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## Inverse Wishart Distribution

### Definition

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

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

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.

### Background

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.

### 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```