Wishart distribution

Wishart
Notation	$X ~ W p (V, n)$
Parameters	$n > p - 1$ degrees of freedom (real) $V > 0$ scale matrix ( $p \times p$ pos. def)
Support	$X (p \times p)$ positive definite matrix
PDF	${\frac {\|\mathbf {X} \|^{(n-p-1)/2}e^{-\operatorname {tr} (\mathbf {V} ^{-1}\mathbf {X} )/2}}{2^{\frac {np}{2}}\|{\mathbf {V} }\|^{n/2}\Gamma _{p}({\frac {n}{2}})}}$ $Γ p$ is the multivariate gamma function $tr$ is the trace function
Mean	$\operatorname {E} [X]=nV$
Mode	$(n - p - 1) V$ for $n \geq p + 1$
Variance	$\operatorname{Var}(\mathbf{X}_{ij}) = n \left (v_{ij}^2+v_{ii}v_{jj} \right )$
Entropy	see below
CF	$\Theta \mapsto \left\|{\mathbf I} - 2i\,{\mathbf\Theta}{\mathbf V}\right\|^{-\frac{n}{2}}$

In statistics, the Wishart distribution is a generalization to multiple dimensions of the chi-squared distribution, or, in the case of non-integer degrees of freedom, of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928.^[1]

It is a family of probability distributions defined over symmetric, nonnegative-definite matrix-valued random variables (“random matrices”). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. In Bayesian statistics, the Wishart distribution is the conjugate prior of the inverse covariance-matrix of a multivariate-normal random-vector.

Definition

Suppose $X$ is an $n \times p$ matrix, each row of which is independently drawn from a $p$ -variate normal distribution with zero mean:

X_{(i)}{=}(x_i^1,\dots,x_i^p)\sim N_p(0,V).

Then the Wishart distribution is the probability distribution of the $p \times p$ random matrix $S = X T X$ known as the scatter matrix. One indicates that $S$ has that probability distribution by writing

S\sim W_p(V,n).

The positive integer $n$ is the number of degrees of freedom. Sometimes this is written $W (V, p, n)$ . For $n \geq p$ the matrix $S$ is invertible with probability $1$ if $V$ is invertible.

If $p = V = 1$ then this distribution is a chi-squared distribution with $n$ degrees of freedom.

Occurrence

The Wishart distribution arises as the distribution of the sample covariance matrix for a sample from a multivariate normal distribution. It occurs frequently in likelihood-ratio tests in multivariate statistical analysis. It also arises in the spectral theory of random matrices and in multidimensional Bayesian analysis.^[2] It is also encountered in wireless communications, while analyzing the performance of Rayleigh fading MIMO wireless channels .^[3]

Probability density function

The Wishart distribution can be characterized by its probability density function as follows:

Let $X$ be a $p \times p$ symmetric matrix of random variables that is positive definite. Let $V$ be a (fixed) positive definite matrix of size $p \times p$ .

Then, if $n \geq p$ , $X$ has a Wishart distribution with $n$ degrees of freedom if it has a probability density function given by

{\displaystyle {\frac {1}{2^{np/2}\left|{\mathbf {V} }\right|^{n/2}\Gamma _{p}\left|{\frac {n}{2}}\right|}}{\left|\mathbf {X} \right|}^{(n-p-1)/2}e^{-(1/2)\operatorname {tr} ({\mathbf {V} }^{-1}\mathbf {X} )}}

where $\left|{{\mathbf X}}\right|$ denotes determinant and $Γ p (\cdot)$ is the multivariate gamma function defined as

\Gamma _{p}\left({\frac {n}{2}}\right)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma \left({\frac {n}{2}}+{\frac {1-j}{2}}\right).

In fact the above definition can be extended to any real $n > p - 1$ . If $n \leq p - 1$ , then the Wishart no longer has a density—instead it represents a singular distribution that takes values in a lower-dimension subspace of the space of $p \times p$ matrices.^[4]

Use in Bayesian statistics

In Bayesian statistics, in the context of the multivariate normal distribution, the Wishart distribution is the conjugate prior to the precision matrix $Ω = Σ -1$ , where $Σ$ is the covariance matrix.

Choice of parameters

The least informative, proper Wishart prior is obtained by setting $n = p$ .

