Inverse-Wishart distribution

The probability density function of the inverse Wishart is:{{Cite book|title=Bayesian Data Analysis, Third Edition|last1=Gelman|first1=Andrew|last2=Carlin|first2=John B.|last3=Stern|first3=Hal S.|last4=Dunson|first4=David B.|last5=Vehtari|first5=Aki|last6=Rubin|first6=Donald B.|date=2013-11-01|publisher=Chapman and Hall/CRC|isbn=9781439840955|edition=3rd|location=Boca Raton|language=en}}

:$$

f_{\mathbf X}({\mathbf X}; {\mathbf \Psi}, \nu) = \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})}

where $\backslash mathbf\{X\}$ and $\{\backslash mathbf\backslash Psi\}$ are $p\backslash times\; p$ positive definite matrices, $|\; \backslash cdot\; |$ is the determinant, and $\backslash Gamma\_p(\backslash cdot)$ is the multivariate gamma function.

If $\{\backslash mathbf\; X\}\backslash sim\; \backslash mathcal\{W\}(\{\backslash mathbf\backslash Sigma\},\backslash nu)$ and $\{\backslash mathbf\backslash Sigma\}$ is of size $p\; \backslash times\; p$, then $\backslash mathbf\{A\}=\{\backslash mathbf\; X\}^\{-1\}$ has an inverse Wishart distribution $\backslash mathbf\{A\}\backslash sim\; \backslash mathcal\{W\}^\{-1\}(\{\backslash mathbf\backslash Sigma\}^\{-1\},\backslash nu)$ .{{Cite book

| author = Kanti V. Mardia, J. T. Kent and J. M. Bibby

| title = Multivariate Analysis

| publisher = Academic Press

| year = 1979

| isbn = 978-0-12-471250-8

}}

Suppose $\{\backslash mathbf\; A\}\backslash sim\; \backslash mathcal\{W\}^\{-1\}(\{\backslash mathbf\backslash Psi\},\backslash nu)$ has an inverse Wishart distribution. Partition the matrices $\{\backslash mathbf\; A\}$ and $\{\backslash mathbf\backslash Psi\}$ conformably with each other

:$$

{\mathbf{A}} = \begin{bmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{21} & \mathbf{A}_{22} \end{bmatrix}, \;

{\mathbf{\Psi}} = \begin{bmatrix} \mathbf{\Psi}_{11} & \mathbf{\Psi}_{12} \\ \mathbf{\Psi}_{21} & \mathbf{\Psi}_{22} \end{bmatrix}

where $\{\backslash mathbf\; A\_\{ij\}\}$ and $\{\backslash mathbf\; \backslash Psi\_\{ij\}\}$ are $p\_\{i\}\backslash times\; p\_\{j\}$ matrices, then we have

1. $\backslash mathbf\; A\_\{11\}$ is independent of $\backslash mathbf\; A\_\{11\}^\{-1\}\; \backslash mathbf\; A\_\{12\}$ and $\{\backslash mathbf\; A\}\_\{22\backslash cdot\; 1\}$, where $\{\backslash mathbf\; A\_\{22\backslash cdot\; 1\}\}\; =\; \{\backslash mathbf\; A\}\_\{22\}\; -\; \{\backslash mathbf\; A\}\_\{21\}\{\backslash mathbf\; A\}\_\{11\}^\{-1\}\{\backslash mathbf\; A\}\_\{12\}$ is the Schur complement of $\{\backslash mathbf\; A\_\{11\}\; \}$ in $\{\backslash mathbf\; A\}$;
2. $\{\backslash mathbf\; A\_\{11\}\; \}\; \backslash sim\; \backslash mathcal\{W\}^\{-1\}(\{\backslash mathbf\; \backslash Psi\_\{11\}\; \},\; \backslash nu-p\_\{2\})$;
3. $\{\backslash mathbf\; A\}\_\{11\}^\{-1\}\; \{\backslash mathbf\; A\}\_\{12\}\; \backslash mid\; \{\backslash mathbf\; A\}\_\{22\backslash cdot\; 1\}\; \backslash sim\; MN\_\{p\_\{1\}\backslash times\; p\_\{2\}\}$

( {\mathbf \Psi}_{11}^{-1} {\mathbf \Psi}_{12}, {\mathbf A}_{22\cdot 1} \otimes {\mathbf \Psi}_{11}^{-1}) , where $MN\_\{p\backslash times\; q\}(\backslash cdot,\backslash cdot)$ is a matrix normal distribution;

