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The posterior distribution (of the objective function), in turn, is used to construct an acquisition function (often also referred to as infill sampling criteria) that determines what the next query point should be.

erences 8 External links History[edit source] The term is generally attributed to Jonas Mockus and is coined in his work from a series of publications on global optimization in the 1970s and 1980s. [2] [3] [4] Strategy[edit source] <span>Since the objective function is unknown, the Bayesian strategy is to treat it as a random function and place a prior over it. The prior captures our beliefs about the behaviour of the function. After gathering the function evaluations, which are treated as data, the prior is updated to form the posterior distribution over the objective function. The posterior distribution, in turn, is used to construct an acquisition function (often also referred to as infill sampling criteria) that determines what the next query point should be. Examples[edit source] Examples of acquisition functions include probability of improvement, expected improvement, Bayesian expected losses, upper confidence bounds (UCB), Thompson s

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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. </

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 <span>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 =

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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. <span>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. <span><body><html>

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 <span>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 =

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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.

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 <span>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 =

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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] <span>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

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 <span>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 =

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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.

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 <span>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 =

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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.

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 <span>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 =

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s away from the observations. Based on this distribution over functions, an acquisition function is computed (green shaded area, bottom panels), which represents the gain from evaluating the unknown function f at different x values; note that <span>the acquisition function is high where the posterior over f has both high mean and large uncertainty. Different acquisition functions can be used such as “expected improvement” or “information-gain”. The peak of the acquisition function (red line) is the best next point to evaluate, and

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the acquisition function is high where the posterior over f has both high mean and large uncertainty.

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stimate in the special case that all errors are Gaussian-distributed. Extensions and generalizations to the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. <span>The underlying model is similar to a hidden Markov model except that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Contents [hide] 1 History 2 Overview of the calculation 3 Example application 4 Technical description and context 5 Underlying dynamical system model 6 Details 6.1 Predict

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The underlying model of Kalman filter is similar to a hidden Markov model except that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions.

stimate in the special case that all errors are Gaussian-distributed. Extensions and generalizations to the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. <span>The underlying model is similar to a hidden Markov model except that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Contents [hide] 1 History 2 Overview of the calculation 3 Example application 4 Technical description and context 5 Underlying dynamical system model 6 Details 6.1 Predict

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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] <span>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

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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

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] <span>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

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rward 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 <span>the probability of observing the remaining observations given any starting point , i.e. . 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:

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] <span>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

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pass, the algorithm computes a set of backward probabilities which provide the probability of observing the remaining observations given any starting point , i.e. . These two sets of probability distributions can then be combined to obtain <span>the distribution over states at any specific point in time given the entire observation sequence: The last step follows from an application of the Bayes' rule and the conditional independence of and given . It remains to be seen, of course, how the forwa

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] <span>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

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bserving the remaining observations given any starting point , i.e. . 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<span>: The last step follows from an application of the Bayes' rule and the conditional independence of and given . It remains to be seen, of course, how the forwar

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] <span>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

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sed to define the Poisson distribution. If a Poisson point process is defined on some underlying space, then the number of points in a bounded region of this space will be a Poisson random variable. [45] Complete independence[edit source] <span>For a collection of disjoint and bounded subregions of the underlying space, the number of points of a Poisson point process in each bounded subregion will be completely independent of all the others. This property is known under several names such as complete randomness, complete independence, [21] or independent scattering [46] [47] and is common to all Poisson point processes.

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For a collection of disjoint and bounded subregions of the underlying space, the number of points of a Poisson point process in each bounded subregion will be completely independent of all the others. </h

sed to define the Poisson distribution. If a Poisson point process is defined on some underlying space, then the number of points in a bounded region of this space will be a Poisson random variable. [45] Complete independence[edit source] <span>For a collection of disjoint and bounded subregions of the underlying space, the number of points of a Poisson point process in each bounded subregion will be completely independent of all the others. This property is known under several names such as complete randomness, complete independence, [21] or independent scattering [46] [47] and is common to all Poisson point processes.