The prior mean of $W p (V, n)$ is $n V$ , suggesting that a reasonable choice for $V -1$ would be $n Σ 0$ , where $Σ 0$ is some prior guess for the covariance matrix.

Properties

Log-expectation

Note the following formula:^[5]

\operatorname {E} [\ln |\mathbf {X} |]=\psi _{p}\left({\frac {n}{2}}\right)+p\ln(2)+\ln |\mathbf {V} |

where $\psi _{p}$ is the multivariate digamma function (the derivative of the log of the multivariate gamma function).

This plays a role in variational Bayes derivations for Bayes networks involving the Wishart distribution.

Log-variance

The following variance computation could be of help in Bayesian statistics:

\operatorname {Var} [\ln |\mathbf {X} |]=\sum _{i=1}^{p}\psi _{1}\left({\frac {n+1-i}{2}}\right)

where $\psi _{{1}}$ is the trigamma function. This comes up when computing the Fisher information of the Wishart random variable.

Entropy

The information entropy of the distribution has the following formula:^[5]

\operatorname{H}[\mathbf{X}] = -\ln \left (B(\mathbf{V},n) \right ) -\frac{n-p-1}{2} \operatorname{E}[\ln|\mathbf{X}|] + \frac{np}{2}

where $B (V, n)$ is the normalizing constant of the distribution:

B(\mathbf {V} ,n)={\frac {1}{\left|\mathbf {V} \right|^{n/2}}}2^{np/2}\Gamma _{p}\left({\frac {n}{2}}\right).

This can be expanded as follows:

{\begin{aligned}\operatorname {H} [\mathbf {X} ]&={\frac {n}{2}}\ln |\mathbf {V} |+{\frac {np}{2}}\ln 2+\ln \Gamma _{p}\left({\frac {n}{2}}\right)-{\frac {n-p-1}{2}}\operatorname {E} [\ln |\mathbf {X} |]+{\frac {np}{2}}\\[8pt]&={\frac {n}{2}}\ln |\mathbf {V} |+{\frac {np}{2}}\ln 2+\ln \Gamma _{p}\left({\frac {n}{2}}\right)-{\frac {n-p-1}{2}}\left(\psi _{p}\left({\frac {n}{2}}\right)+p\ln 2+\ln |\mathbf {V} |\right)+{\frac {np}{2}}\\[8pt]&={\frac {n}{2}}\ln |\mathbf {V} |+{\frac {np}{2}}\ln 2+\ln \Gamma _{p}\left({\frac {n}{2}}\right)-{\frac {n-p-1}{2}}\psi _{p}\left({\frac {n}{2}}\right)-{\frac {n-p-1}{2}}\left(p\ln 2+\ln |\mathbf {V} |\right)+{\frac {np}{2}}\\[8pt]&={\frac {p+1}{2}}\ln |\mathbf {V} |+{\frac {1}{2}}p(p+1)\ln 2+\ln \Gamma _{p}\left({\frac {n}{2}}\right)-{\frac {n-p-1}{2}}\psi _{p}\left({\frac {n}{2}}\right)+{\frac {np}{2}}\end{aligned}}

Cross-entropy

The cross entropy of two Wishart distributions $p_{0}$ with parameters $n_{0},V_{0}$ and $p_{1}$ with parameters $n_{1},V_{1}$ is

