Inverse-Wishart distribution

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In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability density function defined on matrices. In Bayesian statistics it is used as the conjugate for the covariance matrix of a multivariate normal distribution.

We say follows an inverse Wishart distribution, denoted as , if its probability density function is written as follows:

where is a matrix. The matrix is assumed to be positive definite.

Theorems

Distribution of the inverse of a Wishart-distributed matrix

If and is , then has an inverse Wishart distribution with probability density function:

.

where and is the multivariate gamma function.[1]

Marginal and conditional distributions from an inverse Wishart-distributed matrix

Suppose has an inverse Wishart distribution. Partition the matrices and conformably with each other

where and are matrices, then we have

i) is independent of and , where ;

ii) ;

iii) , where is a matrix normal distribution;

iv)

Conjugate distribution

If and has the a priori distribution then the conditional distribution of is .

Expectation

If , then

Related distributions

A univariate specialization of the inverse-Wishart distribution is the inverse-gamma distribution. With (i.e. univariate) and , and the probability density function of the inverse-Wishart distribution becomes

i.e., the inverse-gamma distribution, where is the ordinary Gamma function.

A generalization is the normal-inverse-Wishart distribution.

See also

References

  1. Kanti V. Mardia, J. T. Kent and J. M. Bibby (1979). Multivariate Analysis. Academic Press. ISBN 0-12-471250-9.

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