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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)).

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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. 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 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 the observer. <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. 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 1729672908044

Tags

#multivariate-normal-distribution

Question

Y = c + BX is an affine transformation,
then Y has a multivariate normal distribution with expected value [...]

Answer

c + Bμ

Corollaries: sums of Gaussian are Gaussian, marginals of Gaussian are Gaussian.

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If Y = c + BX is an affine transformation of where c is an vector of constants and B is a constant matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB T . Corollaries: sums of Gaussian are Gaussian, marginals of Gaussian are Gaussian.

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Multivariate normal distribution - Wikipedia
{\displaystyle {\boldsymbol {\Sigma }}'={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{13}\\{\boldsymbol {\Sigma }}_{31}&{\boldsymbol {\Sigma }}_{33}\end{bmatrix}}} . Affine transformation[edit source] If Y = c + BX is an affine transformation of X ∼ N ( μ , Σ ) , {\displaystyle \mathbf {X} \ \sim {\mathcal {N}}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}),} where c is an M × 1 {\displaystyle M\times 1} vector of constants and B is a constant M × N {\displaystyle M\times N} matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB T i.e., Y ∼ N ( c + B μ , B Σ B T ) {\displaystyle \mathbf {Y} \sim {\mathcal {N}}\left(\mathbf {c} +\mathbf {B} {\boldsymbol {\mu }},\mathbf {B} {\boldsymbol {\Sigma }}\mathbf {B} ^{\rm {T}}\right)} . In particular, any subset of the X i has a marginal distribution that is also multivariate normal. To see this, consider the following example: to extract the subset (X 1 , X 2 , X 4 )

Flashcard 1729703841036

Tags

#probability

Question

The negative binomial distribution also arises as a [...] of Poisson distributions

Answer

continuous mixture

Can be used to model over dispersed count observations, known as Gamma-Poisson distribution.

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The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution.

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Negative binomial distribution - Wikipedia
) . {\displaystyle \operatorname {Poisson} (\lambda )=\lim _{r\to \infty }\operatorname {NB} \left(r,{\frac {\lambda }{\lambda +r}}\right).} Gamma–Poisson mixture[edit source] The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution. That is, we can view the negative binomial as a Poisson(λ) distribution, where λ is itself a random variable, distributed as a gamma distribution with shape = r and scale θ = p/(1 − p) or correspondingly rate β = (1 − p)/p. To display the intuition behind this statement, consider two independent Poisson processes, “Success” and “Failure”, with intensities p and 1 − p. Together, the Success and Failure pr

Annotation 1729791921420

#linear-algebra #matrix-decomposition

In linear algebra, the Cholesky decomposition or Cholesky factorization is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful e.g. for efficient numerical solutions and Monte Carlo simulations.

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Cholesky decomposition - Wikipedia
ikipedia ocultar siempre | ocultar ahora Cholesky decomposition From Wikipedia, the free encyclopedia Jump to: navigation, search In linear algebra, the Cholesky decomposition or Cholesky factorization is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful e.g. for efficient numerical solutions and Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems

Annotation 1729827048716

#linear-algebra #matrix-decomposition

The Cholesky decomposition of a Hermitian positive-definite matrix A is a decomposition of the form

where L is a lower triangular matrix with real and positive diagonal entries, and L* denotes the conjugate transpose of L.

Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition.^[2] If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL* if the diagonal entries of L are allowed to be zero.^[3]

When A has real entries, L has real entries as well, and the factorization may be written A = LL^T.^[4]

The Cholesky decomposition is unique when A is positive definite; there is only one lower triangular matrix L with strictly positive diagonal entries such that A = LL*. However, the decomposition need not be unique when A is positive semidefinite.

The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite.

