Wishart
Notation	X ~Wp(V,n)
Parameters	ndegrees of freedom (real) V > 0scale matrix (p ×ppos. def)
Support	X (p ×p)positive definite matrix
PDF	${\displaystyle f_{\mathbf{X}}(\mathbf{X})=\frac{|\mathbf{X}{|}^{(n-p-1)/2}{e}^{-\mathrm{tr}({\mathbf{V}}^{-1}\mathbf{X})/2}}{2^{(np)/2}|\mathbf{V}{|}^{n/2}{\mathrm{\Gamma }}_p(\frac{n}{2})}}$ Γp is themultivariate gamma function tr is thetrace function
Mean	${\displaystyle \mathrm{E}[\mathbf{X}]=n\mathbf{V}}$
Mode	(n −p − 1)V forn ≥p + 1
Variance	${\displaystyle \mathrm{Var}({\mathbf{X}}_{ij})=n\left(v_{ij}^2+v_{ii}{v}_{jj}\right)}$
Entropy	see below
CF	${\displaystyle \mathrm{\Theta }\mapsto {\left|\mathbf{I}-2i\mathbf{\Theta }\mathbf{V}\right|}^{-\frac{n}{2}}}$

Instatistics, theWishart distribution is a generalization of thegamma distribution to multiple dimensions. It is named in honor ofJohn Wishart, who first formulated the distribution in 1928.^[1] Other names includeWishart ensemble (inrandom matrix theory, probability distributions over matrices are usually called "ensembles"), orWishart–Laguerre ensemble (since its eigenvalue distribution involveLaguerre polynomials), or LOE, LUE, LSE (in analogy withGOE, GUE, GSE).^[2]

It is a family ofprobability distributions defined over symmetric,positive-definiterandom matrices (i.e.matrix-valuedrandom variables). These distributions are of great importance in theestimation of covariance matrices inmultivariate statistics. InBayesian statistics, the Wishart distribution is theconjugate prior of theinversecovariance-matrix of amultivariate-normal random vector.^[3]

SupposeG is ap ×n matrix, each column of which isindependently drawn from ap-variate normal distribution with zero mean:

${\displaystyle G=(g_1,\dots ,g_n)\sim {\mathcal{N}}_p(0,V).}$

It means :${\displaystyle g_i=(g_{i,1},\dots ,g_{i,p}{)}^T\stackrel{iid}{\sim }{\mathcal{N}}_p(0,V)\mathrm{\forall }i\in \{1,\dots ,n\}}$

Then the Wishart distribution is theprobability distribution of thep ×p random matrix^[4]

${\displaystyle S=G{G}^T=\sum _{i=1}^n{g}_i{g}_i^T}$

known as thescatter matrix. One indicates thatS has that probability distribution by writing

${\displaystyle S\sim W_p(V,n).}$

The positive integern is the number ofdegrees of freedom. Sometimes this is writtenW(V,p,n). Forn ≥p the matrixS is invertible with probability1 ifV is invertible.

Ifp =V = 1 then this distribution is achi-squared distribution withn degrees of freedom.

The Wishart distribution arises as the distribution of the sample covariance matrix for a sample from amultivariate normal distribution. It occurs frequently inlikelihood-ratio tests in multivariate statistical analysis. It also arises in the spectral theory ofrandom matrices^{[citation needed]} and in multidimensional Bayesian analysis.^[5] It is also encountered in wireless communications, while analyzing the performance ofRayleigh fadingMIMO wireless channels.^[6]

The Wishart distribution can becharacterized by itsprobability density function as follows:

LetX be ap ×p symmetric matrix of random variables that ispositive semi-definite. LetV be a (fixed) symmetric positive definite matrix of sizep ×p.

Then, ifn ≥p,X has a Wishart distribution withn degrees of freedom if it has theprobability density function

${\displaystyle f_{\mathbf{X}}(\mathbf{X})=\frac{1}{2^{np/2}{\left|\mathbf{V}\right|}^{n/2}{\mathrm{\Gamma }}_p\left(\frac{n}{2}\right)}{\left|\mathbf{X}\right|}^{(n-p-1)/2}{e}^{-\frac{1}{2}\mathrm{tr}({\mathbf{V}}^{-1}\mathbf{X})}}$

where${\displaystyle \left|\mathbf{X}\right|}$ is thedeterminant of${\displaystyle \mathbf{X}}$ andΓ_p is themultivariate gamma function defined as

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

The density above is not the joint density of all the${\displaystyle p^2}$ elements of the random matrixX (such${\displaystyle p^2}$-dimensional density does not exist because of the symmetry constrains${\displaystyle X_{ij}=X_{ji}}$), it is rather the joint density of${\displaystyle p(p+1)/2}$ elements${\displaystyle X_{ij}}$ for${\displaystyle i\le j}$ (,^[1] page 38). Also, the density formula above applies only to positive definite matrices${\displaystyle \mathbf{x};}$ for other matrices the density is equal to zero.

In fact the above definition can be extended to any realn >p − 1. Ifn ≤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 ofp ×p matrices.^[8]

The joint-eigenvalue density for the eigenvalues${\displaystyle {\lambda }_1,\dots ,{\lambda }_p\ge 0}$ of a random matrix${\displaystyle \mathbf{X}\sim W_p(\mathbf{I},n)}$ is,^[9][10]

where${\displaystyle c_{n,p}}$ is a constant. The spectral density can be marginalized to yield the density of a single eigenvalue, by evaluating aSelberg integral.

