Inverse Wishart distribution¶
If \(\Sigma^{-1} \sim Wi(S,v)\) than \(\Sigma \sim IW(S^{-1}, v + D + 1)\) where IW is the inverse Wishart, and it can be viewed as the generalization of Inverse Gamma distribution.
It is defined if \(v > D -1\)
\[IW(\Sigma| S, v) = \frac{1}{Z_{IW}} |\Sigma|^{-(v + D +1)/2} \exp (-\frac{1}{2} tr(S^{-1}\Sigma^{-1})) \]
Moments¶
\[
X \sim IW(S^{-1}, v+D+1)
\]
Mean¶
\[E[X] = \frac{S^{-1}}{v - D - 1} \]
Mode¶
\[\frac{S^{-1}}{v - D + 1} \]
Connection to Inverse Gamma¶
In case of D = 1 $\( IW(\sigma^{2}| S^{-1}, v) = IG(\sigma^{2}| v/2, S/2)\)$