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Cholesky decomposition - Wikipedia
s 7 Generalization 8 Implementations in programming languages 9 See also 10 Notes 11 References 12 External links 12.1 History of science 12.2 Information 12.3 Computer code 12.4 Use of the matrix in simulation 12.5 Online calculators Statement[edit source] The Cholesky decomposition of a Hermitian positive-definite matrix A is a decomposition of the form A = L L ∗ , {\displaystyle \mathbf {A} =\mathbf {LL} ^{*},} where L is a lower triangular matrix with real and positive diagonal entries, and L* denotes the conjugate transpose of L. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition. [2] If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL* if the diagonal entries of L are allowed to be zero. [3] When A has real entries, L has real entries as well, and the factorization may be written A = LL T . [4] The Cholesky decomposition is unique when A is positive definite; there is only one lower triangular matrix L with strictly positive diagonal entries such that A = LL*. However, the decomposition need not be unique when A is positive semidefinite. The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite. LDL decomposition[edit source] A closely related variant of the classical Cholesky decomposition is the LDL decomposition, A =

Annotation 1730486078732

#linear-algebra #matrix-decomposition

The Cholesky decomposition of a Hermitian positive-definite matrix A is a decomposition of the form

where L is a lower triangular matrix with real and positive diagonal entries, and L* denotes the conjugate transpose of L.

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The Cholesky decomposition of a Hermitian positive-definite matrix A is a decomposition of the form where L is a lower triangular matrix with real and positive diagonal entries, and L* denotes the conjugate transpose of L. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition. [2] If the matrix A is Her

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Flashcard 1730517011724

Tags

#linear-algebra #matrix-decomposition

Question

The Cholesky decomposition is of the form [...]

Answer

where L is a lower triangular matrix with real and positive diagonal entries, and L* denotes the conjugate transpose of L.

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Flashcard 1730593557772

Tags

#linear-algebra #matrix-decomposition

Question

In Cholesky decomposition of the form L is a [...]

Answer

lower triangular matrix with real and positive diagonal entries

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Flashcard 1731009580300

Tags

#linear-algebra #matrix-decomposition

Question

Cholesky decomposition factorises a Hermitian, positive-definite matrix into the product of [...] and its conjugate transpose,

Answer

a lower triangular matrix

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In linear algebra, the Cholesky decomposition or Cholesky factorization is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful e.g. for efficient numerical solutions and Monte Carlo simulations.

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Flashcard 1735828311308

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#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.

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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. 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

Annotation 1738860268812

#metric-space

Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element x₀ in X and define a sequence {x_n} by x_n = T(x_n−1), then x_n → x* .

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Banach fixed-point theorem - Wikipedia
x ) , T ( y ) ) ≤ q d ( x , y ) {\displaystyle d(T(x),T(y))\leq qd(x,y)} for all x, y in X. Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element x 0 in X and define a sequence {x n } by x n = T(x n−1 ), then x n → x*. Remark 1. The following inequalities are equivalent and describe the speed of convergence: d

Flashcard 1738868919564

Tags

#metric-space

Question

When a complete metric space admits a contraction mapping T : X → X. The fixed point for the map can be found as follows: start with [...] and define a sequence {x_n} by x_n = T(x_n−1), then x_n → x* .

Answer

an arbitrary element x₀ in X

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Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element x 0 in X and define a sequence {x n } by x n = T(x n−1 ), then x n → x* . <body><html>

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Flashcard 1739046391052

Tags

#graphical-models

Question

A Markov random field is a model over [...].

Answer

an undirected graph

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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 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 1739186113804

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#linear-algebra

Question

a bilinear form on a vector space V is a bilinear map [...] ,

Answer

V × V → K

K is the field of scalars.
An inner product is obviously a bilinear form

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In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K , where K is the field of scalars.

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Bilinear form - Wikipedia
Bilinear form - Wikipedia Bilinear form From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one

Flashcard 1739352313100

Tags

#linear-algebra #matrix-decomposition

Question

The Cholesky decomposition only works properly for [...] matrices

Answer

Hermitian positive-definite

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Annotation 1739354934540

#linear-algebra #matrix-decomposition

The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite.

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position is unique when A is positive definite; there is only one lower triangular matrix L with strictly positive diagonal entries such that A = LL*. However, the decomposition need not be unique when A is positive semidefinite. The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite. <body><html>

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Flashcard 1739356507404

Tags

#linear-algebra #matrix-decomposition

Question

Cholesky decomposability implies that if A can be written as LL* for some invertible L then A is [...]

Answer

Hermitian and positive definite.

L can be lower triangular or otherwise,

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The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite.