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at the beginning of our exposition there must be mathematical words or symbols which we do not deﬁne in terms of others but merely take as given: they are called primitives. And proof must start somewhere, just as deﬁnition must. If we are to avoid an inﬁnite regress, there must be some propositions that are not proved but can be used in the proofs of the

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At the core of attitudes to the axiomatic method that may be called realist is the view that ‘undeﬁned’ does not entail ‘meaningless’ and so it may be possible to provide a meaning for the primitive terms of our theory in advance of laying down the axioms: perhaps they are previously understood terms of ordinary langu

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d volume of Euclidean geometry to suitable subsets of the n -dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. <span>Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller"

[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma

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Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive:

[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma

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Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive:

[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma

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Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive:

[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma

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inequality Venn diagram Tree diagram v t e In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. [3] <span>The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e.g. t

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The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space.

inequality Venn diagram Tree diagram v t e In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. [3] <span>The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e.g. t

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on subsumes both of these and also works for distributions which are neither discrete nor absolutely continuous; the expected value of a random variable is the integral of the random variable with respect to its probability measure. [1] [2] <span>The expected value does not exist for random variables having some distributions with large "tails", such as the Cauchy distribution. [3] For random variables such as these, the long-tails of the distribution prevent the sum/integral from converging. The expected value is a key aspect of how one characterizes a probability distribution; it is one type of location parameter. By contrast, the variance is a measure of dispersion of t

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The expected value does not exist for random variables having some distributions with large "tails", such as the Cauchy distribution. [3] For random variables such as these, the long-tails of the distribution prevent the sum/integral from converging.

on subsumes both of these and also works for distributions which are neither discrete nor absolutely continuous; the expected value of a random variable is the integral of the random variable with respect to its probability measure. [1] [2] <span>The expected value does not exist for random variables having some distributions with large "tails", such as the Cauchy distribution. [3] For random variables such as these, the long-tails of the distribution prevent the sum/integral from converging. The expected value is a key aspect of how one characterizes a probability distribution; it is one type of location parameter. By contrast, the variance is a measure of dispersion of t

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One way to unify the discrete and continuous cases is to use the P and E operators (and related operators like var, cov, and cor) exclusively rather than writing sums and integrals.

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One way to unify the discrete and continuous cases is to use the P and E operators (and related operators like var, cov, and cor) exclusively rather than writing sums and integrals.

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Note that P is a different function for each different probability distri- bution

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P gives probabilities of events, so it is a function A↦P(A) that maps events to real numbers.

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E gives expectations of random variables, so it is a function X↦E(X) that maps random variables to real numbers.

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Let Ω be an arbitrary set. A sigma-algebra for Ω is a family of subsets of Ω that contains Ω and is closed under complements and countable unions and intersections.

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Let Ω be an arbitrary set. A sigma-algebra for Ω is a family of subsets of Ω that contains Ω and is closed under complements and countable unions and intersections.

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Let Ω be an arbitrary set. A sigma-algebra for Ω is a family of subsets of Ω that contains Ω and is closed under complements and countable unions and intersections.

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The smallest sigma-algebra is {∅, Ω}. It must contain Ω by definition, and it must contain ∅ because it is Ω c . Unions and intersections of Ω and ∅ give us the same sets back, no new sets.

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The largest sigma-algebra is the set of all subsets of Ω, called the power set of Ω.

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The largest sigma-algebra is the set of all subsets of Ω, called the power set of Ω.

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A set Ω equipped with a sigma-algebra is called a measurable space and usually denoted as a pair (Ω,). In this context, the elements of are called measurable sets.

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A set Ω equipped with a sigma-algebra is called a measurable space and usually denoted as a pair (Ω,). In this context, the elements of are called measurable sets.

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A set Ω equipped with a sigma-algebra is called a measurable space and usually denoted as a pair . In this context, the elements of are called measurable sets.

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A positive measure on a measurable space (Ω,) is a function µ : A → R ∪ {∞} that satisfies µ(A) ≥ 0, A ∈ .

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A signed measure on a measurable space (Ω,) is a function µ : A → R that is countably additive

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A signed measure on a measurable space is a function µ : A → R that is countably additive

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Counting measure is a positive measure counts the number of points in a set

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Counting measure is a positive measure counts the number of points in a set

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Lebesgue measure on R corresponds to the dx of ordinary calculus: μ(A)=∫Adx whenever A is a set over which the Riemann integral is defined.

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A measurable space (Ω, A) equipped with a measure µ, either a positive measure or a signed measure, is called a measure space and usually denoted as a triple (Ω, A, µ).

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If P and Q are probability measures, then P − Q is a signed measure, so we need signed measures to compare probability measures. If µ and ν are signed measures and a and b are real numbers, then aµ+bν is a signed measure, so the family of all signed measures on a measurable space is a vector space. The latter exp

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If P and Q are probability measures, then P − Q is a signed measure, so we need signed measures to compare probability measures.

