian-process
A key fact of Gaussian processes is that they can be completely defined by their second-order statistics.
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Gaussian process - Wikipedia
μ ℓ {\displaystyle \mu _{\ell }} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] <span>A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of t




#gaussian-process
if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion.
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Gaussian process - Wikipedia
} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, <span>if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] Stationarity refers to the process' beha




#gaussian-process
Stationarity refers to the process' behaviour regarding the separation of any two points x and x' . If the process is stationary, it depends on their separation, xx', while if non-stationary it depends on the actual position of the points x and x'.
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Gaussian process - Wikipedia
initeness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] <span>Stationarity refers to the process' behaviour regarding the separation of any two points x and x' . If the process is stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. If the process depends only on |x − x'|, the Euclidean distance (not the dire




#gaussian-process

If the process depends only on |xx'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous;[7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer.

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Gaussian process - Wikipedia
stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne




#gaussian-process
If we expect that for "near-by" input points x and x' their corresponding output points y and y' to be "near-by" also, then the assumption of continuity is present.
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Gaussian process - Wikipedia
r the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] <span>If we expect that for "near-by" input points x and x' their corresponding output points y and y' to be "near-by" also, then the assumption of continuity is present. If we wish to allow for significant displacement then we might choose a rougher covariance function. Extreme examples of the behaviour is the Ornstein–Uhlenbeck covariance function and




#gaussian-process

Periodicity refers to inducing periodic patterns within the behaviour of the process. Formally, this is achieved by mapping the input x to a two dimensional vector u(x) = (cos(x), sin(x)).

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Gaussian process - Wikipedia
en we might choose a rougher covariance function. Extreme examples of the behaviour is the Ornstein–Uhlenbeck covariance function and the squared exponential where the former is never differentiable and the latter infinitely differentiable. <span>Periodicity refers to inducing periodic patterns within the behaviour of the process. Formally, this is achieved by mapping the input x to a two dimensional vector u(x) = (cos(x), sin(x)). Usual covariance functions[edit source] [imagelink] The effect of choosing different kernels on the prior function distribution of the Gaussian process. Left is a squared expon




#gaussian-process
Importantly, a complicated covariance function can be defined as a linear combination of other simpler covariance functions in order to incorporate different insights about the data-set at hand.
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Gaussian process - Wikipedia
{\displaystyle \nu } and Γ ( ν ) {\displaystyle \Gamma (\nu )} is the gamma function evaluated at ν {\displaystyle \nu } . <span>Importantly, a complicated covariance function can be defined as a linear combination of other simpler covariance functions in order to incorporate different insights about the data-set at hand. Clearly, the inferential results are dependent on the values of the hyperparameters θ (e.g. ℓ and σ) defining the model's behaviour. A popular choice for θ is to provide maximum a pos




Flashcard 1729609469196

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#gaussian-process
Question
Importantly, a complicated covariance function can be defined as a [...] of other simpler covariance functions in order to incorporate different insights about the data-set at hand.
Answer
linear combination

Perhaps more than linear combination.

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Importantly, a complicated covariance function can be defined as a linear combination of other simpler covariance functions in order to incorporate different insights about the data-set at hand.

Original toplevel document

Gaussian process - Wikipedia
{\displaystyle \nu } and Γ ( ν ) {\displaystyle \Gamma (\nu )} is the gamma function evaluated at ν {\displaystyle \nu } . <span>Importantly, a complicated covariance function can be defined as a linear combination of other simpler covariance functions in order to incorporate different insights about the data-set at hand. Clearly, the inferential results are dependent on the values of the hyperparameters θ (e.g. ℓ and σ) defining the model's behaviour. A popular choice for θ is to provide maximum a pos







Flashcard 1729611042060

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#gaussian-process
Question

Periodicity maps the input x to a two dimensional vector [...]

Answer
u(x) = (cos(x), sin(x)).

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Periodicity refers to inducing periodic patterns within the behaviour of the process. Formally, this is achieved by mapping the input x to a two dimensional vector u(x) = (cos(x), sin(x)).

Original toplevel document

Gaussian process - Wikipedia
en we might choose a rougher covariance function. Extreme examples of the behaviour is the Ornstein–Uhlenbeck covariance function and the squared exponential where the former is never differentiable and the latter infinitely differentiable. <span>Periodicity refers to inducing periodic patterns within the behaviour of the process. Formally, this is achieved by mapping the input x to a two dimensional vector u(x) = (cos(x), sin(x)). Usual covariance functions[edit source] [imagelink] The effect of choosing different kernels on the prior function distribution of the Gaussian process. Left is a squared expon







Flashcard 1729612614924

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#gaussian-process
Question
If we expect that for "near-by" input points x and x' their corresponding output points y and y' to be "near-by" also, then the assumption of [...] is present.
Answer
continuity

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If we expect that for "near-by" input points x and x' their corresponding output points y and y' to be "near-by" also, then the assumption of continuity is present.

Original toplevel document

Gaussian process - Wikipedia
r the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] <span>If we expect that for "near-by" input points x and x' their corresponding output points y and y' to be "near-by" also, then the assumption of continuity is present. If we wish to allow for significant displacement then we might choose a rougher covariance function. Extreme examples of the behaviour is the Ornstein–Uhlenbeck covariance function and







#gaussian-process
If the process depends only on |xx'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic.
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If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the be

Original toplevel document

Gaussian process - Wikipedia
stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne




Flashcard 1729615760652

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#gaussian-process
Question
isotropic process depend only on distance, not [...]
Answer
direction

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If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic.

Original toplevel document

Gaussian process - Wikipedia
stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne







Flashcard 1729617333516

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#gaussian-process
Question
a stochastic process is called [...] if it depends only on distance but not the direction
Answer
isotropic

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If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic.

Original toplevel document

Gaussian process - Wikipedia
stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne







#gaussian-process
A process that is concurrently stationary and isotropic is considered to be homogeneous;[7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer.
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If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer.

Original toplevel document

Gaussian process - Wikipedia
stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne




Flashcard 1729620479244

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#gaussian-process
Question
homogeneous process behaves the same regardless the location of [...].
Answer
the observer

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A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of <span>the observer. <span><body><html>

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Gaussian process - Wikipedia
stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne







Flashcard 1729622052108

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#gaussian-process
Question
Stationarity process' behaviour depends on the distance between points, not their [...].
Answer
the actual positions

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y> Stationarity refers to the process' behaviour regarding the separation of any two points x and x' . If the process is stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. <body><html>

Original toplevel document

Gaussian process - Wikipedia
initeness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] <span>Stationarity refers to the process' behaviour regarding the separation of any two points x and x' . If the process is stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. If the process depends only on |x − x'|, the Euclidean distance (not the dire







#gaussian-process
if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour.
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if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion.

Original toplevel document

Gaussian process - Wikipedia
} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, <span>if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] Stationarity refers to the process' beha




Flashcard 1729625197836

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#gaussian-process
Question
if a Gaussian process is assumed to have mean zero, defining [...] completely defines the process' behaviour.
Answer

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if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour.

Original toplevel document

Gaussian process - Wikipedia
} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, <span>if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] Stationarity refers to the process' beha







Flashcard 1729626770700

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#gaussian-process
Question
In Gaussian process, the non-negative definiteness of the covariance function enables its [...] using the Karhunen–Loeve expansion.
Answer
spectral decomposition

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head> if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. <html>

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Gaussian process - Wikipedia
} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, <span>if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] Stationarity refers to the process' beha







Flashcard 1729628343564

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#gaussian-process
Question
Gaussian processes can be completely defined by their [...].
Answer
second-order statistics

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A key fact of Gaussian processes is that they can be completely defined by their second-order statistics.

Original toplevel document

Gaussian process - Wikipedia
μ ℓ {\displaystyle \mu _{\ell }} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] <span>A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of t







Flashcard 1729634897164

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#gaussian-process
Question
Viewed as a machine-learning algorithm, a Gaussian process uses [...] and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data.
Answer

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Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data.

Original toplevel document

Gaussian process - Wikipedia
f them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. <span>Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data. The prediction is not just an estimate for that point, but also has uncertainty information—it is a one-dimensional Gaussian distribution (which is the marginal distribution at that poi







Flashcard 1729636470028

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#gaussian-process
Question
Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of [...] to predict the value for an unseen point from training data.
Answer
the similarity between points

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Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data.

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Gaussian process - Wikipedia
f them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. <span>Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data. The prediction is not just an estimate for that point, but also has uncertainty information—it is a one-dimensional Gaussian distribution (which is the marginal distribution at that poi







Flashcard 1729641712908

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The eigendecomposition can be derived from [...] and thus which yields .
Answer
the fundamental property of eigenvectors:

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The eigendecomposition can be derived from the fundamental property of eigenvectors: and thus which yields .

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Eigendecomposition of a matrix - Wikipedia
, {\displaystyle v_{i}\,\,(i=1,\dots ,N),} can also be used as the columns of Q. That can be understood by noting that the magnitude of the eigenvectors in Q gets canceled in the decomposition by the presence of Q −1 . <span>The decomposition can be derived from the fundamental property of eigenvectors: A v = λ v {\displaystyle \mathbf {A} \mathbf {v} =\lambda \mathbf {v} } and thus A Q = Q Λ {\displaystyle \mathbf {A} \mathbf {Q} =\mathbf {Q} \mathbf {\Lambda } } which yields A = Q Λ Q − 1 {\displaystyle \mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{-1}} . Example[edit source] Taking a 2 × 2 real matrix A = [







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The eigendecomposition decomposes matrix A to [...]
Answer
.

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The eigendecomposition can be derived from the fundamental property of eigenvectors: and thus which yields .