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Annotation 1739358080268

#linear-algebra #matrix-decomposition

If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL* if the diagonal entries of L are allowed to be zero.

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eal and positive diagonal entries, and L* denotes the conjugate transpose of L. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition. [2] If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL* if the diagonal entries of L are allowed to be zero. [3] When A has real entries, L has real entries as well, and the factorization may be written A = LL T . [4] The Cholesky decomposition is unique when A is pos

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Flashcard 1739359653132

Tags

#linear-algebra #matrix-decomposition

Question

If the matrix A is [...], then it still has a decomposition of the form A = LL* if the diagonal entries of L are allowed to be zero.

Answer

Hermitian and positive semi-definite

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If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL* if the diagonal entries of L are allowed to be zero.

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Flashcard 1739361225996

Tags

#linear-algebra #matrix-decomposition

Question

If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL* if [...].

Answer

the diagonal entries of L are allowed to be zero

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If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL* if the diagonal entries of L are allowed to be zero.

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Flashcard 1739928505612

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#forward-backward-algorithm #hmm

Question

In the first pass, the forward–backward algorithm computes [...]

Answer

.

the distribution over hidden states given the observations up to the point.

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The forward-backward algorithm In the first pass, the forward–backward algorithm computes a set of forward probabilities which provide, for all , the probability of ending up in any particular state given the first observations in the sequence, i.e. . In the second pass, the algorithm computes a set of backward probabilities which provide the probability of observing the remaining observations given any starting point , i.e. . Thes

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Forward–backward algorithm - Wikipedia
cific instance of this class. Contents [hide] 1 Overview 2 Forward probabilities 3 Backward probabilities 4 Example 5 Performance 6 Pseudocode 7 Python example 8 See also 9 References 10 External links Overview[edit source] In the first pass, the forward–backward algorithm computes a set of forward probabilities which provide, for all k ∈ { 1 , … , t } {\displaystyle k\in \{1,\dots ,t\}} , the probability of ending up in any particular state given the first k {\displaystyle k} observations in the sequence, i.e. P ( X k | o 1 : k ) {\displaystyle P(X_{k}\ |\ o_{1:k})} . In the second pass, the algorithm computes a set of backward probabilities which provide the probability of observing the remaining observations given any starting point k {\displaystyle k} , i.e. P ( o k + 1 : t | X k ) {\displaystyle P(o_{k+1:t}\ |\ X_{k})} . These two sets of probability distributions can then be combined to obtain the distribution over states at any specific point in time given the entire observation sequence: P ( X k | o 1 : t ) = P ( X k | o 1 : k , o k + 1 : t ) ∝ P ( o k + 1 : t | X k ) P ( X k | o 1 : k ) {\displaystyle P(X_{k}\ |\ o_{1:t})=P(X_{k}\ |\ o_{1:k},o_{k+1:t})\propto P(o_{k+1:t}\ |\ X_{k})P(X_{k}|o_{1:k})} The last step follows from an application of the Bayes' rule and the conditional independence of o k + 1 : t {\displaystyle o_{k+1:t}} and o 1 : k {\displaystyle o_{1:k}} given X k {\displaystyle X_{k}} . As outlined above, the algorithm involves three steps: computing forward probabilities computing backward probabilities computing smoothed values. The forward and backward steps m

Flashcard 1756493647116

Tags

#has-images #lagrange-multiplier #optimization

[unknown IMAGE 1756479753484]

Question

In x is a D dimensional variable. then \( g(x)=0 \) represents a [...]

Answer

(D−1) dimensional surface

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In optimization with equality contraint and a D dimensional variable x. The constraint equation g(x)=0 then represents a (D−1) dimensional surface in x-space

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cannot see any pdfs

Flashcard 1758485417228

Tags

#function-space

Question

A function between two topological spaces X and Y is continuous if for every open set V ⊆ Y, [...].

Answer

the inverse image is an open subset of X

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A function between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image is an open subset of X.

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Continuous function - Wikipedia
intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology). A function f : X → Y {\displaystyle f\colon X\rightarrow Y} between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image f − 1 ( V ) = { x ∈ X | f ( x ) ∈ V } {\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}} is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology T X ), but the continuity of f depends on the topologies used on X and Y. This is equivalent to

Flashcard 1758518185228

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#topology

Question

continuous deformations includes [...] , but not tearing or gluing.