{\begin{aligned}H(p_{0},p_{1})&=\operatorname {E} _{p_{0}}[-\log p_{1}]\\[8pt]&=\operatorname {E} _{p_{0}}\left[-\log {\frac {|\mathbf {X} |^{(n_{1}-p-1)/2}e^{-\operatorname {tr} (\mathbf {V} _{1}^{-1}\mathbf {X} )/2}}{2^{n_{1}p/2}|\mathbf {V} _{1}|^{n_{1}/2}\Gamma _{p}({\tfrac {n_{1}}{2}})}}\right]\\[8pt]&={\tfrac {n_{1}p}{2}}\log 2+{\tfrac {n_{1}}{2}}\log |\mathbf {V} _{1}|+\log \Gamma _{p}({\tfrac {n_{1}}{2}})-{\tfrac {n_{1}-p-1}{2}}\operatorname {E} _{p_{0}}[\log |\mathbf {X} |]+{\tfrac {1}{2}}\operatorname {E} _{p_{0}}[\operatorname {tr} (\mathbf {V} _{1}^{-1}\mathbf {X} )]\\[8pt]&={\tfrac {n_{1}p}{2}}\log 2+{\tfrac {n_{1}}{2}}\log |\mathbf {V} _{1}|+\log \Gamma _{p}({\tfrac {n_{1}}{2}})-{\tfrac {n_{1}-p-1}{2}}\left(\psi _{p}({\tfrac {n_{0}}{2}})+p\log 2+\log |\mathbf {V} _{0}|\right)+{\tfrac {1}{2}}\operatorname {tr} (\mathbf {V} _{1}^{-1}n_{0}\mathbf {V} _{0})\\[8pt]&=-{\tfrac {n_{1}}{2}}\log |\mathbf {V} _{1}^{-1}\mathbf {V} _{0}|+{\tfrac {p+1}{2}}\log |\mathbf {V} _{0}|+{\tfrac {n_{0}}{2}}\operatorname {tr} (\mathbf {V} _{1}^{-1}\mathbf {V} _{0})+\log \Gamma _{p}({\tfrac {n_{1}}{2}})-{\tfrac {n_{1}-p-1}{2}}\psi _{p}({\tfrac {n_{0}}{2}})+{\tfrac {p(p+1)}{2}}\log 2\end{aligned}}

Note that when $p_{0}=p_{1}$ we recover the entropy.

KL-divergence

The Kullback–Leibler divergence of $p_{1}$ from $p_{0}$ is

{\begin{aligned}D_{KL}(p_{0}\|p_{1})&=H(p_{0},p_{1})-H(p_{0})\\[6pt]&=-{\frac {n_{1}}{2}}\log |\mathbf {V} _{1}^{-1}\mathbf {V} _{0}|+{\frac {n_{0}}{2}}(\operatorname {tr} (\mathbf {V} _{1}^{-1}\mathbf {V} _{0})-p)+\log {\frac {\Gamma _{p}\left({\frac {n_{1}}{2}}\right)}{\Gamma _{p}\left({\frac {n_{0}}{2}}\right)}}+{\tfrac {n_{0}-n_{1}}{2}}\psi _{p}\left({\frac {n_{0}}{2}}\right)\end{aligned}}

Characteristic function

The characteristic function of the Wishart distribution is

\Theta \mapsto \left|{\mathbf {I} }-2i\,{\mathbf {\Theta } }{\mathbf {V} }\right|^{-n/2}.

In other words,

\Theta \mapsto \operatorname {E} \left[\exp \left(i\operatorname {tr} (\mathbf {X} {\mathbf {\Theta } })\right)\right]=\left|{\mathbf {I} }-2i{\mathbf {\Theta } }{\mathbf {V} }\right|^{-n/2}

where $E[\cdot]$ denotes expectation. (Here $Θ$ and $I$ are matrices the same size as $V$ ( $I$ is the identity matrix); and $i$ is the square root of −1).^[6]

Theorem

If a $p \times p$ random matrix $X$ has a Wishart distribution with $m$ degrees of freedom and variance matrix $V$ — write $\mathbf{X}\sim\mathcal{W}_p({\mathbf V},m)$ — and $C$ is a $q \times p$ matrix of rank $q$ , then ^[7]

\mathbf{C}\mathbf{X}{\mathbf C}^T \sim \mathcal{W}_q\left({\mathbf C}{\mathbf V}{\mathbf C}^T,m\right).

Corollary 1

If $z$ is a nonzero $p \times 1$ constant vector, then:^[7]

{\mathbf z}^T\mathbf{X}{\mathbf z}\sim\sigma_z^2\chi_m^2.

In this case, $\chi_m^2$ is the chi-squared distribution and $\sigma_z^2={\mathbf z}^T{\mathbf V}{\mathbf z}$ (note that $\sigma_z^2$ is a constant; it is positive because $V$ is positive definite).

Corollary 2

Consider the case where $z T = (0, ..., 0, 1, 0, ..., 0)$ (that is, the $j$ -th element is one and all others zero). Then corollary 1 above shows that

w_{jj}\sim\sigma_{jj}\chi^2_m

gives the marginal distribution of each of the elements on the matrix's diagonal.

