The inverse Wishart distribution
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The inverse Wishart distribution (otherwise called the
inverted Wishart distribution) is a probability
distribution that is used in the Bayesian analysis of realvalued positive
definite matrices (e.g. matrices of the type that arise in risk management
contexts). It is a conjugate prior for the covariance matrix of a multivariate
normal distribution.
It has the following characteristics, where is a matrix,
is a positive
definite matrix and is the multivariate
gamma function.
Parameters (and constraints on parameters):

( =
degrees of freedom, real)
( = inverse scale
matrix, positive definite)

Support (i.e. values that it can take)

, i.e. is positive
definite, an matrix

Probability density function


Mean


If the elements of are and the elements of
are then
The main use of the inverse Wishart distribution appears to
arise in Bayesian statistics. Suppose we want to make an inference about a
covariance matrix, , whose prior has a distribution. If
the observation set where the are independent variate
Normal (i.e. Gaussian) random variables drawn from a distribution then
the conditional distribution , i.e. the
probability of given , has a distribution, where
is the sample
covariance matrix.
The univariate special case of the inverse Wishart
distribution is the inverse gamma
distribution. With , , , we have:
where is the ordinary
(i.e. univariate) Gamma function, see MnGamma.
For other probability distributions see here.
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