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Eigendecomposition of a matrix - Wikipedia
, {\displaystyle v_{i}\,\,(i=1,\dots ,N),} can also be used as the columns of Q. That can be understood by noting that the magnitude of the eigenvectors in Q gets canceled in the decomposition by the presence of Q −1 . <span>The decomposition can be derived from the fundamental property of eigenvectors: A v = λ v {\displaystyle \mathbf {A} \mathbf {v} =\lambda \mathbf {v} } and thus A Q = Q Λ {\displaystyle \mathbf {A} \mathbf {Q} =\mathbf {Q} \mathbf {\Lambda } } which yields A = Q Λ Q − 1 {\displaystyle \mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{-1}} . Example[edit source] Taking a 2 × 2 real matrix A = [







Flashcard 1729646955788

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Question

[...] is the factorization of a matrix into a canonical form

Answer
eigendecomposition

Also called spectral decomposition

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In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonal

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Eigendecomposition of a matrix - Wikipedia
| ocultar ahora Eigendecomposition of a matrix From Wikipedia, the free encyclopedia (Redirected from Eigendecomposition) Jump to: navigation, search <span>In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. Contents [hide] 1 Fundamental theory of matrix eigenvectors and eigenvalues 2 Eigendecomposition of a matrix 2.1 Example 2.2 Matrix inverse via eigendecomposition 2.2.1 Pr







Flashcard 1729648528652

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eigendecomposition is sometimes also called [...]


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In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in

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Eigendecomposition of a matrix - Wikipedia
| ocultar ahora Eigendecomposition of a matrix From Wikipedia, the free encyclopedia (Redirected from Eigendecomposition) Jump to: navigation, search <span>In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. Contents [hide] 1 Fundamental theory of matrix eigenvectors and eigenvalues 2 Eigendecomposition of a matrix 2.1 Example 2.2 Matrix inverse via eigendecomposition 2.2.1 Pr







Flashcard 1729650101516

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#linear-algebra #matrix-decomposition
Question

eigendecomposition factorises a matrix into a [...] form

Answer
canonical

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In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way.

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Eigendecomposition of a matrix - Wikipedia
| ocultar ahora Eigendecomposition of a matrix From Wikipedia, the free encyclopedia (Redirected from Eigendecomposition) Jump to: navigation, search <span>In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. Contents [hide] 1 Fundamental theory of matrix eigenvectors and eigenvalues 2 Eigendecomposition of a matrix 2.1 Example 2.2 Matrix inverse via eigendecomposition 2.2.1 Pr







Flashcard 1729683393804

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#matrix-inversion
Question

The pseudoinverse can be computed using [...].

Answer
singular value decomposition

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The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition.

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Moore–Penrose inverse - Wikipedia
tion (see below under § Applications). Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra. <span>The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition. Contents [hide] 1 Notation 2 Definition 3 Properties 3.1 Existence and uniqueness 3.2 Basic properties 3.2.1 Identities 3.3 Reduction to Hermitian case 3.4 Products 3.5







Flashcard 1729684966668

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A common use of the pseudoinverse is to compute a [...] to a system of linear equations that lacks a unique solution
Answer
'best fit' (least squares) solution

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A common use of the pseudoinverse is to compute a 'best fit' (least squares) solution to a system of linear equations that lacks a unique solution

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Moore–Penrose inverse - Wikipedia
tegral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse. <span>A common use of the pseudoinverse is to compute a 'best fit' (least squares) solution to a system of linear equations that lacks a unique solution (see below under § Applications). Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the







#singular-value-decomposition
In linear algebra, the singular-value decomposition (SVD) generalises the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any matrix via an extension of the polar decomposition.
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Singular-value decomposition - Wikipedia
nto three simple transformations: an initial rotation V ∗ , a scaling Σ along the coordinate axes, and a final rotation U. The lengths σ 1 and σ 2 of the semi-axes of the ellipse are the singular values of M, namely Σ 1,1 and Σ 2,2 . <span>In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. Formally, the singular-value decomposition of an m × n {\d




#singular-value-decomposition
Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is a rectangular diagonal matrix with non-negative real numbers on the diagonal, and is an real or complex unitary matrix.
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Singular-value decomposition - Wikipedia
ositive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. <span>Formally, the singular-value decomposition of an m × n {\displaystyle m\times n} real or complex matrix M {\displaystyle \mathbf {M} } is a factorization of the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } is an m × m {\displaystyle m\times m} real or complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is a m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {\displaystyle \mathbf {V} } is an n × n {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries σ i {\displaystyle \sigma _{i}} of




Flashcard 1729714326796

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#singular-value-decomposition
Question
[...] generalises eigendecomposition of a positive semidefinite normal matrix to any matrix
Answer
singular-value decomposition (SVD)

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In linear algebra, the singular-value decomposition (SVD) generalises the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any matrix via an extension of the polar deco

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Singular-value decomposition - Wikipedia
nto three simple transformations: an initial rotation V ∗ , a scaling Σ along the coordinate axes, and a final rotation U. The lengths σ 1 and σ 2 of the semi-axes of the ellipse are the singular values of M, namely Σ 1,1 and Σ 2,2 . <span>In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. Formally, the singular-value decomposition of an m × n {\d







#singular-value-decomposition
Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form
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Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is a rectangular diagonal matrix with non-negative real numbers on the diagonal, and is an real or complex unitary matrix.

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Singular-value decomposition - Wikipedia
ositive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. <span>Formally, the singular-value decomposition of an m × n {\displaystyle m\times n} real or complex matrix M {\displaystyle \mathbf {M} } is a factorization of the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } is an m × m {\displaystyle m\times m} real or complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is a m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {\displaystyle \mathbf {V} } is an n × n {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries σ i {\displaystyle \sigma _{i}} of




Flashcard 1729718258956

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singular-value decomposition factorises an matrix M to the form [...]
Answer

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Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form

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Singular-value decomposition - Wikipedia
ositive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. <span>Formally, the singular-value decomposition of an m × n {\displaystyle m\times n} real or complex matrix M {\displaystyle \mathbf {M} } is a factorization of the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } is an m × m {\displaystyle m\times m} real or complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is a m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {\displaystyle \mathbf {V} } is an n × n {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries σ i {\displaystyle \sigma _{i}} of







Flashcard 1729720618252

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#singular-value-decomposition
Question
With a factorization of the form , , , represent [...]
Answer
real or complex unitary matrix
real or complex unitary matrix.
rectangular diagonal matrix

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Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is a rectangular diagonal matrix with non-negative real numbers on the diagonal, and is an real or complex unitary matrix.

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Singular-value decomposition - Wikipedia
ositive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. <span>Formally, the singular-value decomposition of an m × n {\displaystyle m\times n} real or complex matrix M {\displaystyle \mathbf {M} } is a factorization of the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } is an m × m {\displaystyle m\times m} real or complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is a m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {\displaystyle \mathbf {V} } is an n × n {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries σ i {\displaystyle \sigma _{i}} of







#variational-inference
Variational Bayes can be seen as an extension of the EM (expectation-maximization) algorithm from maximum a posteriori estimation (MAP estimation) of the single most probable value of each parameter to fully Bayesian estimation which computes (an approximation to) the entire posterior distribution of the parameters and latent variables.
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Variational Bayesian methods - Wikipedia
om. In particular, whereas Monte Carlo techniques provide a numerical approximation to the exact posterior using a set of samples, Variational Bayes provides a locally-optimal, exact analytical solution to an approximation of the posterior. <span>Variational Bayes can be seen as an extension of the EM (expectation-maximization) algorithm from maximum a posteriori estimation (MAP estimation) of the single most probable value of each parameter to fully Bayesian estimation which computes (an approximation to) the entire posterior distribution of the parameters and latent variables. As in EM, it finds a set of optimal parameter values, and it has the same alternating structure as does EM, based on a set of interlocked (mutually dependent) equations that cannot be s




Flashcard 1730164690188

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[...] can be seen as an extension of the expectation-maximization algorithm
Answer
Variational inference

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Variational Bayes can be seen as an extension of the EM (expectation-maximization) algorithm from maximum a posteriori estimation (MAP estimation) of the single most probable value of each parameter to f

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Variational Bayesian methods - Wikipedia
om. In particular, whereas Monte Carlo techniques provide a numerical approximation to the exact posterior using a set of samples, Variational Bayes provides a locally-optimal, exact analytical solution to an approximation of the posterior. <span>Variational Bayes can be seen as an extension of the EM (expectation-maximization) algorithm from maximum a posteriori estimation (MAP estimation) of the single most probable value of each parameter to fully Bayesian estimation which computes (an approximation to) the entire posterior distribution of the parameters and latent variables. As in EM, it finds a set of optimal parameter values, and it has the same alternating structure as does EM, based on a set of interlocked (mutually dependent) equations that cannot be s







#variational-inference
Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.
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Variational Bayesian methods - Wikipedia
f references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (September 2010) (Learn how and when to remove this template message) <span>Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various




#variational-inference

Variational Bayesian methods are primarily used for two purposes:

  1. To provide an analytical approximation to the posterior probability of the unobserved variables, in order to do statistical inference over these variables.
  2. To derive a lower bound for the marginal likelihood (sometimes called the "evidence") of the observed data (i.e. the marginal probability of the data given the model, with marginalization performed over unobserved variables). This is typically used for performing model selection, the general idea being that a higher marginal likelihood for a given model indicates a better fit of the data by that model and hence a greater probability that the model in question was the one that generated the data. (See also the Bayes factor article.)
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Variational Bayesian methods - Wikipedia
s sorts of relationships among the three types of random variables, as might be described by a graphical model. As is typical in Bayesian inference, the parameters and latent variables are grouped together as "unobserved variables". <span>Variational Bayesian methods are primarily used for two purposes: To provide an analytical approximation to the posterior probability of the unobserved variables, in order to do statistical inference over these variables. To derive a lower bound for the marginal likelihood (sometimes called the "evidence") of the observed data (i.e. the marginal probability of the data given the model, with marginalization performed over unobserved variables). This is typically used for performing model selection, the general idea being that a higher marginal likelihood for a given model indicates a better fit of the data by that model and hence a greater probability that the model in question was the one that generated the data. (See also the Bayes factor article.) In the former purpose (that of approximating a posterior probability), variational Bayes is an alternative to Monte Carlo sampling methods — particularly, Markov chain Monte Carlo met




#deep-gaussian-process
DGPs can perform input warping or dimensionality compression or expansion, and automatically learn to construct a kernel that works well for the data at hand. As a result, learning in this model provides a flexible form of Bayesian kernel design.
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#variational-inference
For many applications, variational Bayes produces solutions of comparable accuracy to Gibbs sampling at greater speed.
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Variational Bayesian methods - Wikipedia
he parameters and latent variables. As in EM, it finds a set of optimal parameter values, and it has the same alternating structure as does EM, based on a set of interlocked (mutually dependent) equations that cannot be solved analytically. <span>For many applications, variational Bayes produces solutions of comparable accuracy to Gibbs sampling at greater speed. However, deriving the set of equations used to iteratively update the parameters often requires a large amount of work compared with deriving the comparable Gibbs sampling equations. Th




#singular-value-decomposition
SVD as change of coordinates

The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : KnKm one can find orthonormal bases of Kn and Km such that T maps the i-th basis vector of Kn to a non-negative multiple of the i-th basis vector of Km , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries.
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Singular-value decomposition - Wikipedia
m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map




#matrix-inversion
A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition.[1][9][15] If is the singular value decomposition of A , then . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv , the tolerance is taken to be t = ε⋅max(m,n)⋅max(Σ) , where ε is the machine epsilon.
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Moore–Penrose inverse - Wikipedia
A {\displaystyle A} and A ∗ {\displaystyle A^{*}} . Singular value decomposition (SVD)[edit source] <span>A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If A = U Σ V ∗ {\displaystyle A=U\Sigma V^{*}} is the singular value decomposition of A, then A + = V Σ + U ∗ {\displaystyle A^{+}=V\Sigma ^{+}U^{*}} . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv , the tolerance is taken to be t = ε⋅max(m,n)⋅max(Σ), where ε is the machine epsilon. The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implem




#matrix-inversion

Moore-Penrose Pseudo-inverse

exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula.