1. $\{\backslash mathbf\; A\}\_\{22\backslash cdot\; 1\}\; \backslash sim\; \backslash mathcal\{W\}^\{-1\}(\{\backslash mathbf\; \backslash Psi\}\_\{22\backslash cdot\; 1\},\; \backslash nu)$, where $\{\backslash mathbf\; \backslash Psi\_\{22\backslash cdot\; 1\}\}\; =\; \{\backslash mathbf\; \backslash Psi\}\_\{22\}\; -\; \{\backslash mathbf\; \backslash Psi\}\_\{21\}\{\backslash mathbf\; \backslash Psi\}\_\{11\}^\{-1\}\{\backslash mathbf\; \backslash Psi\}\_\{12\}$;

Suppose we wish to make inference about a covariance matrix $\{\backslash mathbf\{\backslash Sigma\}\}$ whose prior $\{p(\backslash mathbf\{\backslash Sigma\})\}$ has a $\backslash mathcal\{W\}^\{-1\}(\{\backslash mathbf\backslash Psi\},\backslash nu)$ distribution. If the observations $\backslash mathbf\{X\}=[\backslash mathbf\{x\}\_1,\backslash ldots,\backslash mathbf\{x\}\_n]$ are independent p-variate Gaussian variables drawn from a $N(\backslash mathbf\{0\},\{\backslash mathbf\; \backslash Sigma\})$ distribution, then the conditional distribution $\{p(\backslash mathbf\{\backslash Sigma\}\backslash mid\backslash mathbf\{X\})\}$ has a $\backslash mathcal\{W\}^\{-1\}(\{\backslash mathbf\; A\}+\{\backslash mathbf\backslash Psi\},n+\backslash nu)$ distribution, where $\{\backslash mathbf\{A\}\}=\backslash mathbf\{X\}\backslash mathbf\{X\}^T$.

Because the prior and posterior distributions are the same family, we say the inverse Wishart distribution is conjugate to the multivariate Gaussian.

Due to its conjugacy to the multivariate Gaussian, it is possible to marginalize out (integrate out) the Gaussian's parameter $\backslash mathbf\{\backslash Sigma\}$, using the formula $p(x)\; =\; \backslash frac\{\; p(x\; |\; \backslash Sigma)\; p(\backslash Sigma)\}\{p(\backslash Sigma\; |\; x)\}$ and the linear algebra identity $v^T\; \backslash Omega\; v\; =\; \backslash text\{tr\}(\; \backslash Omega\; v\; v^T)$:

: $f\_\{\backslash mathbf\; X\backslash ,\backslash mid\backslash ,\backslash Psi,\backslash nu\}\; (\backslash mathbf\; x)\; =\; \backslash int\; f\_\{\backslash mathbf\; X\backslash ,\backslash mid\backslash ,\backslash mathbf\backslash Sigma\backslash ,=\backslash ,\backslash sigma\}(\backslash mathbf\; x)\; f\_\{\backslash mathbf\backslash Sigma\backslash ,\backslash mid\backslash ,\backslash mathbf\backslash Psi,\backslash nu\}\; (\backslash sigma)\backslash ,d\backslash sigma\; =\; \backslash frac\{|\backslash mathbf\{\backslash Psi\}|^\{\backslash nu/2\}\; \backslash Gamma\_p\backslash left(\backslash frac\{\backslash nu+n\}\{2\}\backslash right)\}\{\backslash pi^\{np/2\}|\backslash mathbf\{\backslash Psi\}+\backslash mathbf\{A\}|^\{(\backslash nu+n)/2\}\; \backslash Gamma\_p(\backslash frac\{\backslash nu\}\{2\})\}$

(this is useful because the variance matrix $\backslash mathbf\{\backslash Sigma\}$ is not known in practice, but because $\{\backslash mathbf\backslash Psi\}$ is known a priori, and $\{\backslash mathbf\; A\}$ can be obtained from the data, the right hand side can be evaluated directly). The inverse-Wishart distribution as a prior can be constructed via existing transferred prior knowledge.{{cite journal|last1=Shahrokh Esfahani|first1=Mohammad|last2=Dougherty|first2=Edward|title=Incorporation of Biological Pathway Knowledge in the Construction of Priors for Optimal Bayesian Classification|journal= IEEE/ACM Transactions on Computational Biology and Bioinformatics|date=2014|volume=11|issue=1|pages=202–218|doi=10.1109/tcbb.2013.143|pmid=26355519|s2cid=10096507 }}

The following is based on Press, S. J. (1982) "Applied Multivariate Analysis", 2nd ed. (Dover Publications, New York), after reparameterizing the degree of freedom to be consistent with the p.d.f. definition above.