Answer

streching, crumpling and bending

Mind the scrub!

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In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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Topology - Wikipedia
ogy (disambiguation). For a topology of a topos or category, see Lawvere–Tierney topology and Grothendieck topology. [imagelink] Möbius strips, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important

Flashcard 1758563798284

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#topology

Question

[...] is the process whereby a two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure.

Answer

crumpling

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In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges a

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Crumpling - Wikipedia
ikipedia Crumpling From Wikipedia, the free encyclopedia Jump to: navigation, search "Crumpled" redirects here. For the deformation feature, see Crumple zone. In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges and facets with variable density. The geometry of crumpled structures is the subject of some interest the mathematical community within the discipline of topology. [1] Crumpled paper balls have been studied and found t

Flashcard 1759103814924

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#hilbert-space

Question

Hilbert spaces arise naturally and frequently in mathematics and physics, typically as [...spaces...]

Answer

infinite-dimensional function spaces.

Think Fourier analysis

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Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces.

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Hilbert space - Wikipedia
n abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tool

Annotation 1759639637260

#inner-product-space

The Cauchy–Schwarz inequality states that for all vectors and of an inner product space it is true that

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Cauchy–Schwarz inequality - Wikipedia
al space) 3.2 R n (n-dimensional Euclidean space) 3.3 L 2 4 Applications 4.1 Analysis 4.2 Geometry 4.3 Probability theory 5 Generalizations 6 See also 7 Notes 8 References 9 External links Statement of the inequality[edit source] The Cauchy–Schwarz inequality states that for all vectors u {\displaystyle u} and v {\displaystyle v} of an inner product space it is true that | ⟨ u , v ⟩ | 2 ≤ ⟨ u , u ⟩ ⋅ ⟨ v , v ⟩ , {\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product. Examples of inner products include the real and complex dot product, see the examples in inner product. Equivalently, by taking the square root of both sides, and referring to the norms

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Lebesgue spaces are sometimes called [...]

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L^p spaces

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In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces,

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Lp space - Wikipedia
Lp space - Wikipedia L p space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910). L p

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#inner-product-space

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The [...] states that for all vectors and of an inner product space it is true that

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Cauchy–Schwarz inequality

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The Cauchy–Schwarz inequality states that for all vectors and of an inner product space it is true that

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After defining an inner product on the set of random variables using the expectation of their product,

⟨ X , Y ⟩ := E ⁡ ( X Y ) , {\displaystyle \langle X,Y\rangle :=\operatorname {E} (XY),}

then the Cauchy–Schwarz inequality becomes

| E ⁡ ( X Y ) | 2 ≤ E ⁡ ( X 2 ) E ⁡ ( Y 2 ) . {\displaystyle |\operatorname {E} (XY)|^{2}\leq \operatorname {E} (X^{2})\operatorname {E} (Y^{2}).}

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Cauchy–Schwarz inequality - Wikipedia
( X ) . {\displaystyle \operatorname {Var} (Y)\geq {\frac {\operatorname {Cov} (Y,X)\operatorname {Cov} (Y,X)}{\operatorname {Var} (X)}}.} After defining an inner product on the set of random variables using the expectation of their product, ⟨ X , Y ⟩ := E ⁡ ( X Y ) , {\displaystyle \langle X,Y\rangle :=\operatorname {E} (XY),} then the Cauchy–Schwarz inequality becomes | E ⁡ ( X Y ) | 2 ≤ E ⁡ ( X 2 ) E ⁡ ( Y 2 ) . {\displaystyle |\operatorname {E} (XY)|^{2}\leq \operatorname {E} (X^{2})\operatorname {E} (Y^{2}).} To prove the covariance inequality using the Cauchy–Schwarz inequality, let μ = E ⁡ ( X ) {

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#calculus

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The infinitesimal approach fell out of favor in the 19th century because it was difficult to make [...]

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the notion of an infinitesimal precise.

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The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise.

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Calculus - Wikipedia
infinitesimals. The symbols dx and dy were taken to be infinitesimal, and the derivative d y / d x {\displaystyle dy/dx} was simply their ratio. The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals. In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a near

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William Shakespeare was born in [...]