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这 26 条婚姻法的小常识，你结没结婚都应该知道 - 博海拾贝 - 萝卜网 博海拾贝 关于 联系 每日博海拾贝 萝卜网关闭公告 订阅 微博 腾讯微博 微信 诸暨 | 最优购| 烧饼博客 这 26 条婚姻法的小常识，你结没结婚都应该知道 梁萧 发布于 9小时前 分类：文摘 作者：蔡思斌 如何在婚姻家庭中保护自己权益的话题越来越受到关注，因为再美好的婚姻亦随时有可能触礁，而一旦触及便会牵涉到感情、财产、孩子等诸多问题，故了解一些婚姻法的小常识是相当有必要的。 1. 大多数人都把举办婚礼当作是自己婚姻生活的开始，殊不知只举办婚礼而没有领结婚证，却够不上法律意义上的结婚，结婚必须进行结婚登记。 以 1994 年 2 月 1 日《婚姻登记管理条例》实施为起算点，在此之后的同居已没有事实婚姻一说。 2. 同居期间买房所有权以登记为准，如二人联名的未必各得一半，要看各自出资比例的。如果后面结婚了基本上就一人一半，不再考虑原出资比例了。 3. 结婚给付彩礼是传统习俗，但未办理结婚登记是可以要求返还的。即便已经办理结婚登记的，离婚时在一定条件下亦有可能要求返还。比如登记后没有同居，比如对方是贫困家庭，倾家荡产来送彩礼造成贫困等。 4. 婚前个人财产婚后经过一定期限就转化为夫妻共同财产的规定早已失效，你就结婚五十年，人家婚前财产仍然是他自己的。除非双方特别约定，否则婚前个人财产仍属于个人财产而不会自动转化。当然，如果都混同一起无从区分那又是另外一回事了。 5. 夫妻在婚姻关系存续期间所取得的财产归夫妻双方共同所有（另有书面约定的除外），离婚时一般平均分配，有时也会考虑过错方多分。多分其实没有多分多少了，别想得太多，能多个一成已经是非常多了。想想看，对方长期出轨，与异性长期同居才多分个二成。你这一般性的过错能多分多少呢。 6. 购房属于家庭重大支出，但因为购房时间、资金来源等不同因素，一旦离婚有可能争议不断，故建议在房产证上加上自己名字，如此可将风险降到最低。 7. 夫妻共同财产不能隐瞒，若离婚后发现对方有隐藏的共同财产，另一方依然可向人民法院提起诉讼请求分割。如果是不动产分割的，是不受诉讼时效限定的。 8. 很多离婚当事人特别是女性在离婚时往往处于被动地位，主要在于其不清楚家庭的收支情况，房子、车子、存款只知道有很多但具体情况一概不知，故了解并掌握家庭财产情况十分必要，至少可以防患于未然，不至于在离婚时执行不能或被对方恶意转移。所以，平时可以长长心，配偶一方常用的银行账户、证券开户机构及用户码、公司名称、保管箱存放地点及编号等多了解一些不过分吧。 9. 很多人误以为分居两年或更长时间后便可以自动离婚。但根据我国婚姻法规定，夫妻离婚只能通过协议离婚或诉讼离婚进行，不存在自动离婚一说。想离，不能协商的，就赶快起诉吧。毕竟分居二年很多地方法院都不会直接判离的，还是连续离婚诉讼比较靠谱。 10. 一般情况下，夫妻一方向他人借的钱，由夫妻共同财产偿还。即便离婚亦不能逃脱，但如果他人知道夫妻双方已签订协议约定财产归各自所有的，则由借款一方偿还。这就考验你的智商了，这实质上很不容易的，没有债权人会这么傻瓜的吧。 11. 若一方有出轨现象的，应及时保留证据，短信、录音、录像、往来信件、日记、悔过书、保证书、通话记录清单、QQ、微信聊天记录等都可以作为证据使用。 另外，一定要抓住黄金 72 小时。一般当事人出轨被发现的，出于愧疚的心理，在事发初期对于财产、孩子抚养权、离婚问题会作出相对多的让步，如果你决定离婚的，那就速战速决。当然也不能太贪心了，激起对方反感，内疚悔恨情绪一过找了律师咨询的话那就落不了什么好了。 12. 