Let $W\; \backslash sim\; \backslash mathcal\{W\}(\backslash mathbf\; \backslash Psi^\{-1\},\; \backslash nu)$ with $\backslash nu\; \backslash ge\; p$ and $X\; \backslash doteq\; W^\{-1\}$, so that $X\; \backslash sim\; \backslash mathcal\{W\}^\{-1\}(\backslash mathbf\; \backslash Psi,\; \backslash nu)$.

The mean, for $\backslash nu\backslash geq\; p+2$:{{rp|91}}

:$\backslash operatorname\; E(\backslash mathbf\; X)\; =\; \backslash frac\{\backslash mathbf\backslash Psi\}\{\backslash nu-p-1\}.$

The variance of each element of $\backslash mathbf\{X\}$:

:$$

\operatorname{Var}(x_{ij}) = \frac{(\nu-p+1)\psi_{ij}^2 + (\nu-p-1)\psi_{ii}\psi_{jj}}

{(\nu-p)(\nu-p-1)^2(\nu-p-3)}

The variance of the diagonal uses the same formula as above with $i=j$, which simplifies to:

:$$

\operatorname{Var}(x_{ii}) = \frac{2\psi_{ii}^2}{(\nu-p-1)^2(\nu-p-3)}.

The covariance of elements of $\backslash mathbf\{X\}$ are given by:

:$$

\operatorname{Cov}(x_{ij},x_{k\ell}) = \frac{2\psi_{ij}\psi_{k\ell} + (\nu-p-1) (\psi_{ik}\psi_{j\ell} + \psi_{i\ell} \psi_{kj})}{(\nu-p)(\nu-p-1)^2(\nu-p-3)}

The same results are expressed in Kronecker product form by von Rosen{{Cite journal|last=Rosen|first=Dietrich von|date=1988|title=Moments for the Inverted Wishart Distribution|journal=Scand. J. Stat.|volume=15|pages=97–109|jstor=}} as follows:

:$$

\begin{align}

\mathbf{E} \left ( W^{-1} \otimes W^{-1} \right ) & = c_1 \Psi \otimes \Psi

+ c_2 Vec (\Psi) Vec (\Psi)^T + c_2 K_{pp} \Psi \otimes \Psi \\

\mathbf{Cov}_\otimes \left ( W^{-1} ,W^{-1} \right ) & = (c_1 - c_3 ) \Psi \otimes \Psi

+ c_2 Vec (\Psi) Vec (\Psi)^T + c_2 K_{pp} \Psi \otimes \Psi

\end{align}

where

:$$

\begin{align}

c_2 & = \left [ (\nu-p)(\nu-p-1)(\nu-p-3) \right ]^{-1} \\

c_1 & = (\nu-p-2) c_2 \\

c_3 & = (\nu -p-1)^{-2},

\end{align}

:$K\_\{pp\}\; \backslash text\{\; is\; a\; \}\; p^2\; \backslash times\; p^2$ commutation matrix

:$$

\mathbf{Cov}_\otimes \left ( W^{-1},W^{-1} \right ) = \mathbf{E} \left ( W^{-1} \otimes W^{-1} \right ) - \mathbf{E} \left ( W^{-1} \right ) \otimes \mathbf{E} \left ( W^{-1} \right ).

There appears to be a typo in the paper whereby the coefficient of $K\_\{pp\}\; \backslash Psi\; \backslash otimes\; \backslash Psi$ is given as $c\_1$ rather than $c\_2$, and that the expression for the mean square inverse Wishart, corollary 3.1, should read

:$$

\mathbf{E} \left [ W^{-1} W^{-1} \right ] = (c_1+c_2) \Sigma^{-1} \Sigma^{-1} + c_2 \Sigma^{-1} \mathbf{tr}(\Sigma^{-1}).