In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as:

This particular pseudoinverse constitutes a left inverse, since, in this case, .

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3; ( AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); ( A + is a weak inverse for the multiplicative semigroup); ( AA + is Hermitian); and ( A + A is also Hermitian). <span>Moore-Penrose Pseudo-inverse exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula. In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as: This particular pseudoinverse constitutes a left inverse, since, in this case, . When has linearly independent rows (matrix is invertible), can be computed as: This is a right inverse, as . <span><body><html>

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Moore–Penrose inverse - Wikipedia
; K ) {\displaystyle I_{n}\in \mathrm {M} (n,n;K)} denotes the n × n {\displaystyle n\times n} identity matrix. Definition[edit source] <span>For A ∈ M ( m , n ; K ) {\displaystyle A\in \mathrm {M} (m,n;K)} , a pseudoinverse of A {\displaystyle A} is defined as a matrix A + ∈ M ( n , m ; K ) {\displaystyle A^{+}\in \mathrm {M} (n,m;K)} satisfying all of the following four criteria: [8] [9] A A + A = A {\displaystyle AA^{+}A=A\,\!} (AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); A + A A + = A + {\displaystyle A^{+}AA^{+}=A^{+}\,\!} (A + is a weak inverse for the multiplicative semigroup); ( A A + ) ∗ = A A + {\displaystyle (AA^{+})^{*}=AA^{+}\,\!} (AA + is Hermitian); and ( A + A ) ∗ = A + A {\displaystyle (A^{+}A)^{*}=A^{+}A\,\!} (A + A is also Hermitian). A + {\displaystyle A^{+}} exists for any matrix A {\displaystyle A} , but when the latter has full rank, A + {\displaystyle A^{+}} can be expressed as a simple algebraic formula. In particular, when A {\displaystyle A} has linearly independent columns (and thus matrix A ∗ A {\displaystyle A^{*}A} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = ( A ∗ A ) − 1 A ∗ . {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}\,.} This particular pseudoinverse constitutes a left inverse, since, in this case, A + A = I {\displaystyle A^{+}A=I} . When A {\displaystyle A} has linearly independent rows (matrix A A ∗ {\displaystyle AA^{*}} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = A ∗ ( A A ∗ ) − 1 . {\displaystyle A^{+}=A^{*}(AA^{*})^{-1}\,.} This is a right inverse, as A A + = I {\displaystyle AA^{+}=I} . Properties[edit source] Proofs for some of these facts may be found on a separate page Proofs involving the Moore–Penrose inverse. Existence and uniqueness[edit source] The pseu




Flashcard 1731444477196

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when has [...] the Moore-Penrose inverse is a left inverse

Answer
linearly independent columns

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ml> Moore-Penrose Pseudo-inverse exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula. In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as: This particular pseudoinverse constitutes a left inverse, since, in this case, . <html>

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Moore–Penrose inverse - Wikipedia
; K ) {\displaystyle I_{n}\in \mathrm {M} (n,n;K)} denotes the n × n {\displaystyle n\times n} identity matrix. Definition[edit source] <span>For A ∈ M ( m , n ; K ) {\displaystyle A\in \mathrm {M} (m,n;K)} , a pseudoinverse of A {\displaystyle A} is defined as a matrix A + ∈ M ( n , m ; K ) {\displaystyle A^{+}\in \mathrm {M} (n,m;K)} satisfying all of the following four criteria: [8] [9] A A + A = A {\displaystyle AA^{+}A=A\,\!} (AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); A + A A + = A + {\displaystyle A^{+}AA^{+}=A^{+}\,\!} (A + is a weak inverse for the multiplicative semigroup); ( A A + ) ∗ = A A + {\displaystyle (AA^{+})^{*}=AA^{+}\,\!} (AA + is Hermitian); and ( A + A ) ∗ = A + A {\displaystyle (A^{+}A)^{*}=A^{+}A\,\!} (A + A is also Hermitian). A + {\displaystyle A^{+}} exists for any matrix A {\displaystyle A} , but when the latter has full rank, A + {\displaystyle A^{+}} can be expressed as a simple algebraic formula. In particular, when A {\displaystyle A} has linearly independent columns (and thus matrix A ∗ A {\displaystyle A^{*}A} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = ( A ∗ A ) − 1 A ∗ . {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}\,.} This particular pseudoinverse constitutes a left inverse, since, in this case, A + A = I {\displaystyle A^{+}A=I} . When A {\displaystyle A} has linearly independent rows (matrix A A ∗ {\displaystyle AA^{*}} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = A ∗ ( A A ∗ ) − 1 . {\displaystyle A^{+}=A^{*}(AA^{*})^{-1}\,.} This is a right inverse, as A A + = I {\displaystyle AA^{+}=I} . Properties[edit source] Proofs for some of these facts may be found on a separate page Proofs involving the Moore–Penrose inverse. Existence and uniqueness[edit source] The pseu







Flashcard 1731448147212

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#matrix-inversion
Question

The left Moore-Penrose Pseudo-inverse is [...]

Answer

This is the one for linear models


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e exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula. In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as<span>: This particular pseudoinverse constitutes a left inverse, since, in this case, . <span><body><html>

Original toplevel document

Moore–Penrose inverse - Wikipedia
; K ) {\displaystyle I_{n}\in \mathrm {M} (n,n;K)} denotes the n × n {\displaystyle n\times n} identity matrix. Definition[edit source] <span>For A ∈ M ( m , n ; K ) {\displaystyle A\in \mathrm {M} (m,n;K)} , a pseudoinverse of A {\displaystyle A} is defined as a matrix A + ∈ M ( n , m ; K ) {\displaystyle A^{+}\in \mathrm {M} (n,m;K)} satisfying all of the following four criteria: [8] [9] A A + A = A {\displaystyle AA^{+}A=A\,\!} (AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); A + A A + = A + {\displaystyle A^{+}AA^{+}=A^{+}\,\!} (A + is a weak inverse for the multiplicative semigroup); ( A A + ) ∗ = A A + {\displaystyle (AA^{+})^{*}=AA^{+}\,\!} (AA + is Hermitian); and ( A + A ) ∗ = A + A {\displaystyle (A^{+}A)^{*}=A^{+}A\,\!} (A + A is also Hermitian). A + {\displaystyle A^{+}} exists for any matrix A {\displaystyle A} , but when the latter has full rank, A + {\displaystyle A^{+}} can be expressed as a simple algebraic formula. In particular, when A {\displaystyle A} has linearly independent columns (and thus matrix A ∗ A {\displaystyle A^{*}A} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = ( A ∗ A ) − 1 A ∗ . {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}\,.} This particular pseudoinverse constitutes a left inverse, since, in this case, A + A = I {\displaystyle A^{+}A=I} . When A {\displaystyle A} has linearly independent rows (matrix A A ∗ {\displaystyle AA^{*}} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = A ∗ ( A A ∗ ) − 1 . {\displaystyle A^{+}=A^{*}(AA^{*})^{-1}\,.} This is a right inverse, as A A + = I {\displaystyle AA^{+}=I} . Properties[edit source] Proofs for some of these facts may be found on a separate page Proofs involving the Moore–Penrose inverse. Existence and uniqueness[edit source] The pseu







Flashcard 1731451292940

Tags
#singular-value-decomposition
Question
geometrically SVD finds [...] for every linear map T : KnKm
Answer
orthonormal bases of Kn and Km

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SVD as change of coordinates The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases,

Original toplevel document

Singular-value decomposition - Wikipedia
m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map







Flashcard 1731453127948

Tags
#singular-value-decomposition
Question
Geometrically SVD finds orthonormal bases of Kn and Km for every linear map T : KnKm such that T maps the i-th basis vector of Kn to a non-negative multiple of the i-th basis vector of Km , and sends the left-over basis vectors to [...].
Answer
zero

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ollows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to <span>zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. <span><body><html>

Original toplevel document

Singular-value decomposition - Wikipedia
m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map







Flashcard 1731454700812

Tags
#singular-value-decomposition
Question
With SVD geometrically every linear map T : KnKm is represented by a diagonal matrix with [...] entries.
Answer
non-negative real diagonal

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h that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with <span>non-negative real diagonal entries. <span><body><html>

Original toplevel document

Singular-value decomposition - Wikipedia
m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map







#logic
According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.
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The rise and fall and rise of logic | Aeon Essays
the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara




Flashcard 1731479866636

Tags
#spanish
Question
use conditional simple to [...]:

¿Podrías pasarme ese plato, por favor?
Answer
ask politely

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Verbs: The Conditional Simple Usage: To ask politely: ¿Podrías pasarme ese plato, por favor? (Could you pass me that plate, please?).

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Verbs: The Conditional Simple Usage: To ask politely: ¿Podrías pasarme ese plato, por favor? (Could you pass me that plate, please?). To express wishes: ¡Me encantaría ir de viaje a Australia! (I would love to go on a trip to Australia!). To suggest: Creo que deberías ir al médico a verte ese dolor de espalda (I think







Flashcard 1731508440332

Tags
#logic
Question
According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were [...], the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.
Answer
the ancient Greek period

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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.

