Inverse-Wishart distribution

Inverse-Wishart
Notation
Parameters	degrees of freedom (real); , scale matrix (pos. def.)
Support	is p × p positive definite
PDF	is the multivariate gamma function; is the trace function;
Mean	For
Mode
Variance	see below

In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution.

We say $\mathbf {X}$ follows an inverse Wishart distribution, denoted as $\mathbf {X} \sim {\mathcal {W}}^{-1}(\mathbf {\Psi } ,\nu )$ , if its inverse $\mathbf {X} ^{-1}$ has a Wishart distribution ${\mathcal {W}}(\mathbf {\Psi } ^{-1},\nu )$ . Important identities have been derived for the inverse-Wishart distribution.^[2]

^ A. O'Hagan, and J. J. Forster (2004). Kendall's Advanced Theory of Statistics: Bayesian Inference. Vol. 2B (2 ed.). Arnold. ISBN 978-0-340-80752-1.
^ Haff, LR (1979). "An identity for the Wishart distribution with applications". Journal of Multivariate Analysis. 9 (4): 531–544. doi:10.1016/0047-259x(79)90056-3.

[Ohagan-1] A. O'Hagan, and J. J. Forster (2004). Kendall's Advanced Theory of Statistics: Bayesian Inference. Vol. 2B (2 ed.). Arnold. ISBN 978-0-340-80752-1.

[identity-haff-2] Haff, LR (1979). "An identity for the Wishart distribution with applications". Journal of Multivariate Analysis. 9 (4): 531–544. doi:10.1016/0047-259x(79)90056-3.

[1]

[2]

Notation	${\mathcal {W}}^{-1}({\mathbf {\Psi } },\nu )$
Parameters	$\nu >p-1$ degrees of freedom (real) $\mathbf {\Psi } >0$ , $p\times p$ scale matrix (pos. def.)
Support	$\mathbf {X}$ is p × p positive definite
PDF	${\frac {\left\|\mathbf {\Psi } \right\|^{\nu /2}}{2^{\nu p/2}\Gamma _{p}({\frac {\nu }{2}})}}\left\|\mathbf {X} \right\|^{-(\nu +p+1)/2}e^{-{\frac {1}{2}}\operatorname {tr} (\mathbf {\Psi } \mathbf {X} ^{-1})}$ $\Gamma _{p}$ is the multivariate gamma function $\operatorname {tr}$ is the trace function
Mean	${\frac {\mathbf {\Psi } }{\nu -p-1}}$ For $\nu >p+1$
Mode	${\frac {\mathbf {\Psi } }{\nu +p+1}}$ ^[1]^: 406
Variance	see below