Wishart

Parameters

Scaling \(\mathbf{A}\) (symmetric \(n\times n\) matrix) and degrees of freedom (\(\nu \)). This only has a proper density if \(\nu > n - 1\) and \(\mathbf{A}\) is positive definite

Kernel

\(\exp \left( -\frac{1}{2}trace\left( \mathbf{A}^{-1}\mathbf{X}\right) \right) \left\vert \mathbf{X}\right\vert ^{\frac{1}{2}\left( \nu -n-1\right) }\)

Support

Positive definite symmetric matrices

Mean

\(\nu \mathbf{A}\)

Main Uses

Prior, exact and approximate posterior for the precision matrix (inverse of covariance matrix) of residuals in a multivariate regression, though that is mainly in the inverse form since that would be the distribution of the covariance matrix itself.

Draws

%RANWISHART(n,nu) draws a single \(n\times n\) Wishart matrix with \(\mathbf{A}=\mathbf{I}\) and degrees of freedom \(\nu \).

%RANWISHARTF(F,nu) draws a single \(n\times n \) Wishart matrix with \(\mathbf{A}=\mathbf{F}\mathbf{F^{\prime }}\) and degrees of freedom \(\nu \). \(\mathbf{F}\) can be any factor of \(\mathbf{A}\), but would typically be computed as the Cholesky factor using %DECOMP