Original toplevel document

The rise and fall and rise of logic | Aeon Essays
the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara







Flashcard 1731510013196

Tags
#logic
Question
According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, [...], and the mathematical period of the 19th and 20th centuries.
Answer
the medieval scholastic period

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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.

Original toplevel document

The rise and fall and rise of logic | Aeon Essays
the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara







Flashcard 1731511586060

Tags
#logic
Question
According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and [...] of the 19th and 20th centuries.
Answer
the mathematical period

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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.

Original toplevel document

The rise and fall and rise of logic | Aeon Essays
the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara







#matrix-inversion
A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition.[1][9][15] If is the singular value decomposition of A , then .
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A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If is the singular value decomposition of A , then . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in p

Original toplevel document

Moore–Penrose inverse - Wikipedia
A {\displaystyle A} and A ∗ {\displaystyle A^{*}} . Singular value decomposition (SVD)[edit source] <span>A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If A = U Σ V ∗ {\displaystyle A=U\Sigma V^{*}} is the singular value decomposition of A, then A + = V Σ + U ∗ {\displaystyle A^{+}=V\Sigma ^{+}U^{*}} . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv , the tolerance is taken to be t = ε⋅max(m,n)⋅max(Σ), where ε is the machine epsilon. The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implem




Flashcard 1731520498956

Tags
#matrix-inversion
Question
If is the singular value decomposition of A , then the pseudoinverse of A is [...]
Answer
.

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A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If is the singular value decomposition of A , then .

Original toplevel document

Moore–Penrose inverse - Wikipedia
A {\displaystyle A} and A ∗ {\displaystyle A^{*}} . Singular value decomposition (SVD)[edit source] <span>A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If A = U Σ V ∗ {\displaystyle A=U\Sigma V^{*}} is the singular value decomposition of A, then A + = V Σ + U ∗ {\displaystyle A^{+}=V\Sigma ^{+}U^{*}} . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv , the tolerance is taken to be t = ε⋅max(m,n)⋅max(Σ), where ε is the machine epsilon. The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implem







Flashcard 1731525217548

Tags
#deep-gaussian-process
Question
pseudo datapoint based approximation methods for DGPs has a computational complexity of [...]
Answer
\(O(NLM^ 2 ) \)

where N is the number of datapoints, L is the number of layers, and M is the number of pseudo datapoints.

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pseudo datapoint based approximation methods for DGPs trade model complexity for a lower computational complexity of \(O(NLM^ 2 ) \) where N is the number of datapoints, L is the number of layers, and M is the number of pseudo datapoints.

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Flashcard 1731526790412

Tags
#deep-gaussian-process
Question
DGPs can perform [...] or dimensionality compression or expansion
Answer
input warping

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DGPs can perform input warping or dimensionality compression or expansion, and automatically learn to construct a kernel that works well for the data at hand. As a result, learning in this model provides a flexible f

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Flashcard 1731528363276

Tags
#deep-gaussian-process
Question
DGPs can perform input warping or [...]
Answer
dimensionality compression or expansion

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DGPs can perform input warping or dimensionality compression or expansion, and automatically learn to construct a kernel that works well for the data at hand. As a result, learning in this model provides a flexible form of Bayesian kernel design. </sp

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Flashcard 1731529936140

Tags
#deep-gaussian-process
Question
DGPs can automatically learn to [...] that works well for the data at hand.
Answer
construct a kernel

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DGPs can perform input warping or dimensionality compression or expansion, and automatically learn to construct a kernel that works well for the data at hand. As a result, learning in this model provides a flexible form of Bayesian kernel design.

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Flashcard 1731535703308

Tags
#variational-inference
Question
Variational Bayesian methods are a family of techniques for approximating [...] arising in Bayesian inference and machine learning.
Answer
intractable integrals

In Bayesian inference this manifests as calculating marginal posteriors

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Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.

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Variational Bayesian methods - Wikipedia
f references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (September 2010) (Learn how and when to remove this template message) <span>Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various







#dynamic-programming
In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by
  1. breaking it down into a collection of simpler subproblems,
  2. solving each of those subproblems just once, and
  3. storing their solutions.
The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space.
(Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.)
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Dynamic programming - Wikipedia
This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna




Flashcard 1731673328908

Tags
#dynamic-programming
Question
The first step in dynamic programming it to [...]
Answer
break the problem down into a collection of simpler subproblems

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head><head> In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the p

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Dynamic programming - Wikipedia
This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna







Flashcard 1731674901772

Tags
#dynamic-programming
Question
dynamic programming aims to solve each subproblems [...] and storing their solutions.
Answer
just once

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ter science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, <span>solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving c

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Dynamic programming - Wikipedia
This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna







Flashcard 1731676474636

Tags
#dynamic-programming
Question
dynamic programming solves each of the subproblems just once, and [...].
Answer
storing their solutions

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nomics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and <span>storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expens

Original toplevel document

Dynamic programming - Wikipedia
This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna







Flashcard 1731678047500

Tags
#dynamic-programming
Question
In dynamic programming, when the same subproblem occurs, instead of [...], one simply looks up the previously computed solution
Answer
recomputing its solution

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is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of <span>recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem

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Dynamic programming - Wikipedia
This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna







Flashcard 1731679620364

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#dynamic-programming
Question
In dynamic programming, the next time when the same subproblem occurs, instead of recomputing its solution, one simply [...]
Answer
looks up the previously computed solution

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blem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply <span>looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on t

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Dynamic programming - Wikipedia
This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna







Flashcard 1731681455372

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#dynamic-programming
Question
The purpose of dynamic programming is to [...].
Answer
trade space for time

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simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby <span>saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) <span><body><html>

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Dynamic programming - Wikipedia
This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna







Flashcard 1731683028236

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#dynamic-programming
Question
In dynamic programming each of the subproblem solutions is [...], typically based on the values of its input parameters, so as to facilitate its lookup.
Answer
indexed in some way

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occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is <span>indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) <span><body><html>

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Dynamic programming - Wikipedia
This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna







Flashcard 1731684601100

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#dynamic-programming
Question
subproblem solutions are typically indexed by [...] to facilitate lookup.
Answer
input values

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tion, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on <span>the values of its input parameters, so as to facilitate its lookup.) <span><body><html>

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Dynamic programming - Wikipedia
This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna







#finance
In finance, an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option.
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Option (finance) - Wikipedia
y Real estate Reinsurance Over-the-counter (off-exchange) Forwards Options Spot market Swaps Trading Participants Regulation Clearing Related areas Banks and banking Finance corporate personal public v t e <span>In finance, an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price (market price) of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or




#finance
An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed.
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Option (finance) - Wikipedia
or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. <span>An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed. The seller may grant an option to a buyer as part of another transaction, such as a share issue or as part of an employee incentive scheme, otherwise a buyer would pay a premium to th




Flashcard 1731692465420

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#finance
Question
an option gives the buyer [...], to buy or sell
Answer
the right, but not the obligation

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In finance, an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option.

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Option (finance) - Wikipedia
y Real estate Reinsurance Over-the-counter (off-exchange) Forwards Options Spot market Swaps Trading Participants Regulation Clearing Related areas Banks and banking Finance corporate personal public v t e <span>In finance, an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price (market price) of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or







Flashcard 1731694038284

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#finance
Question
an option gives the buyer the right to [...] at a specified strike price on a specified date, depending on the form of the option.
Answer
buy or sell an underlying asset or instrument

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In finance, an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option.

Original toplevel document

Option (finance) - Wikipedia
y Real estate Reinsurance Over-the-counter (off-exchange) Forwards Options Spot market Swaps Trading Participants Regulation Clearing Related areas Banks and banking Finance corporate personal public v t e <span>In finance, an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price (market price) of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or







Flashcard 1731695611148

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#finance
Question
[...] is an option to buy
Answer
a call

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An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed. <

Original toplevel document

Option (finance) - Wikipedia
or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. <span>An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed. The seller may grant an option to a buyer as part of another transaction, such as a share issue or as part of an employee incentive scheme, otherwise a buyer would pay a premium to th







Flashcard 1731697184012

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#finance
Question
[...] is an option to sell
Answer
a put

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An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed. <html>

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Option (finance) - Wikipedia
or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. <span>An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed. The seller may grant an option to a buyer as part of another transaction, such as a share issue or as part of an employee incentive scheme, otherwise a buyer would pay a premium to th







Flashcard 1731698756876

Tags
#finance
Question
the [...] option is more frequently discussed.
Answer
call

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An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but <span>the call option is more frequently discussed. <span><body><html>

Original toplevel document

Option (finance) - Wikipedia
or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. <span>An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed. The seller may grant an option to a buyer as part of another transaction, such as a share issue or as part of an employee incentive scheme, otherwise a buyer would pay a premium to th







#finance
The Black–Scholes / ˌ b l æ k ˈ ʃ oʊ l z / [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments.
  1. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce
  2. the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that
  3. the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate).
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Black–Scholes model - Wikipedia
Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]




Flashcard 1731703737612

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#finance
Question
The Black–Scholes model is a mathematical model of a financial market containing [...] instruments
Answer
derivative investment

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The Black–Scholes / ˌ b l æ k ˈ ʃ oʊ l z / [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of E

Original toplevel document

Black–Scholes model - Wikipedia
Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]







Flashcard 1731706096908

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#finance
Question
mathematically, the Black–Scholes equation is a [...]

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The Black–Scholes / ˌ b l æ k ˈ ʃ oʊ l z / [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a

Original toplevel document

Black–Scholes model - Wikipedia
Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]







Flashcard 1731707669772

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#finance
Question
the Black–Scholes formula estimates the price of [...]