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1564

Look, here comes the tall shrew!

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William Shakespeare ( / ˈ ʃ eɪ k s p ɪər / ; 26 April 1564 (baptised) – 23 April 1616) [a] was an English poet, playwright and actor, widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist.

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William Shakespeare - Wikipedia
Era Elizabethan era Jacobean era Movement English Renaissance Spouse(s) Anne Hathaway ( m. 1582) Children Susanna Hall Hamnet Shakespeare Judith Quiney Parent(s) John Shakespeare Mary Arden Signature [imagelink] William Shakespeare (/ˈʃeɪkspɪər/; 26 April 1564 (baptised) – 23 April 1616) [a] was an English poet, playwright and actor, widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist. [2] [3] [4] He is often called England's national poet and the "Bard of Avon". [5] [b] His extant works, including collaborations, consist of approximately 39 plays, [c] 15

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The Hundred Years' War started in [...]

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1337

The war between the tame and the meek (Christians).

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The Hundred Years' War was a series of conflicts waged from 1337 to 1453 by the House of Plantagenet, rulers of the Kingdom of England, against the House of Valois, rulers of the Kingdom of France, over the succession to the French throne. </

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Hundred Years' War - Wikipedia
Anglo-French wars 1202–04 1213–14 1215–17 1242–43 1294–1303 1337–1453 (1337–60, 1369–89, 1415–53) 1496-98 1512–14 1522–26 1542–46 1557–59 1627–29 1666–67 1689–97 1702–13 1744–48 1744–1763 1754–63 1778–83 1793–1802 1803–14 1815 The Hundred Years' War was a series of conflicts waged from 1337 to 1453 by the House of Plantagenet, rulers of the Kingdom of England, against the House of Valois, rulers of the Kingdom of France, over the succession to the French throne. Each side drew many allies into the war. It was one of the most notable conflicts of the Middle Ages, in which five generations of kings from two rival dynasties fought for the throne o

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The Hundred Years' War ended in [...]

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1453

England would hate losing its heirloom.

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The Weimar Republic began in [...].

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1919

I'd say the government is tip-top!

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The Weimar Republic (German: Weimarer Republik [ˈvaɪmaʁɐ ʁepuˈbliːk] ( [imagelink] listen ) ) is an unofficial, historical designation for the German state as it existed between 1919 and 1933.

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#ancient-history #history #roman-empire #rome #wiki

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the Roman Senate formally granted Octavian overarching power and the new title Augustus in [...]

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27 BC

Octavian had the Senate at its neck.

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Octavian's power was then unassailable and in 27 BC the Roman Senate formally granted him overarching power and the new title Augustus, effectively marking the end of the Roman Republic.

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Roman Empire - Wikipedia
perpetual dictator and then assassinated in 44 BC. Civil wars and executions continued, culminating in the victory of Octavian, Caesar's adopted son, over Mark Antony and Cleopatra at the Battle of Actium in 31 BC and the annexation of Egypt. Octavian's power was then unassailable and in 27 BC the Roman Senate formally granted him overarching power and the new title Augustus, effectively marking the end of the Roman Republic. The imperial period of Rome lasted approximately 1,500 years compared to the 500 years of the Republican era. The first two centuries of the empire's existence were a period of unprec

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for a function and set V ⊆ Y, the inverse image of Y is defined as [...]

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A function between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image is an open subset of X.

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The theory of [...] for linear state space systems is elegant and simple

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prediction

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The theory of prediction for linear state space systems is elegant and simple

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Linear State Space Models – Quantitative Economics
t]=Gμt The variance-covariance matrix of ytyt is easily shown to be (19)¶ Var[yt]=Var[Gxt+Hvt]=GΣtG′+HH′Var[yt]=Var[Gxt+Hvt]=GΣtG′+HH′ The distribution of ytyt is therefore yt∼N(Gμt,GΣtG′+HH′)yt∼N(Gμt,GΣtG′+HH′) Prediction¶ The theory of prediction for linear state space systems is elegant and simple Forecasting Formulas – Conditional Means¶ The natural way to predict variables is to use conditional distributions For example, the optimal forecast of xt+1xt+1 given informatio

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