To show how the interacting terms become sparse when the covariance is diagonal, let $\backslash Psi\; =\; \backslash mathbf\; I\_\{3\; \backslash times\; 3\}$ and introduce some arbitrary parameters $u,\; v,\; w$:

:$$

\mathbf{E} \left ( W^{-1} \otimes W^{-1} \right ) = u \Psi \otimes \Psi

+ v \, \mathrm{vec}(\Psi) \, \mathrm{vec}(\Psi)^T + w K_{pp} \Psi \otimes \Psi.

where $\backslash mathrm\{vec\}$ denotes the matrix vectorization operator. Then the second moment matrix becomes

:$$

\mathbf{E} \left ( W^{-1} \otimes W^{-1} \right ) = \begin{bmatrix}

u+v+w & \cdot & \cdot & \cdot & v & \cdot & \cdot & \cdot & v \\

\cdot & u & \cdot & w & \cdot & \cdot & \cdot & \cdot & \cdot \\

\cdot & \cdot & u & \cdot & \cdot & \cdot & w & \cdot & \cdot \\

\cdot & w & \cdot & u & \cdot & \cdot & \cdot & \cdot & \cdot \\

v & \cdot & \cdot & \cdot & u+v+w & \cdot & \cdot & \cdot & v \\

\cdot & \cdot & \cdot & \cdot & \cdot & u & \cdot & w & \cdot \\

\cdot & \cdot & w & \cdot & \cdot & \cdot & u & \cdot & \cdot \\

\cdot & \cdot & \cdot & \cdot & \cdot & w & \cdot & u & \cdot \\

v & \cdot & \cdot & \cdot & v & \cdot & \cdot & \cdot & u+v+w \\

\end{bmatrix}

which is non-zero only when involving the correlations of diagonal elements of $W^\{-1\}$, all other elements are mutually uncorrelated, though not necessarily statistically independent. The variances of the Wishart product are also obtained by Cook et al.{{Cite journal|last1=Cook|first1=R D|last2=Forzani|first2=Liliana|author2-link= Liliana Forzani |editor-first1=Brian |editor-last1=Cook |date=August 2019|title=On the mean and variance of the generalized inverse of a singular Wishart matrix|url=https://www.researchgate.net/publication/254211710|journal=Electronic Journal of Statistics|volume=5|doi=10.4324/9780429344633 |isbn=9780429344633 |s2cid=146200569 }} in the singular case and, by extension, to the full rank case.

Muirhead{{Cite book |last=Muirhead |first=Robb |title=Aspects of Multivariate Statistical Theory |publisher=Wiley |year=1982 |isbn=0-471-76985-1 |location=USA |pages=93 |language=English}} shows in Theorem 3.2.8 that if $A^\{p\; \backslash times\; p\}$ is distributed as $\backslash mathcal\{W\}\_p\; (\backslash nu,\backslash Sigma\; )$ and $V$ is an arbitrary vector, independent of $A$ then $V^T\; A\; V\; \backslash sim\; \backslash mathcal\{\; W\; \}\_1(\backslash nu,\; A^T\; \backslash Sigma\; A)$ and $\backslash frac\; \{\; V^T\; A\; V\; \}\{\; V^T\; \backslash Sigma\; V\; \}\; \backslash sim\; \backslash chi^2\_\{\backslash nu-1\}$, one degree of freedom being relinquished by estimation of the sample mean in the latter. Similarly, Bodnar et.al. further find that $\backslash frac\; \{\; V^T\; A^\{-1\}\; V\; \}\{\; V^T\; \backslash Sigma^\{-1\}\; V\; \}\; \backslash sim\; \backslash text\{Inv-\}\backslash chi^2\_\{\backslash nu\; -\; p\; +\; 1\}$ and setting $V=\; (1,\backslash ,0,\; \backslash cdots\; ,0)^T$ the marginal distribution of the leading diagonal element is thus

: $\backslash frac\; \{\; [\; A^\{-1\}\; ]\_\{1,1\}\; \}\{\; [\; \backslash Sigma^\{-1\}]\_\{1,1\}\; \}\; \backslash sim\; \backslash frac\{2^\{-k/2\}\}\{\backslash Gamma(k/2)\}\; x^\{-k/2-1\}\; e^\{-1/(2\; x)\},\; \backslash ;\backslash ;\; k\; =\; \backslash nu\; -\; p\; +\; 1$

and by rotating $V$ end-around a similar result applies to all diagonal elements $[\; A^\{-1\}\; ]\_\{i,i\}$.

