The inverse Wishart distribution represents positive definite matrices. This can be extremely useful in Bayesian analysis, as the inverse Wishart can serve as a prior for any covariance matrix .

## Basic Facts

Let be a DxD positive definite matrix. We say is drawn from an inverse Wishart with parameters , when we have the following probability density function

where

- : scalar degrees of freedom
- : DxD positive definite matrix

The expected value (mean) of under this p.d.f. is

which is only defined for . So for , we have , etc.

## Visualizations

To gain intuition about the influence of different parameters, here I show i.i.d. draws from an inverse Wishart with inverse scale matrix set to the 2-dimensional identity matrix. Each plot considers a different value of , the degrees of freedom.

Here, we use **2** degrees of freedom, the minimum permissible with a two-dimensional scale matrix.

Moving to **8** degrees of freedom… we can see the shrinking size of the covariance matrix

Finally, here’s **32** degrees of freedom. There is very little variability here, as we’re converging to the mean (a scaled-down version of the identity matrix).

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