Inverse-Wishart distribution

Inverse-Wishart
Notation	$\mathcal{W}^{-1}({\mathbf\Psi},\nu)$
Parameters	$\nu > p-1$ degrees of freedom (real) $\mathbf{\Psi} > 0$ scale matrix (pos. def)
Support	$\mathbf {X}$ is positive definite
PDF	$\frac{\left\|{\mathbf\Psi}\right\|^{\frac{\nu}{2}}}{2^{\frac{\nu p}{2}}\Gamma_p(\frac{\nu}{2})} \left\|\mathbf{X}\right\|^{-\frac{\nu+p+1}{2}}e^{-\frac{1}{2}\operatorname{tr}({\mathbf\Psi}\mathbf{X}^{-1})}$ $\Gamma_p$ is the multivariate gamma function $\mathrm{tr}$ 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

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 Inverse-Wishart distribution.^[2]

Density

The probability density function of the inverse Wishart is:

\frac{\left|{\mathbf\Psi}\right|^{\frac{\nu}{2}}}{2^{\frac{\nu p}{2}}\Gamma_p(\frac{\nu}{2})} \left|\mathbf{X}\right|^{-\frac{\nu+p+1}{2}}e^{-\frac{1}{2}\operatorname{tr}({\mathbf\Psi}\mathbf{X}^{-1})}

where $\mathbf {X}$ and ${\mathbf\Psi}$ are $p\times p$ positive definite matrices, and Γ_p(·) is the multivariate gamma function.

Theorems

Distribution of the inverse of a Wishart-distributed matrix

If ${\mathbf A}\sim \mathcal{W}({\mathbf\Sigma},\nu)$ and ${\mathbf\Sigma}$ is of size $p\times p$ , then $\mathbf{X}={\mathbf A}^{-1}$ has an inverse Wishart distribution $\mathbf{X}\sim \mathcal{W}^{-1}({\mathbf\Sigma}^{-1},\nu)$ .^[3]

Marginal and conditional distributions from an inverse Wishart-distributed matrix

Suppose ${\mathbf A}\sim \mathcal{W}^{-1}({\mathbf\Psi},\nu)$ has an inverse Wishart distribution. Partition the matrices ${\mathbf A}$ and ${\mathbf\Psi}$ 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 ${\mathbf A_{ij}}$ and ${\mathbf \Psi_{ij}}$ are $p_{i}\times p_{j}$ matrices, then we have

i) ${\mathbf A_{11} }$ is independent of ${\mathbf A}_{11}^{-1}{\mathbf A}_{12}$ and ${\mathbf A}_{22\cdot 1}$ , where ${\mathbf A_{22\cdot 1}} = {\mathbf A}_{22} - {\mathbf A}_{21}{\mathbf A}_{11}^{-1}{\mathbf A}_{12}$ is the Schur complement of ${\mathbf A_{11} }$ in ${\mathbf A}$ ;

ii) ${\mathbf A_{11} } \sim \mathcal{W}^{-1}({\mathbf \Psi_{11} }, \nu-p_{2})$ ;

iii) ${\mathbf A}_{11}^{-1} {\mathbf A}_{12}| {\mathbf A}_{22\cdot 1} \sim MN_{p_{1}\times p_{2}} ( {\mathbf \Psi}_{11}^{-1} {\mathbf \Psi}_{12}, {\mathbf A}_{22\cdot 1} \otimes {\mathbf \Psi}_{11}^{-1})$ , where $MN_{p\times q}(\cdot,\cdot)$ is a matrix normal distribution;

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

Conjugate distribution

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

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 $\mathbf{\Sigma}$ .

$P(\mathbf{X}|\mathbf{\Psi},\nu) = \int P(\mathbf{X}|\mathbf{\Sigma})P(\mathbf{\Sigma}|\mathbf{\Psi},\nu) d\mathbf{\Sigma} = \frac{|\mathbf{\Psi}|^{\frac{\nu}{2}}\Gamma_p\left(\frac{\nu+n}{2}\right)}{\pi^{\frac{np}{2}}|\mathbf{\Psi}+\mathbf{A}|^{\frac{\nu+n}{2}}\Gamma_p(\frac{\nu}{2})}$

(this is useful because the variance matrix $\mathbf{\Sigma}$ is not known in practice, but because ${\mathbf\Psi}$ is known a priori, and ${\mathbf A}$ 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.^[4]

Moments

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.

The mean:^[3]^:85

E(\mathbf X) = \frac{\mathbf\Psi}{\nu-p-1}.

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

\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 $\mathbf {X}$ are given by:

\operatorname{Cov}(x_{ij},x_{kl}) = \frac{2\psi_{ij}\psi_{kl} + (\nu-p-1) (\psi_{ik}\psi_{jl} + \psi_{il}\psi_{kj})}{(\nu-p)(\nu-p-1)^2(\nu-p-3)}

Related distributions

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

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

i.e., the inverse-gamma distribution, where $\Gamma_1(\cdot)$ is the ordinary Gamma function.

A generalization is the inverse multivariate gamma distribution.

Another generalization has been termed the generalized inverse Wishart distribution, $\mathcal{GW}^{-1}$ . A $p \times p$ positive definite matrix $\mathbf {X}$ is said to be distributed as $\mathcal{GW}^{-1}(\mathbf{\Psi},\nu,\mathbf{S})$ if $\mathbf{Y} = \mathbf{X}^{1/2}\mathbf{S}^{-1}\mathbf{X}^{1/2}$ is distributed as $\mathcal{W}^{-1}(\mathbf{\Psi},\nu)$ . Here $\mathbf{X}^{1/2}$ denotes the symmetric matrix square root of $\mathbf {X}$ , the parameters $\mathbf{\Psi},\mathbf{S}$ are $p \times p$ positive definite matrices, and the parameter $\nu$ is a positive scalar larger than $2p$ . Note that when $\mathbf {S}$ is equal to an identity matrix, $\mathcal{GW}^{-1}(\mathbf{\Psi},\nu,\mathbf{S}) = \mathcal{W}^{-1}(\mathbf{\Psi},\nu)$ . This generalized inverse Wishart distribution has been applied to estimating the distributions of multivariate autoregressive processes.^[5]

A different type of generalization is the normal-inverse-Wishart distribution, essentially the product of a multivariate normal distribution with an inverse Wishart distribution.

References

↑ A. O'Hagan, and J. J. Forster (2004). Kendall's Advanced Theory of Statistics: Bayesian Inference. 2B (2 ed.). Arnold. ISBN 0-340-80752-0.
↑ 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 Kanti V. Mardia, J. T. Kent and J. M. Bibby (1979). Multivariate Analysis. Academic Press. ISBN 0-12-471250-9.
↑ Shahrokh Esfahani, Mohammad; Dougherty, Edward (2014). "Incorporation of Biological Pathway Knowledge in the Construction of Priors for Optimal Bayesian Classification". IEEE Transactions on Bioinformatics and Computational Biology. 11 (1): 202–218. doi:10.1109/tcbb.2013.143.
↑ Triantafyllopoulos, K. (2011). "Real-time covariance estimation for the local level model". Journal of Time Series Analysis. 32 (2): 93–107. doi:10.1111/j.1467-9892.2010.00686.x.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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