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is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives <span>a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate)

Original toplevel document

Black–Scholes model - Wikipedia
Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]







Flashcard 1731709242636

Tags
#finance
Question
under the Black–Scholes–Merton model, the option [...] regardless of the risk of the security and its expected return.
Answer
the option has a unique price

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nt instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that <span>the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). <span><body><html>

Original toplevel document

Black–Scholes model - Wikipedia
Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]







Flashcard 1731710815500

Tags
#finance
Question
In the Black–Scholes model the option has a unique price regardless of [...]
Answer
the risk of the security and its expected return

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rential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of <span>the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). <span><body><html>

Original toplevel document

Black–Scholes model - Wikipedia
Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]







Flashcard 1731712388364

Tags
#finance
Question
Under the Black–Scholes–Merton model, the option's unique price is decided by [...]
Answer
the risk-neutral rate

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mula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with <span>the risk-neutral rate). <span><body><html>

Original toplevel document

Black–Scholes model - Wikipedia
Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]







#stochastics
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values.
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Martingale (probability theory) - Wikipedia
h For the martingale betting strategy, see martingale (betting system). [imagelink] Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. <span>In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Contents [hide] 1 History 2 Definitions 2.1 Martingale sequences with respect to another sequence 2.2 General definition 3 Examples of martingales 4 Submartingales, super




Flashcard 1731718679820

Tags
#stochastics
Question
for a [...], at a particular time in the realized sequence, the expectation of the next value is equal to the present observed value, even given knowledge of all prior observed values.
Answer
martingale

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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to th

Original toplevel document

Martingale (probability theory) - Wikipedia
h For the martingale betting strategy, see martingale (betting system). [imagelink] Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. <span>In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Contents [hide] 1 History 2 Definitions 2.1 Martingale sequences with respect to another sequence 2.2 General definition 3 Examples of martingales 4 Submartingales, super







Flashcard 1731720252684

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#stochastics
Question
a martingale is a [...]

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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior obse

Original toplevel document

Martingale (probability theory) - Wikipedia
h For the martingale betting strategy, see martingale (betting system). [imagelink] Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. <span>In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Contents [hide] 1 History 2 Definitions 2.1 Martingale sequences with respect to another sequence 2.2 General definition 3 Examples of martingales 4 Submartingales, super







Flashcard 1731721825548

Tags
#stochastics
Question
In a martingale [...] is equal to the present observed value even given knowledge of all prior observed values.
Answer
the expectation of the next value in the sequence

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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values.

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Martingale (probability theory) - Wikipedia
h For the martingale betting strategy, see martingale (betting system). [imagelink] Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. <span>In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Contents [hide] 1 History 2 Definitions 2.1 Martingale sequences with respect to another sequence 2.2 General definition 3 Examples of martingales 4 Submartingales, super







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In a martingale, the expectation of the next value in the sequence equals to [...]
Answer
the present observed value

even given knowledge of all prior observed values.

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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. <body><html>

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Martingale (probability theory) - Wikipedia
h For the martingale betting strategy, see martingale (betting system). [imagelink] Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. <span>In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Contents [hide] 1 History 2 Definitions 2.1 Martingale sequences with respect to another sequence 2.2 General definition 3 Examples of martingales 4 Submartingales, super







#gauss-markov-process

Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes.[1][2] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions.

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translations!] Gauss–Markov process From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Gauss–Markov theorem of mathematical statistics. <span>Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions. Every Gauss–Markov process X(t) possesses the three following properties: If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process If f(t) is




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Gauss–Markov stochastic processes satisfy the requirements for [...]


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Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions. </sp

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Gauss–Markov process - Wikipedia
translations!] Gauss–Markov process From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Gauss–Markov theorem of mathematical statistics. <span>Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions. Every Gauss–Markov process X(t) possesses the three following properties: If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process If f(t) is







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The stationary Gauss–Markov process is known as [...]


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kov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] The stationary Gauss–Markov process (also known as <span>a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions. <span><body><html>

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Gauss–Markov process - Wikipedia
translations!] Gauss–Markov process From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Gauss–Markov theorem of mathematical statistics. <span>Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions. Every Gauss–Markov process X(t) possesses the three following properties: If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process If f(t) is







Flashcard 1731762982156

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#spanish
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[...] significa ‘en la parte anterior’ y [...] equivale a ‘hacia allá’
Answer
Delante Adelante

Examples
Voy yo delante, que sé el camino.
Sigamos adelante.


La forma alante, por otro lado, es incorrecta.

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Delante significa ‘en la parte anterior’, ‘en frente’ o ‘ante alguien’, se usa por lo general para indicar la situación de alguien o algo. Adelante , por su parte, equivale a ‘más allá’, ‘hacia allá’, o ‘hacia enfrente’, y se emplea para indicar la existencia de un movimiento, sea real o figurado. La forma alante , por otro lado, es incorrecta.

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Delante o adelante - Diccionario de Dudas
amp;cj=1"> [imagelink] Palabras Homófonas Palabras Parónimas Fonética y fonología Uso Grafía Léxicas Ver más Latinismos Extranjerismos Barbarismos Ultracorrecciones Dudas de uso Delante o adelante Delante significa ‘en la parte anterior’, ‘en frente’ o ‘ante alguien’, se usa por lo general para indicar la situación de alguien o algo. Adelante , por su parte, equivale a ‘más allá’, ‘hacia allá’, o ‘hacia enfrente’, y se emplea para indicar la existencia de un movimiento, sea real o figurado. La forma alante , por otro lado, es incorrecta. Cuándo usar delante Delante es un adverbio de lugar; se emplea con el significado de ‘en la parte anterior’, ‘en frente’ o ‘en presencia de alguien’. Por lo general, es un adverbio que







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Ornstein–Uhlenbeck covariance function: [...]
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Gaussian process - Wikipedia
2 ) {\displaystyle K_{\operatorname {SE} }(x,x')=\exp {\Big (}-{\frac {\|d\|^{2}}{2\ell ^{2}}}{\Big )}} <span>Ornstein–Uhlenbeck: K OU ( x , x ′ ) = exp ⁡ ( − | d | ℓ ) {\displaystyle K_{\operatorname {OU} }(x,x')=\exp \left(-{\frac {|d|}{\ell }}\right)} Matérn: K Matern ( x , x ′ )







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Periodic covariance function: [...]
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{\displaystyle K_{\operatorname {Matern} }(x,x')={\frac {2^{1-\nu }}{\Gamma (\nu )}}{\Big (}{\frac {{\sqrt {2\nu }}|d|}{\ell }}{\Big )}^{\nu }K_{\nu }{\Big (}{\frac {{\sqrt {2\nu }}|d|}{\ell }}{\Big )}} <span>Periodic: K P ( x , x ′ ) = exp ⁡ ( − 2 sin 2 ⁡ ( d 2 ) ℓ 2 ) {\displaystyle K_{\operatorname {P} }(x,x')=\exp \left(-{\frac {2\sin ^{2}\left({\frac {d}{2}}\right)}{\ell ^{2}}}\right)} Rational quadratic: K RQ ( x , x ′







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Rational quadratic covariance function: [...]
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2 ) {\displaystyle K_{\operatorname {P} }(x,x')=\exp \left(-{\frac {2\sin ^{2}\left({\frac {d}{2}}\right)}{\ell ^{2}}}\right)} <span>Rational quadratic: K RQ ( x , x ′ ) = ( 1 + | d | 2 ) − α , α ≥ 0 {\displaystyle K_{\operatorname {RQ} }(x,x')=(1+|d|^{2})^{-\alpha },\quad \alpha \geq 0} Here d = x − x ′ {\displaystyle d=x-x'} . The parameter ℓ is the character







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J M Bocheński wrote [...] in 1961
Answer
A History of Formal Logic

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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centurie

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The rise and fall and rise of logic | Aeon Essays
the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara







#functional-analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

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Functional analysis - Wikipedia
analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that




#logic
The third of them, in contrast, exemplifies an entirely different rationale for logic, namely as a foundational branch of mathematics
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The rise and fall and rise of logic | Aeon Essays
grand periods in the history of logic, two of them, the ancient period and the medieval scholastic period, were closely connected to the idea that the primary application of logic is for practices of debating such as dialectical disputations. <span>The third of them, in contrast, exemplifies an entirely different rationale for logic, namely as a foundational branch of mathematics, not in any way connected to the ordinary languages in which debates are typically conducted. The hiatus between the second and third periods can be explained by the fall from grace of




#logic
I n the modern period, a number of philosophers came to see the nature of logic in terms of the faculties of mind.
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The rise and fall and rise of logic | Aeon Essays
rving of the title ‘logic’ than the work done under this heading in the early modern period, given that it comes closer to the level of rigour and formal sophistication that came to be associated with logic from the late 19th century onwards. <span>In the modern period, a number of philosophers came to see the nature of logic in terms of the faculties of mind. To be sure, this is again a theme present in medieval scholastic thought (in the work of the 14th-century author Pierre d’Ailly, for example), but in the early modern period it became t




#logic
It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440.
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The rise and fall and rise of logic | Aeon Essays
ich is thoroughly disputational, with Meditations on First Philosophy (1641) by Descartes, a book argued through long paragraphs driven by the first-person singular. The nature of intellectual enquiry shifted with the downfall of disputation. <span>It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440. Before that, books were a rare commodity, and education was conducted almost exclusively by means of oral contact between masters and pupils in the form of expository lectures in which




#logic
The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm.
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The rise and fall and rise of logic | Aeon Essays
s Diafoirus resorts to disputational vocabulary to make a point about love: Distinguo, Mademoiselle; in all that does not concern the possession of the loved one, concedo, I grant it; but in what does regard that possession, nego, I deny it. <span>The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. Despite all this, disputations continued to be practised in certain university contexts for some time – indeed, they live on in the ceremonial character of PhD defences. The point, thou




#logic
The tight connection between traditional logic and debating practices dates back to the classical Hellenistic period.
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The rise and fall and rise of logic | Aeon Essays
cond intentions’, roughly what we call second-order concepts, or concepts of concepts. But as late as in the 16th century, the Spanish theologian Domingo de Soto could write with confidence that ‘dialectic is the art or science of disputing’. <span>The tight connection between traditional logic and debating practices dates back to the classical Hellenistic period. Intellectual activity then was quintessentially a dialogical affair, as registered in Plato’s dialogues. In these dialogues, Socrates regularly engages in the practice of refutation (el




#functional-analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.
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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as tr

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Functional analysis - Wikipedia
analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that




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Functional analysis studies [...] endowed with limit-related structure
Answer

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitab

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Functional analysis - Wikipedia
analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that







Flashcard 1732491742476

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In Functional analysis, the vector spaces are endowed with [...] structures
Answer
limit-related

Norms, inner products, etc.

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

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Functional analysis - Wikipedia
analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that







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Functional analysis study vector spaces endowed with limit-related structure and the [...] defined on them.

