InvertedWishart

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

See Also

• Wishart
• Gaussian
• ChiSquared
• Covariance