The probability density function of the *d*-dimensional
Inverse Wishart distribution is given by

$$y=f({\rm X},\Sigma ,\nu )=\frac{{\left|T\right|}^{\left(\nu /2\right)}{e}^{\left(-\frac{1}{2}\text{trace}\left(T{X}^{-1}\right)\right)}}{{2}^{(\nu d)/2}{\pi}^{(d(d-1))/4}{\left|X\right|}^{(\nu +d+1)/2}\Gamma \left(\nu /2\right)\mathrm{...}\Gamma (\nu -(d-1))/2},$$

where *X* and *T* are *d*-by-*d* symmetric
positive definite matrices, and *ν* is a scalar
greater than or equal to *d*. While it is possible
to define the Inverse Wishart for singular *Τ*,
the density cannot be written as above.

If a random matrix has a Wishart distribution with parameters *T*^{–1} and *ν*,
then the inverse of that random matrix has an inverse Wishart distribution
with parameters *Τ* and *ν*.
The mean of the distribution is given by

$$\frac{1}{\nu -d-1}T$$

where *d* is the
number of rows and columns in *T*.

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

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

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