George Seber points out that the Wishart distribution is not called the “multivariate chi-squared distribution” because the marginal distribution of the off-diagonal elements is not chi-squared. Seber prefers to reserve the term multivariate for the case when all univariate marginals belong to the same family.^[8]

Estimator of the multivariate normal distribution

The Wishart distribution is the sampling distribution of the maximum-likelihood estimator (MLE) of the covariance matrix of a multivariate normal distribution.^[9] A derivation of the MLE uses the spectral theorem.

Bartlett decomposition

The Bartlett decomposition of a matrix $X$ from a $p$ -variate Wishart distribution with scale matrix $V$ and $n$ degrees of freedom is the factorization:

\mathbf{X} = {\textbf L}{\textbf A}{\textbf A}^T{\textbf L}^T,

where $L$ is the Cholesky factor of $V$ , and:

\mathbf A = \begin{pmatrix} c_1 & 0 & 0 & \cdots & 0\\ n_{21} & c_2 &0 & \cdots& 0 \\ n_{31} & n_{32} & c_3 & \cdots & 0\\ \vdots & \vdots & \vdots &\ddots & \vdots \\ n_{p1} & n_{p2} & n_{p3} &\cdots & c_p \end{pmatrix}

where $c_i^2 \sim \chi^2_{n-i+1}$ and $n ij ~ N (0, 1)$ independently.^[10] This provides a useful method for obtaining random samples from a Wishart distribution.^[11]

Marginal distribution of matrix elements

Let $V$ be a $2 \times 2$ variance matrix characterized by correlation coefficient $-1 < ρ < 1$ and $L$ its lower Cholesky factor:

\mathbf{V} = \begin{pmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{pmatrix}, \qquad \mathbf{L} = \begin{pmatrix} \sigma_1 & 0 \\ \rho \sigma_2 & \sqrt{1-\rho^2} \sigma_2 \end{pmatrix}

Multiplying through the Bartlett decomposition above, we find that a random sample from the $2 \times 2$ Wishart distribution is

\mathbf{X} = \begin{pmatrix} \sigma_1^2 c_1^2 & \sigma_1 \sigma_2 \left (\rho c_1^2 + \sqrt{1-\rho^2} c_1 n_{21} \right ) \\ \sigma_1 \sigma_2 \left (\rho c_1^2 + \sqrt{1-\rho^2} c_1 n_{21} \right ) & \sigma_2^2 \left(\left (1-\rho^2 \right ) c_2^2 + \left (\sqrt{1-\rho^2} n_{21} + \rho c_1 \right )^2 \right) \end{pmatrix}

The diagonal elements, most evidently in the first element, follow the $χ 2$ distribution with $n$ degrees of freedom (scaled by $σ 2$ ) as expected. The off-diagonal element is less familiar but can be identified as a normal variance-mean mixture where the mixing density is a $χ 2$ distribution. The corresponding marginal probability density for the off-diagonal element is therefore the variance-gamma distribution

f(x_{12}) = \frac{\left | x_{12} \right |^{\frac{n-1}{2}}}{\Gamma\left(\frac{n}{2}\right) \sqrt{2^{n-1} \pi \left (1-\rho^2 \right ) \left (\sigma_1 \sigma_2 \right )^{n+1}}} \cdot K_{\frac{n-1}{2}} \left(\frac{\left |x_{12} \right |}{\sigma_1 \sigma_2 \left (1-\rho^2 \right )}\right) \exp{\left(\frac{\rho x_{12}}{\sigma_1 \sigma_2 (1-\rho^2)}\right)}

where $K ν (z)$ is the modified Bessel function of the second kind.^[12] Similar results may be found for higher dimensions, but the interdependence of the off-diagonal correlations becomes increasingly complicated. It is also possible to write down the moment-generating function even in the noncentral case (essentially the nth power of Craig (1936)^[13] equation 10) although the probability density becomes an infinite sum of Bessel functions.

The possible range of the shape parameter

It can be shown ^[14] that the Wishart distribution can be defined if and only if the shape parameter $n$ belongs to the set

\Lambda _{p}:=\{0,\ldots ,p-1\}\cup \left(p-1,\infty \right).