若一方将夫妻共同财产赠与小三，该赠与行为因违背社会伦理而无效，赠与的财产可以追回。有时侯，甚至有夫妻共同配合，将送给小三财产追回的。郎教授的例子就很典型，做个小三也是不容易的。 13. 签订「忠诚协议」应就赔偿责任予以明确，若笼统使用「净身出户」有可能被认定无效。不要总想得到太多或全部，适当多一些、不太过分的忠诚协议还是比较容易获得法官支持的。 14. 家暴不仅仅是殴打、捆绑等暴力行为，精神暴力亦属于家庭暴力范畴，遇到家庭暴力应及时保留证据，派出所出警记录、双方笔录、伤情鉴定等均是很有证明力的证据。 男人打女人，这是最大的恶习。依我的办案经验，有了第一次，就绝对会有第二次，n 次。真碰到家暴的，其实第一次就可以选择离婚了，别想着对方能够悔改了。 想想，尝到靠拳头说话的快感，后面还怎么会选择用语言与你沟通哟。不服就打，简单快捷。过了这个底线，后面也就无限可言了。 15. 若一方有赌博恶习的，应当注意保留证据，如派出所出警记录、居委会调解、谈话承认录音等，避免莫名背上赌债。 16. 所谓的青春补偿费是没有法律依据的，但夫妻一方因重婚、与他人同居、实施家庭暴力、虐待家庭成员导致离婚的，无过错方是有权请求损害赔偿的。当然，如果已实际给付又是另外一回事了。 17. 公民有生育的权利，但丈夫不能强迫妻子生或不生，通过诉讼解决也不行。若妻子执意不生孩子，丈夫可通过起诉离婚实现自己的生育权，找别人结婚再生还是可以的。 18. 两周岁以下的小孩，离婚时抚养权一般归女方。而十岁以上的孩子就要考虑孩子自己的选择。孩子的年龄、性别、父母各方面条件、长辈帮忙抚养情况均是处理抚养权的考虑情形。 不过，现在民法通则开始实施了，限制民事行为能力人年龄调整为八岁，后续婚姻法司法解释也有可能作相应修改。 19. 抚养费一般是按月收入的百分之二十至三十的比例支付。这只是理论数据，不是人家月收入十万，就要支付二万的抚养费，还要考虑当地经济水平、生活水平等。 20. 夫妻之间写借条实际上是有效的，这视为特殊的财产约定。当然，还是要有实际履行借款义务证据的，如果是空对空，也是没有效力的。 21. 二个人结婚，还是要给彼此一定的空间。不要去翻看配偶的手机、电脑、钱包，这是最基本的彼此尊重，互相给对方信任感、安全感多好呢。 22. 如果一方不尊重配偶的父母，甚至连招呼都不打，要断绝往来的，这种人还是早离了算。不尊重你的父母长辈，就意味着你在他（她）心目中什么都不是，婚姻迟早要崩的。 23. 谈恋爱挺好，谈一场普普通通的恋爱更好。如果一个人在恋爱期间能穷尽花样讨好你的，那意味着在婚后不喜欢你时也有更多的方式来作你，花式虐你没商量，有时也是很恐怖的。 24. 同理，如果在恋爱期间能用自残来表明心意的，有多远离他（她）多远，伤害自己都下得了手，伤害别人更是不在话下。 25. 婚生子女可以跟父姓也可以跟母姓，看谁先下手为强了。后续要改姓的，除非双方协商一致，否则是无法修改的，即便是离婚协议中有约定可以改姓的条款也不行，派出所同样要求双方同时到场表示同意。所以，若离婚协议中有改姓条款的，最好同时履行完毕，否则对方后面反悔你就没招了。 26. 离婚案件法院在审理过程中，对于夫妻债务，除非双方共同确认或借款时有共同签字确认的，法院才会分割该债务。否则，如只是一方个人借款，另一方否认夫妻共同债务的，法院一般都是不予处理分割的，等债权人另案起诉时再处理。 来源：知乎 未经允许不得转载：博海拾贝 » 这 26 条婚姻法的小常识，你结没结婚都应该知道 标签：婚姻法爱情 相关推荐 [imagelink]结婚前应该了解对方些什么？人们最关心这 10 个问题 [imagelink]女儿的17岁愿望，是我和丈夫离婚 [imagelink]你愿意和外地人结婚吗？ [imagelink]爱情这玩意，说不准的 [im