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
![\[ p( \Sigma | v, S ) = \frac{ |S|^{\frac{v}{2}} }{ 2^{\frac{vD}{2}} \Gamma_D(\frac{v}{2} ) } |\Sigma|^{-\frac{v+D-1}{2} } e^{ -\frac{1}{2} tr( S\Sigma^{-1} ) } \]](https://www.michaelchughes.com/blog/wp-content/ql-cache/quicklatex.com-741504ba164b420bebf719fbf170c0b2_l3.png "Rendered by QuickLaTeX.com")
where
- : scalar degrees of freedom
- : DxD positive definite matrix
The expected value (mean) of under this p.d.f. is
![\[ \mathbb{E}[ \Sigma ] = \frac{ S }{ v - D - 1} \]](https://www.michaelchughes.com/blog/wp-content/ql-cache/quicklatex.com-d68da33ebe8616e95cc61f3f5ee725ef_l3.png "Rendered by QuickLaTeX.com")
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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