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span> Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the <span>linear functions defined on these spaces and respecting these structures in a suitable sense. <span><body><html>

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Functional analysis - Wikipedia
analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that







Flashcard 1732605512972

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#linear-algebra #matrix-decomposition
Question

If matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is [...] and its inverse is given by [...]

Answer
nonsingular

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Eigendecomposition of a matrix - Wikipedia
amp;0\\1&3\\\end{bmatrix}}{\begin{bmatrix}-2c&0\\c&d\\\end{bmatrix}}={\begin{bmatrix}1&0\\0&3\\\end{bmatrix}},[c,d]\in \mathbb {R} } Matrix inverse via eigendecomposition[edit source] Main article: Inverse matrix <span>If matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is nonsingular and its inverse is given by A − 1 = Q Λ − 1 Q − 1 {\displaystyle \mathbf {A} ^{-1}=\mathbf {Q} \mathbf {\Lambda } ^{-1}\mathbf {Q} ^{-1}} Furthermore, because Λ is a diagonal matrix, its inverse is easy to calculate: [ Λ







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[...] has real eigenvalues and the eigenvectors can be chosen to be orthogonal

Answer

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{\displaystyle A=A^{*}} ), which implies that it is also complex normal, the diagonal matrix Λ has only real values, and if A is unitary, Λ takes all its values on the complex unit circle. Real symmetric matrices[edit source] <span>As a special case, for every N×N real symmetric matrix, the eigenvalues are real and the eigenvectors can be chosen such that they are orthogonal to each other. Thus a real symmetric matrix A can be decomposed as A = Q Λ Q T {\displaystyle \mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T}} where Q is an orthogonal matrix, and Λ is a diagonal matrix whose entries are the eigenvalues of A. Useful facts[edit source] Useful facts regarding eigenvalues[edit source] The product of the eigenvalues is equal to the determinant of A det







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A History of Formal Logic (1961) is written by the distinguished [...]
Answer
J M Bocheński

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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.

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The rise and fall and rise of logic | Aeon Essays
the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara







Flashcard 1735816514828

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Question
The fall of [...] culture wasn’t the only cause for the demise of scholastic logic
Answer
disputational

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The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language

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The rise and fall and rise of logic | Aeon Essays
s Diafoirus resorts to disputational vocabulary to make a point about love: Distinguo, Mademoiselle; in all that does not concern the possession of the loved one, concedo, I grant it; but in what does regard that possession, nego, I deny it. <span>The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. Despite all this, disputations continued to be practised in certain university contexts for some time – indeed, they live on in the ceremonial character of PhD defences. The point, thou







Flashcard 1735818087692

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Question
Scholastic logic was also viewed as being tied to Aristotelian conceptions of [...and...]
Answer
language and metaphysics

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The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. <body><html>

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The rise and fall and rise of logic | Aeon Essays
s Diafoirus resorts to disputational vocabulary to make a point about love: Distinguo, Mademoiselle; in all that does not concern the possession of the loved one, concedo, I grant it; but in what does regard that possession, nego, I deny it. <span>The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. Despite all this, disputations continued to be practised in certain university contexts for some time – indeed, they live on in the ceremonial character of PhD defences. The point, thou







Flashcard 1735819660556

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#logic
Question
the introduction of [...] in Europe accelerated the downfall of the disputational culture.
Answer
new printing techniques

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It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440.

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The rise and fall and rise of logic | Aeon Essays
ich is thoroughly disputational, with Meditations on First Philosophy (1641) by Descartes, a book argued through long paragraphs driven by the first-person singular. The nature of intellectual enquiry shifted with the downfall of disputation. <span>It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440. Before that, books were a rare commodity, and education was conducted almost exclusively by means of oral contact between masters and pupils in the form of expository lectures in which







Flashcard 1735821233420

Tags
#history #logic
Question
Johannes Gutenberg introduced new printing techniques in Europe around [...].
Answer
1440

You can't terrorise Aristotle!

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It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440.

Original toplevel document

The rise and fall and rise of logic | Aeon Essays
ich is thoroughly disputational, with Meditations on First Philosophy (1641) by Descartes, a book argued through long paragraphs driven by the first-person singular. The nature of intellectual enquiry shifted with the downfall of disputation. <span>It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440. Before that, books were a rare commodity, and education was conducted almost exclusively by means of oral contact between masters and pupils in the form of expository lectures in which







#logic
Instead, early modern authors emphasise the role of novelty and individual discovery
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The rise and fall and rise of logic | Aeon Essays
tually unthinkable before the wide availability of printed books) was well-established. Moreover, as indicated by the passage from Descartes above, the very term ‘logic’ came to be used for something other than what the scholastics had meant. <span>Instead, early modern authors emphasise the role of novelty and individual discovery, as exemplified by the influential textbook Port-Royal Logic (1662), essentially, the logical version of Cartesianism, based on Descartes’s conception of mental operations and the prima




Flashcard 1735825165580

Tags
#logic
Question
Instead of justification of ideas, early modern authors emphasise the role of [...] and [...]
Answer
novelty and individual discovery

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Instead, early modern authors emphasise the role of novelty and individual discovery

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The rise and fall and rise of logic | Aeon Essays
tually unthinkable before the wide availability of printed books) was well-established. Moreover, as indicated by the passage from Descartes above, the very term ‘logic’ came to be used for something other than what the scholastics had meant. <span>Instead, early modern authors emphasise the role of novelty and individual discovery, as exemplified by the influential textbook Port-Royal Logic (1662), essentially, the logical version of Cartesianism, based on Descartes’s conception of mental operations and the prima







#logic
Descartes hits the nail on the head when he claims that the logic of the Schools (scholastic logic) is not really a logic of discovery. Its chief purpose is justification and exposition , which makes sense particularly against the background of dialectical practices, where interlocutors explain and debate what they themselves already know.
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The rise and fall and rise of logic | Aeon Essays
without judgment about things one does not know. Such logic corrupts good sense rather than increasing it. I mean instead the kind of logic which teaches us to direct our reason with a view to discovering the truths of which we are ignorant. <span>Descartes hits the nail on the head when he claims that the logic of the Schools (scholastic logic) is not really a logic of discovery. Its chief purpose is justification and exposition, which makes sense particularly against the background of dialectical practices, where interlocutors explain and debate what they themselves already know. Indeed, for much of the history of logic, both in ancient Greece and in the Latin medieval tradition, ‘dialectic’ and ‘logic’ were taken to be synonymous. Up to Descartes’s time, the ch




Flashcard 1735828311308

Tags
#logic
Question
Descartes claims the chief purpose of scholastic logic is [...]
Answer
justification and exposition

which makes sense particularly against the background of dialectical practices, where interlocutors explain and debate what they themselves already know.

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Descartes hits the nail on the head when he claims that the logic of the Schools (scholastic logic) is not really a logic of discovery. Its chief purpose is justification and exposition , which makes sense particularly against the background of dialectical practices, where interlocutors explain and debate what they themselves already know.

Original toplevel document

The rise and fall and rise of logic | Aeon Essays
without judgment about things one does not know. Such logic corrupts good sense rather than increasing it. I mean instead the kind of logic which teaches us to direct our reason with a view to discovering the truths of which we are ignorant. <span>Descartes hits the nail on the head when he claims that the logic of the Schools (scholastic logic) is not really a logic of discovery. Its chief purpose is justification and exposition, which makes sense particularly against the background of dialectical practices, where interlocutors explain and debate what they themselves already know. Indeed, for much of the history of logic, both in ancient Greece and in the Latin medieval tradition, ‘dialectic’ and ‘logic’ were taken to be synonymous. Up to Descartes’s time, the ch







Flashcard 1735995559180

Tags
#stochastics
Question
Random walks are usually defined as sums of [...] in Euclidean space
Answer
iid random variables or random vectors

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Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

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Stochastic process - Wikipedia
one, while the value of a tail is zero. [61] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, [62] where each coin flip is a Bernoulli trial. [63] Random walk[edit source] Main article: Random walk <span>Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. [64] [65] [66] [67] [68] But some also use the term to refer to processes that change in continuous time, [69] particularly the Wiener process used in finance, which has led to some c







#abstract-algebra
In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms.
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Algebraic structure - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,




Flashcard 1736192953612

Tags
#abstract-algebra
Question
an [...] is a set with one or more operations defined on it that satisfies a list of axioms.
Answer
algebraic structure

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In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms.

Original toplevel document

Algebraic structure - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,







Flashcard 1736195050764

Tags
#abstract-algebra
Question
The underlying set for an algebraic structure is called a [...]
Answer
carrier set

Think sigma-algebra and measurable space

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In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms.

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Algebraic structure - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,







Flashcard 1736196885772

Tags
#abstract-algebra
Question
an algebraic structure is a set with one or more [...] defined on it that satisfies a list of axioms.
Answer

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In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms.

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Algebraic structure - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,







Flashcard 1736198458636

Tags
#abstract-algebra
Question
an algebraic structure is a set with one or more operations defined on it that satisfies [...].
Answer
a list of axioms

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tml> In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. <html>

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Algebraic structure - Wikipedia
Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,







#abstract-algebra

Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex algebraic structures include vector spaces, modules, and algebras.

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Algebraic structure - Wikipedia
algebra v t e In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] <span>Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex algebraic structures include vector spaces, modules, and algebras. The properties of specific algebraic structures are studied in abstract algebra. The general theory of algebraic structures has been formalized in universal algebra. The language of c




#poisson-process #stochastics
Its name (Poisson Process) derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution.
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Poisson point process - Wikipedia
oint processes, some of which are constructed with the Poisson point process, that seek to capture such interaction. [22] The process is named after French mathematician Siméon Denis Poisson despite Poisson never having studied the process. <span>Its name derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution. The process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics. [23] [24] T




#poisson-process #stochastics
The Poisson point process is often defined on the real line, where it can be considered as a stochastic process.
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Poisson point process - Wikipedia
and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy, [3] biology, [4] ecology, [5] geology, [6] physics, [7] economics, [8] image processing, [9] and telecommunications. [10] [11] <span>The Poisson point process is often defined on the real line, where it can be considered as a stochastic process. In this setting, it is used, for example, in queueing theory [12] to model random events, such as the arrival of customers at a store or phone calls at an exchange, distributed in tim




#poisson-process #stochastics
In the plane, the point process, also known as a spatial Poisson process,[13] can represent the locations of scattered objects such as transmitters in a wireless network,[10][14][15][16] particles colliding into a detector, or trees in a forest.
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Poisson point process - Wikipedia
e, where it can be considered as a stochastic process. In this setting, it is used, for example, in queueing theory [12] to model random events, such as the arrival of customers at a store or phone calls at an exchange, distributed in time. <span>In the plane, the point process, also known as a spatial Poisson process, [13] can represent the locations of scattered objects such as transmitters in a wireless network, [10] [14] [15] [16] particles colliding into a detector, or trees in a forest. [17] In this setting, the process is often used in mathematical models and in the related fields of spatial point processes, [18] stochastic geometry, [1] spatial statistics [18] [1




Flashcard 1736250101004

Tags
#poisson-process #stochastics
Question
if [...] in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution.
Answer
a collection of random points

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Its name (Poisson Process) derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution.