A corresponding result in the complex Wishart case was shown by Brennan and Reed{{Cite journal|last1=Brennan|first1=L E|last2=Reed|first2=I S|date=January 1982|title=An Adaptive Array Signal Processing Algorithm for Communications|journal= IEEE Transactions on Aerospace and Electronic Systems|volume=18 |issue=1|pages=120–130|doi=10.1109/TAES.1982.309212 |bibcode=1982ITAES..18..124B |s2cid=45721922 }} and the uncorrelated inverse complex Wishart $\backslash mathcal\{W\_C\}^\{-1\}(\backslash mathbf\{I\},\backslash nu,p)$ was shown by Shaman{{Cite journal |last=Shaman |first=Paul |date=1980 |title=The Inverted Complex Wishart Distribution and Its Application to Spectral Estimation |url=https://core.ac.uk/download/pdf/82734186.pdf |journal=Journal of Multivariate Analysis |volume=10 |pages=51–59 |doi=10.1016/0047-259X(80)90081-0}} to have diagonal statistical structure in which the leading diagonal elements are correlated, while all other element are uncorrelated.

• A univariate specialization of the inverse-Wishart distribution is the inverse-gamma distribution. With $p=1$ (i.e. univariate) and $\backslash alpha\; =\; \backslash nu/2$, $\backslash beta\; =\; \backslash mathbf\{\backslash Psi\}/2$ and $x=\backslash mathbf\{X\}$ the probability density function of the inverse-Wishart distribution becomes matrix

::$$

p(x\mid\alpha, \beta) = \frac{\beta^\alpha\, x^{-\alpha-1} \exp(-\beta/x)}{\Gamma_1(\alpha)}.

: i.e., the inverse-gamma distribution, where $\backslash Gamma\_1(\backslash cdot)$ is the ordinary Gamma function.

• The Inverse Wishart distribution is a special case of the inverse matrix gamma distribution when the shape parameter $\backslash alpha\; =\; \backslash frac\{\backslash nu\}\{2\}$ and the scale parameter $\backslash beta\; =2$.
• Another generalization has been termed the generalized inverse Wishart distribution, $\backslash mathcal\{GW\}^\{-1\}$. A $p\; \backslash times\; p$ positive definite matrix $\backslash mathbf\{X\}$ is said to be distributed as $\backslash mathcal\{GW\}^\{-1\}(\backslash mathbf\{\backslash Psi\},\backslash nu,\backslash mathbf\{S\})$ if $\backslash mathbf\{Y\}\; =\; \backslash mathbf\{X\}^\{1/2\}\backslash mathbf\{S\}^\{-1\}\backslash mathbf\{X\}^\{1/2\}$ is distributed as $\backslash mathcal\{W\}^\{-1\}(\backslash mathbf\{\backslash Psi\},\backslash nu)$. Here $\backslash mathbf\{X\}^\{1/2\}$ denotes the symmetric matrix square root of $\backslash mathbf\{X\}$, the parameters $\backslash mathbf\{\backslash Psi\},\backslash mathbf\{S\}$ are $p\; \backslash times\; p$ positive definite matrices, and the parameter $\backslash nu$ is a positive scalar larger than $2p$. Note that when $\backslash mathbf\{S\}$ is equal to an identity matrix, $\backslash mathcal\{GW\}^\{-1\}(\backslash mathbf\{\backslash Psi\},\backslash nu,\backslash mathbf\{S\})\; =\; \backslash mathcal\{W\}^\{-1\}(\backslash mathbf\{\backslash Psi\},\backslash nu)$. This generalized inverse Wishart distribution has been applied to estimating the distributions of multivariate autoregressive processes.{{Cite journal | last1 = Triantafyllopoulos | first1 = K. | title = Real-time covariance estimation for the local level model | doi = 10.1111/j.1467-9892.2010.00686.x | journal = Journal of Time Series Analysis | volume = 32 | issue = 2 | pages = 93–107 | year = 2011 | arxiv = 1311.0634 | s2cid = 88512953 }}
• A different type of generalization is the normal-inverse-Wishart distribution, essentially the product of a multivariate normal distribution with an inverse Wishart distribution.
• When the scale matrix is an identity matrix, $\backslash mathcal\{\backslash Psi\}\; =\; \backslash mathbf\{I\}$, and $\backslash mathcal\{\backslash Phi\}$ is an arbitrary orthogonal matrix, replacement of $\backslash mathbf\{X\}$ by $\{\backslash Phi\}\; \backslash mathbf\{X\}\; \backslash mathcal\{\backslash Phi\}^T$ does not change the pdf of $\backslash mathbf\{X\}$ so $\backslash mathcal\{W\}^\{-1\}(\backslash mathbf\{I\},\backslash nu,p)$ belongs to the family of spherically invariant random processes (SIRPs) in some sense.{{clarify|date=March 2022}}

