Inverse Wishart Distribution

Definition

The probability density function of thed-dimensional Inverse Wishart distribution is given by

$y = f (Χ, Σ, ν) = \frac{{| T |}^{(ν / 2)} e^{(- \frac{1}{2} trace (T X^{- 1}))}}{2^{(ν d) / 2} π^{(d (d - 1)) / 4} {| X |}^{(ν + d + 1) / 2} Γ (ν / 2) ... Γ (ν - (d - 1)) / 2},$

whereXandTared-by-dsymmetric positive definite matrices, andνis a scalar greater than or equal tod. While it is possible to define the Inverse Wishart for singularΤ, the density cannot be written as above.

If a random matrix has a Wishart distribution with parametersT^–1andν, then the inverse of that random matrix has an inverse Wishart distribution with parametersΤandν. The mean of the distribution is given by

$\frac{1}{ν - d - 1} T$

wheredis the number of rows and columns inT.

Only random matrix generation is supported for the inverse Wishart, including both singular and nonsingularT.

Background

The inverse Wishart distribution is based on theWishart distribution. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution.

Example

Notice that the sampling variability is quite large when the degrees of freedom is small.

Tau = [1 .5; .5 2]; df = 10; S1 = iwishrnd(Tau,df)*(df-2-1) S1 = 1.7959 0.64107 0.64107 1.5496 df = 1000; S2 = iwishrnd(Tau,df)*(df-2-1) S2 = 0.9842 0.50158 0.50158 2.1682