Original toplevel document

Poisson point process - Wikipedia
oint processes, some of which are constructed with the Poisson point process, that seek to capture such interaction. [22] The process is named after French mathematician Siméon Denis Poisson despite Poisson never having studied the process. <span>Its name derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution. The process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics. [23] [24] T







Flashcard 1736251673868

Tags
#poisson-process #stochastics
Question
In a Poisson process, [...] is a random variable with a Poisson distribution.
Answer
the number of points in a region of finite size

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Its name (Poisson Process) derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution.

Original toplevel document

Poisson point process - Wikipedia
oint processes, some of which are constructed with the Poisson point process, that seek to capture such interaction. [22] The process is named after French mathematician Siméon Denis Poisson despite Poisson never having studied the process. <span>Its name derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution. The process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics. [23] [24] T







Flashcard 1736253246732

Tags
#poisson-process #stochastics
Question
The Poisson point process is often defined on [...], where it can be considered as a stochastic process.
Answer
the real line

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The Poisson point process is often defined on the real line, where it can be considered as a stochastic process.

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Poisson point process - Wikipedia
and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy, [3] biology, [4] ecology, [5] geology, [6] physics, [7] economics, [8] image processing, [9] and telecommunications. [10] [11] <span>The Poisson point process is often defined on the real line, where it can be considered as a stochastic process. In this setting, it is used, for example, in queueing theory [12] to model random events, such as the arrival of customers at a store or phone calls at an exchange, distributed in tim







Flashcard 1736254819596

Tags
#poisson-process #stochastics
Question
In the plane, the point process, also known as a spatial Poisson process,[13] can represent [...] such as transmitters in a wireless network,[10][14][15][16] particles colliding into a detector, or trees in a forest.
Answer
the locations of scattered objects

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In the plane, the point process, also known as a spatial Poisson process, [13] can represent the locations of scattered objects such as transmitters in a wireless network, [10] [14] [15] [16] particles colliding into a detector, or trees in a forest.

Original toplevel document

Poisson point process - Wikipedia
e, where it can be considered as a stochastic process. In this setting, it is used, for example, in queueing theory [12] to model random events, such as the arrival of customers at a store or phone calls at an exchange, distributed in time. <span>In the plane, the point process, also known as a spatial Poisson process, [13] can represent the locations of scattered objects such as transmitters in a wireless network, [10] [14] [15] [16] particles colliding into a detector, or trees in a forest. [17] In this setting, the process is often used in mathematical models and in the related fields of spatial point processes, [18] stochastic geometry, [1] spatial statistics [18] [1







#lists #python
An operation like this that combines a sequence of elements into a single value is sometimes called reduce .
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Lists
of the elements; a variable used this way is sometimes called an accumulator. Adding up the elements of a list is such a common operation that Python provides it as a built-in function, sum: >>> t = [1, 2, 3] >>> sum(t) 6 <span>An operation like this that combines a sequence of elements into a single value is sometimes called reduce. Sometimes you want to traverse one list while building another. For example, the following function takes a list of strings and returns a new list that contains capitalized strings:




#lists #python
An operation like capitalize_all is sometimes called a map because it “maps” a function (in this case the method capitalize ) onto each of the elements in a sequence.
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Lists
ngs: def capitalize_all(t): res = [] for s in t: res.append(s.capitalize()) return res res is initialized with an empty list; each time through the loop, we append the next element. So res is another kind of accumulator. <span>An operation like capitalize_all is sometimes called a map because it “maps” a function (in this case the method capitalize) onto each of the elements in a sequence. Another common operation is to select some of the elements from a list and return a sublist. For example, the following function takes a list of strings and returns a list that cont




#lists #python
An operation like only_upper is called a filter because it selects some of the elements and filters out the others.
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Lists
ontains only the uppercase strings: def only_upper(t): res = [] for s in t: if s.isupper(): res.append(s) return res isupper is a string method that returns True if the string contains only upper case letters. <span>An operation like only_upper is called a filter because it selects some of the elements and filters out the others. Most common list operations can be expressed as a combination of map, filter and reduce. 10.8 Deleting elements There are several ways to delete elements from a list. If you kno




#lists #python

Most common list operations can be expressed as a combination of map, filter and reduce.

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Lists
res.append(s) return res isupper is a string method that returns True if the string contains only upper case letters. An operation like only_upper is called a filter because it selects some of the elements and filters out the others. <span>Most common list operations can be expressed as a combination of map, filter and reduce. 10.8 Deleting elements There are several ways to delete elements from a list. If you know the index of the element you want, you can use pop: >>> t = ['a', 'b', 'c']




Flashcard 1737334590732

Tags
#lists #python
Question
An operation like this that combines a sequence of elements into a single value is sometimes called [...] .
Answer
reduce

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An operation like this that combines a sequence of elements into a single value is sometimes called reduce .

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Lists
of the elements; a variable used this way is sometimes called an accumulator. Adding up the elements of a list is such a common operation that Python provides it as a built-in function, sum: >>> t = [1, 2, 3] >>> sum(t) 6 <span>An operation like this that combines a sequence of elements into a single value is sometimes called reduce. Sometimes you want to traverse one list while building another. For example, the following function takes a list of strings and returns a new list that contains capitalized strings:







Flashcard 1737336950028

Tags
#lists #python
Question
An map operation “maps” a function onto [...] in a sequence.
Answer
each of the elements

like capitalize_all

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An operation like capitalize_all is sometimes called a map because it “maps” a function (in this case the method capitalize ) onto each of the elements in a sequence.

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Lists
ngs: def capitalize_all(t): res = [] for s in t: res.append(s.capitalize()) return res res is initialized with an empty list; each time through the loop, we append the next element. So res is another kind of accumulator. <span>An operation like capitalize_all is sometimes called a map because it “maps” a function (in this case the method capitalize) onto each of the elements in a sequence. Another common operation is to select some of the elements from a list and return a sublist. For example, the following function takes a list of strings and returns a list that cont







Flashcard 1737338522892

Tags
#lists #python
Question
An operation like only_upper is called a [...] because it selects some of the elements and filters out the others.
Answer
filter

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An operation like only_upper is called a filter because it selects some of the elements and filters out the others.

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Lists
ontains only the uppercase strings: def only_upper(t): res = [] for s in t: if s.isupper(): res.append(s) return res isupper is a string method that returns True if the string contains only upper case letters. <span>An operation like only_upper is called a filter because it selects some of the elements and filters out the others. Most common list operations can be expressed as a combination of map, filter and reduce. 10.8 Deleting elements There are several ways to delete elements from a list. If you kno







Flashcard 1737340095756

Tags
#lists #python
Question

Most common list operations can be expressed as a combination of [...].

Answer
map, filter and reduce

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Most common list operations can be expressed as a combination of map, filter and reduce.

Original toplevel document

Lists
res.append(s) return res isupper is a string method that returns True if the string contains only upper case letters. An operation like only_upper is called a filter because it selects some of the elements and filters out the others. <span>Most common list operations can be expressed as a combination of map, filter and reduce. 10.8 Deleting elements There are several ways to delete elements from a list. If you know the index of the element you want, you can use pop: >>> t = ['a', 'b', 'c']







Flashcard 1737953512716

Tags
#state-space-models
Question
inference for [...] can be performed efficiently using the sum-product algorithm
Answer
directed graphs having a tree structure (no loops)

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focus on the two most important examples of state space models, namely the hidden Markov model, in which the latent variables are discrete, and linear dynamical systems, in which the latent variables are Gaussian. Both models are described by <span>directed graphs having a tree structure (no loops) for which inference can be performed efficiently using the sum-product algorithm <span><body><html>

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Flashcard 1737961377036

Tags
#state-space-models
Question
[...] assumes equal conditional distributions \(p(x_n |x_{n−1} ) \) for all n
Answer
stationary time series

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In most applications of such (Markovian) models, the conditional distributions p(xn|xn−1) that define the model will be constrained to be equal, corresponding to the assumption of a stationary time series. The model is then known as a homogeneous Markov chain.

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cannot see any pdfs







#measure-theory
The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, goes from 0 to 1 as goes from 0 to 1, and takes on every value in between.
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Cantor function - Wikipedia
= 1/4. 200/243 becomes 0.21102 (or 0.211012222...) in base 3. The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. When read in base 2, this corresponds to 3/4, so c(200/243) = 3/4. Properties[edit] <span>The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, c ( x ) {\textstyle c(x)} goes from 0 to 1 as x {\textstyle x} goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is uniformly continuous (precisely, it is Hölder continuous of exponent α = log 2/log 3) but not absolut




Flashcard 1738558541068

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#measure-theory
Question
The [...] function challenges naive intuitions about continuity and measure
Answer
Cantor

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The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, goes from 0 to 1 as goes from 0 to 1, and take

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Cantor function - Wikipedia
= 1/4. 200/243 becomes 0.21102 (or 0.211012222...) in base 3. The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. When read in base 2, this corresponds to 3/4, so c(200/243) = 3/4. Properties[edit] <span>The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, c ( x ) {\textstyle c(x)} goes from 0 to 1 as x {\textstyle x} goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is uniformly continuous (precisely, it is Hölder continuous of exponent α = log 2/log 3) but not absolut







Flashcard 1738560113932

Tags
#measure-theory
Question
although the Cantor function is [...], goes from 0 to 1 as goes from 0 to 1, and takes on every value in between.
Answer
continuous everywhere and has zero derivative almost everywhere

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The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, goes from 0 to 1 as goes from 0 to 1, and takes on every value in between.