This set is named after Gindikin, who introduced it^[15] in the seventies in the context of gamma distributions on homogeneous cones. However, for the new parameters in the discrete spectrum of the Gindikin ensemble, namely,

\Lambda _{p}^{*}:=\{0,\ldots ,p-1\},

the corresponding Wishart distribution has no Lebesgue density.

Relationships to other distributions

The Wishart distribution is related to the Inverse-Wishart distribution, denoted by $W_p^{-1}$ , as follows: If $X ~ W p (V, n)$ and if we do the change of variables $C = X -1$ , then $\mathbf{C}\sim W_p^{-1}(\mathbf{V}^{-1},n)$ . This relationship may be derived by noting that the absolute value of the Jacobian determinant of this change of variables is $| C | p +1$ , see for example equation (15.15) in.^[16]
In Bayesian statistics, the Wishart distribution is a conjugate prior for the precision parameter of the multivariate normal distribution, when the mean parameter is known.^[17]
A generalization is the multivariate gamma distribution.
A different type of generalization is the normal-Wishart distribution, essentially the product of a multivariate normal distribution with a Wishart distribution.

References

↑ Wishart, J. (1928). "The generalised product moment distribution in samples from a normal multivariate population". Biometrika. 20A (1–2): 32–52. doi:10.1093/biomet/20A.1-2.32. JFM 54.0565.02. JSTOR 2331939.
↑ Gelman, Andrew (2003). Bayesian Data Analysis (2nd ed.). Boca Raton, Fla.: Chapman & Hall. p. 582. ISBN 158488388X. Retrieved 3 June 2015.
↑ Zanella, A.; Chiani, M.; Win, M.Z. (April 2009). "On the marginal distribution of the eigenvalues of wishart matrices". IEEE Transactions on Communications. 57 (4): 1050–1060. doi:10.1109/TCOMM.2009.04.070143.
↑ Uhlig, H. (1994). "On Singular Wishart and Singular Multivariate Beta Distributions". The Annals of Statistics. 22: 395–405. doi:10.1214/aos/1176325375.
1 2 C.M. Bishop, Pattern Recognition and Machine Learning, Springer 2006, p. 693.
↑ Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis (3rd ed.). Hoboken, N. J.: Wiley Interscience. p. 259. ISBN 0-471-36091-0.
1 2 Rao, C. R. (1965). Linear Statistical Inference and its Applications. Wiley. p. 535.
↑ Seber, George A. F. (2004). Multivariate Observations. Wiley. ISBN 978-0471691211.
↑ Chatfield, C.; Collins, A. J. (1980). Introduction to Multivariate Analysis. London: Chapman and Hall. pp. 103–108. ISBN 0-412-16030-7.
↑ Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis (3rd ed.). Hoboken, N. J.: Wiley Interscience. p. 257. ISBN 0-471-36091-0.
↑ Smith, W. B.; Hocking, R. R. (1972). "Algorithm AS 53: Wishart Variate Generator". Journal of the Royal Statistical Society, Series C. 21 (3): 341–345. JSTOR 2346290.
↑ Pearson, Karl; Jeffery, G. B.; Elderton, Ethel M. (December 1929). "On the Distribution of the First Product Moment-Coefficient, in Samples Drawn from an Indefinitely Large Normal Population". Biometrika. Biometrika Trust. 21: 164–201. doi:10.2307/2332556. JSTOR 2332556.
↑ Craig, Cecil C. (1936). "On the Frequency Function of xy". Ann. Math. Statist. 7: 1–15. doi:10.1214/aoms/1177732541.
↑ Peddada and Richards, Shyamal Das; Richards, Donald St. P. (1991). "Proof of a Conjecture of M. L. Eaton on the Characteristic Function of the Wishart Distribution,". Annals of Probability. 19 (2): 868–874. doi:10.1214/aop/1176990455.
↑ Gindikin, S.G. (1975). "Invariant generalized functions in homogeneous domains,". Funct. Anal. Appl. 9 (1): 50–52. doi:10.1007/BF01078179.
↑ Dwyer, Paul S. (1967). "Some Applications of Matrix Derivatives in Multivariate Analysis". J. Amer. Statist. Assoc. 62 (318): 607–625. JSTOR 2283988.
↑ Bishop, C. M. (2006). Pattern Recognition and Machine Learning. Springer.

External links

A C++ library for random matrix generator

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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