Difference between revisions of "InvertedWishart"
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| − | = InvertedWishart( psi, n, I, J ) = | + | == InvertedWishart(psi, n, I, J) == |
| + | The inverted Wishart distribution represents a distribution over [[covariance]] matrices, i.e., each sample from the [[InvertedWishart]] is a covariance matrix. It is conjugate to the [[Wishart]] distribution, which means it can be updated after observing data and computing its sample covariance, such that the Posterior is still a [[InvertedWishart]] distribution. Because of this conjugacy property, it is therefore usually used as a Bayesian prior distribution for covariance. The parameter, Psi, must be a positive definite matrix. | ||
| − | + | Suppose you represent the prior distribution of covariance using an inverted Wishart distribution: InvertedWishart(Psi, m). You observe some data, X[I, R], where R := 1..N indexes each datapoint and I is the vector dimension, and compute <code>A = Sum(X*X[I = J], R)</code>, where <code>A</code> is called the scatter matrix. The assumption is made that the data is generated, by nature, from a [[Gaussian]] distribution with the "true" covariance. The matrix <code>A</code> is an observation that gives you information about the true covariance matrix, so can use this to obtain a Bayesian posterior distribution on the true covariance given by: | |
| − | + | :<code>InverseWishart(A + Psi, n + m)</code> | |
| − | Suppose you represent the prior distribution of covariance using an inverted Wishart distribution: InvertedWishart(Psi,m). You observe some data, X[I,R], where R:=1..N indexes each datapoint and I is the vector dimension, and compute A = Sum( X*X[I=J], R), where A is called the scatter matrix. The assumption is made that the data is generated, by nature, from a Gaussian distribution with the "true" covariance. The matrix A is an observation that gives you information about the true covariance matrix, so can use this to obtain a Bayesian posterior distribution on the true covariance given by: | ||
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| + | == Library == | ||
Distribution Variations.ana | Distribution Variations.ana | ||
= See Also = | = See Also = | ||
| − | |||
* [[Wishart]] | * [[Wishart]] | ||
| − | * [[LDens_InvertedWishart]] | + | * [[LDens_InvertedWishart]] |
| + | * [[LDens_Wishart]] | ||
* [[Gaussian]] | * [[Gaussian]] | ||
* [[ChiSquared]] | * [[ChiSquared]] | ||
* [[Covariance]] | * [[Covariance]] | ||
Revision as of 23:32, 26 January 2016
InvertedWishart(psi, n, I, J)
The inverted Wishart distribution represents a distribution over covariance matrices, i.e., each sample from the InvertedWishart is a covariance matrix. It is conjugate to the Wishart distribution, which means it can be updated after observing data and computing its sample covariance, such that the Posterior is still a InvertedWishart distribution. Because of this conjugacy property, it is therefore usually used as a Bayesian prior distribution for covariance. The parameter, Psi, must be a positive definite matrix.
Suppose you represent the prior distribution of covariance using an inverted Wishart distribution: InvertedWishart(Psi, m). You observe some data, X[I, R], where R := 1..N indexes each datapoint and I is the vector dimension, and compute A = Sum(X*X[I = J], R), where A is called the scatter matrix. The assumption is made that the data is generated, by nature, from a Gaussian distribution with the "true" covariance. The matrix A is an observation that gives you information about the true covariance matrix, so can use this to obtain a Bayesian posterior distribution on the true covariance given by:
InverseWishart(A + Psi, n + m)
Library
Distribution Variations.ana
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