Original toplevel document

Cantor function - Wikipedia
= 1/4. 200/243 becomes 0.21102 (or 0.211012222...) in base 3. The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. When read in base 2, this corresponds to 3/4, so c(200/243) = 3/4. Properties[edit] <span>The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, c ( x ) {\textstyle c(x)} goes from 0 to 1 as x {\textstyle x} goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is uniformly continuous (precisely, it is Hölder continuous of exponent α = log 2/log 3) but not absolut







Flashcard 1738581871884

Tags
#measure-theory #probability-theory
Question
Historically, the dividing line for probability is [...] when Andrey Kolmogorov published Foundations of the Theory of Probability
Answer
1933

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Historically, the dividing line is 1933 when Grundbegriffe der Wahrscheinlichkeitsrech- nung (Foundations of the Theory of Probability) by Andrey Kolmogorov was published

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#graphical-models
A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables.
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Graphical model - Wikipedia
list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (May 2017) (Learn how and when to remove this template message) <span>A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning. [imagelink] An example of a graphical model. Each arrow indicates




#graphical-models

In a Bayesian network, the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are then the joint probability satisfies

where is the set of parents of node . In other words, the joint distribution factors into a product of conditional distributions.

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Graphical model - Wikipedia
the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce. [1] Bayesian network[edit source] Main article: Bayesian network <span>If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} then the joint probability satisfies P [ X 1 , … , X n ] = ∏ i = 1 n P [ X i | p a i ] {\displaystyle P[X_{1},\ldots ,X_{n}]=\prod _{i=1}^{n}P[X_{i}|pa_{i}]} where p a i {\displaystyle pa_{i}} is the set of parents of node X i {\displaystyle X_{i}} . In other words, the joint distribution factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the random variables




#bayesian-network
Formally, Bayesian networks are DAGs whose:
  1. nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses.
  2. Edges represent conditional dependencies;
  3. nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other.
  4. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node.
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Bayesian network - Wikipedia
ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables




#graphical-models
A Markov random field, also known as a Markov network, is a model over an undirected graph.
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Graphical model - Wikipedia
ne learning models like hidden Markov models, neural networks and newer models such as variable-order Markov models can be considered special cases of Bayesian networks. Markov random field[edit source] Main article: Markov random field <span>A Markov random field, also known as a Markov network, is a model over an undirected graph. A graphical model with many repeated subunits can be represented with plate notation. Other types[edit source] A factor graph is an undirected bipartite graph connecting variables a




Flashcard 1739036953868

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#graphical-models
Question
In a graphical model, a graph expresses [...] between random variables.
Answer
the conditional dependence structure

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A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables.

Original toplevel document

Graphical model - Wikipedia
list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (May 2017) (Learn how and when to remove this template message) <span>A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning. [imagelink] An example of a graphical model. Each arrow indicates







Flashcard 1739039313164

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#graphical-models
Question

In a Bayesian network, the network structure of the model is a [...]


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In a Bayesian network, the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are then the joint probability satisfies where is

Original toplevel document

Graphical model - Wikipedia
the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce. [1] Bayesian network[edit source] Main article: Bayesian network <span>If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} then the joint probability satisfies P [ X 1 , … , X n ] = ∏ i = 1 n P [ X i | p a i ] {\displaystyle P[X_{1},\ldots ,X_{n}]=\prod _{i=1}^{n}P[X_{i}|pa_{i}]} where p a i {\displaystyle pa_{i}} is the set of parents of node X i {\displaystyle X_{i}} . In other words, the joint distribution factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the random variables







Flashcard 1739041934604

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#graphical-models
Question

if the events are the joint probability of a Bayesian network satisfies[...]

Answer

is the set of parents of node .


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In a Bayesian network, the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are then the joint probability <span>satisfies where is the set of parents of node . In other words, the joint distribution factors into a product of conditional distributions. <span><body><html>

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Graphical model - Wikipedia
the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce. [1] Bayesian network[edit source] Main article: Bayesian network <span>If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} then the joint probability satisfies P [ X 1 , … , X n ] = ∏ i = 1 n P [ X i | p a i ] {\displaystyle P[X_{1},\ldots ,X_{n}]=\prod _{i=1}^{n}P[X_{i}|pa_{i}]} where p a i {\displaystyle pa_{i}} is the set of parents of node X i {\displaystyle X_{i}} . In other words, the joint distribution factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the random variables







Flashcard 1739044556044

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#graphical-models
Question

In a Bayesian network, the joint distribution factors into [...].

Answer
a product of conditional distributions

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factorization of the joint probability of all random variables. More precisely, if the events are then the joint probability satisfies where is the set of parents of node . In other words, the joint distribution factors into <span>a product of conditional distributions. <span><body><html>

Original toplevel document

Graphical model - Wikipedia
the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce. [1] Bayesian network[edit source] Main article: Bayesian network <span>If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} then the joint probability satisfies P [ X 1 , … , X n ] = ∏ i = 1 n P [ X i | p a i ] {\displaystyle P[X_{1},\ldots ,X_{n}]=\prod _{i=1}^{n}P[X_{i}|pa_{i}]} where p a i {\displaystyle pa_{i}} is the set of parents of node X i {\displaystyle X_{i}} . In other words, the joint distribution factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the random variables







Flashcard 1739046391052

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#graphical-models
Question
A Markov random field is a model over [...].
Answer

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A Markov random field, also known as a Markov network, is a model over an undirected graph.

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Graphical model - Wikipedia
ne learning models like hidden Markov models, neural networks and newer models such as variable-order Markov models can be considered special cases of Bayesian networks. Markov random field[edit source] Main article: Markov random field <span>A Markov random field, also known as a Markov network, is a model over an undirected graph. A graphical model with many repeated subunits can be represented with plate notation. Other types[edit source] A factor graph is an undirected bipartite graph connecting variables a







Flashcard 1739048488204

Tags
#graphical-models
Question
A [...] is a model over an undirected graph.
Answer
Markov random field

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A Markov random field, also known as a Markov network, is a model over an undirected graph.

Original toplevel document

Graphical model - Wikipedia
ne learning models like hidden Markov models, neural networks and newer models such as variable-order Markov models can be considered special cases of Bayesian networks. Markov random field[edit source] Main article: Markov random field <span>A Markov random field, also known as a Markov network, is a model over an undirected graph. A graphical model with many repeated subunits can be represented with plate notation. Other types[edit source] A factor graph is an undirected bipartite graph connecting variables a







Flashcard 1739051896076

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#bayesian-network
Question
In Bayesian network graphs [...] represent variables
Answer
nodes

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Formally, Bayesian networks are DAGs whose: nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that ar

Original toplevel document

Bayesian network - Wikipedia
ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables







Flashcard 1739053468940

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#bayesian-network
Question
Graphically, Bayesian networks use [...] to represent conditional dependencies
Answer
Edges

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html> Formally, Bayesian networks are DAGs whose: nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are condition

Original toplevel document

Bayesian network - Wikipedia
ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables







Flashcard 1739055041804

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#bayesian-network
Question
nodes that are not connected represent variables that are [...].
Answer

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ntities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are <span>conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probabil

Original toplevel document

Bayesian network - Wikipedia
ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables







Flashcard 1739056614668

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#bayesian-network
Question
In a Bayesian network each node takes [...] as input
Answer
a set of values from parent nodes

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re not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, <span>a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. <span><body><html>

Original toplevel document

Bayesian network - Wikipedia
ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables







Flashcard 1739058187532

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#bayesian-network
Question
In Bayesian networks each node represents a variable with [...]
Answer
a probability distribution

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ayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) <span>the probability (or probability distribution, if applicable) of the variable represented by the node. <span><body><html>

Original toplevel document

Bayesian network - Wikipedia
ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables







#d-separation
d-separation is a criterion for deciding, from a given a causal graph, whether a set X of variables is independent of another set Y, given a third set Z.
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Unknown title
CAUSALITY - Discussion d-SEPARATION WITHOUT TEARS (At the request of many readers) Introduction d-separation is a criterion for deciding, from a given a causal graph, whether a set X of variables is independent of another set Y, given a third set Z. The idea is to associate "dependence" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness"




#d-separation
The idea (of d-separation) is to associate "dependence" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness" or "separation".
The only twist on this simple idea is to define what we mean by "connecting path", given that
  1. we are dealing with a system of directed arrows
  2. in the graph some vertices correspond to measured variables, whose values are known precisely.
To account for the orientations of the arrows we use the terms "d-separated" and "d-connected" (d connotes "directional").
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Unknown title
d-SEPARATION WITHOUT TEARS (At the request of many readers) Introduction d-separation is a criterion for deciding, from a given a causal graph, whether a set X of variables is independent of another set Y, given a third set Z. <span>The idea is to associate "dependence" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness" or "separation". The only twist on this simple idea is to define what we mean by "connecting path", given that we are dealing with a system of directed arrows in which some vertices (those residing in Z) correspond to measured variables, whose values are known precisely. To account for the orientations of the arrows we use the terms "d-separated" and "d-connected" (d connotes "directional"). We start by considering separation between two singleton variables, x and y; the extension to sets of variables is straightforward (i.e., two sets are separated if and only if each el




Flashcard 1739068411148

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#d-separation
Question
The idea of d-separation is complicated by defining exactly what is a [...], given that the edges are directed and some variables are already measured.
Answer
connecting path

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nce" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness" or "separation". The only twist on this simple idea is to define what we mean by <span>"connecting path", given that we are dealing with a system of directed arrows in which some vertices (those residing in Z) correspond to measured variables, whose values are known precisely. To account f

Original toplevel document

Unknown title
d-SEPARATION WITHOUT TEARS (At the request of many readers) Introduction d-separation is a criterion for deciding, from a given a causal graph, whether a set X of variables is independent of another set Y, given a third set Z. <span>The idea is to associate "dependence" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness" or "separation". The only twist on this simple idea is to define what we mean by "connecting path", given that we are dealing with a system of directed arrows in which some vertices (those residing in Z) correspond to measured variables, whose values are known precisely. To account for the orientations of the arrows we use the terms "d-separated" and "d-connected" (d connotes "directional"). We start by considering separation between two singleton variables, x and y; the extension to sets of variables is straightforward (i.e., two sets are separated if and only if each el