: Thus, an arbitrary p-vector $V$ with $l\_2$ length $V^TV\; =\; 1$ can be rotated into the vector $\backslash mathbf\{\backslash Phi\}V\; =\; [1\; \backslash ;\; 0\; \backslash ;\; 0\; \backslash cdots]^T$ without changing the pdf of $V^T\; \backslash mathbf\{X\}\; V$, moreover $\backslash mathbf\{\backslash Phi\}$ can be a permutation matrix which exchanges diagonal elements. It follows that the diagonal elements of $\backslash mathbf\{X\}$ are identically inverse chi squared distributed, with pdf $f\_\{x\_\{11\}\}$ in the previous section though they are not mutually independent. The result is known in optimal portfolio statistics, as in Theorem 2 Corollary 1 of Bodnar et al,{{Cite journal|last1=Bodnar|first1=T.|last2=Mazur|first2=S.|last3=Podgórski|first3=K.|date=January 2015|title=Singular Inverse Wishart Distribution with Application to Portfolio Theory|url=https://journals.lub.lu.se/stat/article/download/15033/13602/|journal=Department of Statistics, Lund University|volume=(Working Papers in Statistics; Nr. 2)|pages=1–17}} where it is expressed in the inverse form $\backslash frac\{V^T\; \backslash mathbf\{\backslash Psi\}\; V\}\{V^T\; \backslash mathbf\; \{X\}\; V\}\; \backslash sim\; \backslash chi^2\_\{\backslash nu-p+1\}$.

• As is the case with the Wishart distribution linear transformations of the distribution yield a modified inverse Wishart distribution. If $\backslash mathbf\{X^\{p\; \backslash times\; p\; \}\}\; \backslash sim\; \backslash mathcal\{W\}^\{-1\}\_p\backslash left(\{\backslash mathbf\; \backslash Psi\},\; \backslash nu\; \backslash right).$ and $\{\backslash mathbf\; \backslash Theta\}^\{p\; \backslash times\; p\}$ are full rank matrices then{{Cite journal |last1=Bodnar |first1=T |last2=Mazur |first2=S |last3=Podgorski |first3=K |date=2015 |title=Singular Inverse Wishart Distribution with Application to Portfolio Theory |url=https://www.sciencedirect.com/science/article/pii/S0047259X15002353 |journal=Journal of Multivariate Analysis |volume=143 |pages=314–326|doi=10.1016/j.jmva.2015.09.021 }} $\backslash mathbf\{\backslash Theta\}\backslash mathbf\{X\}\{\backslash mathbf\; \backslash Theta\}^T\; \backslash sim\; \backslash mathcal\{W\}^\{-1\}\_p\backslash left(\{\backslash mathbf\; \backslash Theta\}\{\backslash mathbf\; \backslash Psi\; \}\; \{\backslash mathbf\; \backslash Theta\}^T,\; \backslash nu\; \backslash right).$
• If $\backslash mathbf\{X^\{p\; \backslash times\; p\; \}\}\; \backslash sim\; \backslash mathcal\{W\}^\{-1\}\_p\backslash left(\{\backslash mathbf\; \backslash Psi\},\; \backslash nu\; \backslash right).$ and $\{\backslash mathbf\; \backslash Theta\}\; ^\{\; m\; \backslash times\; p\; \}$ is of full rank $m$ then $\backslash mathbf\{\backslash Theta\}\backslash mathbf\{X\}\{\backslash mathbf\; \backslash Theta\}^T\; \backslash sim\; \backslash mathcal\{W\}^\{-1\}\_m\; \backslash left(\{\backslash mathbf\; \backslash Theta\}\{\backslash mathbf\; \backslash Psi\; \}\; \{\backslash mathbf\; \backslash Theta\}^T,\; \backslash nu\; \backslash right).$

• Inverse matrix gamma distribution
• Matrix normal distribution
• Wishart distribution
• Complex inverse Wishart distribution

{{Reflist}}

{{ProbDistributions|multivariate}}

Category:Continuous distributions

Category:Multivariate continuous distributions

Category:Conjugate prior distributions

Category:Exponential family distributions