The spectral density can be integrated to give the probability that all eigenvalues of a Wishart random matrix lie within an interval.^[11]

InBayesian statistics, in the context of themultivariate normal distribution, the Wishart distribution is the conjugate prior to theprecision matrixΩ =Σ⁻¹, whereΣ is the covariance matrix.^{[12]: 135 [13]} The use of Normal-Wishart conjugate priors (for mean and precision) is particularly common forvector autoregression models.^[14]

The least informative, proper Wishart prior is obtained by settingn =p.^{[citation needed]}

A common choice forV leverages the fact that the mean ofX ~W_p(V,n) isnV. ThenV is chosen so thatnV equals an initial guess forX. For instance, when estimating a precision matrixΣ⁻¹ ~W_p(V,n) a reasonable choice forV would ben⁻¹Σ₀⁻¹, whereΣ₀ is some prior estimate for the covariance matrixΣ.

The following formula plays a role invariational Bayes derivations forBayes networksinvolving the Wishart distribution. From equation (2.63),^[15]

${\displaystyle \mathrm{E}[\ln \left|\mathbf{X}\right|]={\psi }_p\left(\frac{n}{2}\right)+p\ln (2)+\ln |\mathbf{V}|}$

where${\displaystyle {\psi }_p}$ is the multivariate digamma function (the derivative of the log of themultivariate gamma function).

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

${\displaystyle \mathrm{Var}\left[\ln \left|\mathbf{X}\right|\right]=\sum _{i=1}^p{\psi }_1\left(\frac{n+1-i}{2}\right)}$

where${\displaystyle {\psi }_1}$ is the trigamma function. This comes up when computing the Fisher information of the Wishart random variable.

Theinformation entropy of the distribution has the following formula:^[12]: 693

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

whereB(V,n) is thenormalizing constant of the distribution:

${\displaystyle B(\mathbf{V},n)=\frac{1}{{\left|\mathbf{V}\right|}^{n/2}{2}^{np/2}{\mathrm{\Gamma }}_p\left(\frac{n}{2}\right)}.}$

This can be expanded as follows:

Thecross-entropy of two Wishart distributions${\displaystyle p_0}$ with parameters${\displaystyle n_0,V_0}$ and${\displaystyle p_1}$ with parameters${\displaystyle n_1,V_1}$ is

Note that when${\displaystyle p_0=p_1}$ and${\displaystyle n_0=n_1}$ we recover the entropy.

TheKullback–Leibler divergence of${\displaystyle p_1}$ from${\displaystyle p_0}$ is

Thecharacteristic function of the Wishart distribution is

${\displaystyle \mathrm{\Theta }\mapsto \mathrm{E}\left[\exp \left(i\mathrm{tr}\left(\mathbf{X}\mathbf{\Theta }\right)\right)\right]={\left|1-2i\mathbf{\Theta }\mathbf{V}\right|}^{-n/2}}$

whereE[⋅] denotes expectation. (HereΘ is any matrix with the same dimensions asV,1 indicates the identity matrix, andi is a square root of −1).^[10] Properly interpreting this formula requires a little care, because noninteger complex powers aremultivalued; whenn is noninteger, the correct branch must be determined viaanalytic continuation.^[16]

If ap ×p random matrixX has a Wishart distribution withm degrees of freedom and variance matrixV — write${\displaystyle \mathbf{X}\sim {\mathcal{W}}_p(\mathbf{V},m)}$ — andC is aq ×p matrix ofrankq, then^[17]

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

Ifz is a nonzerop × 1 constant vector, then:^[17]

${\displaystyle {\sigma }_z^{-2}{\mathbf{z}}^T\mathbf{X}\mathbf{z}\sim {\chi }_m^2.}$

In this case,${\displaystyle {\chi }_m^2}$ is thechi-squared distribution and${\displaystyle {\sigma }_z^2={\mathbf{z}}^T\mathbf{V}\mathbf{z}}$ (note that${\displaystyle {\sigma }_z^2}$ is a constant; it is positive becauseV is positive definite).

Consider the case wherez^T = (0, ..., 0, 1, 0, ..., 0) (that is, thej-th element is one and all others zero). Then corollary 1 above shows that

${\displaystyle {\sigma }_{jj}^{-1}{w}_{jj}\sim {\chi }_m^2}$

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 theoff-diagonal elements is not chi-squared. Seber prefers to reserve the termmultivariate for the case when all univariate marginals belong to the same family.^[18]

The Wishart distribution is thesampling distribution of themaximum-likelihood estimator (MLE) of thecovariance matrix of amultivariate normal distribution.^[19] Aderivation of the MLE uses thespectral theorem.

TheBartlett decomposition of a matrixX from ap-variate Wishart distribution with scale matrixV andn degrees of freedom is the factorization:

${\displaystyle \mathbf{X}=\mathbf{\text{L}}\mathbf{\text{A}}{\mathbf{\text{A}}}^T{\mathbf{\text{L}}}^T,}$

whereL is theCholesky factor ofV, and:

where${\displaystyle c_i^2\sim {\chi }_{n-i+1}^2}$ andn_ij ~N(0, 1) independently.^[20] This provides a useful method for obtaining random samples from a Wishart distribution.^[21]

LetV be a2 × 2 variance matrix characterized bycorrelation coefficient−1 <ρ < 1 andL its lower Cholesky factor:

Multiplying through the Bartlett decomposition above, we find that a random sample from the2 × 2 Wishart distribution is

The diagonal elements, most evidently in the first element, follow theχ² distribution withn degrees of freedom (scaled byσ²) as expected. The off-diagonal element is less familiar but can be identified as anormal variance-mean mixture where the mixing density is aχ² distribution. The corresponding marginal probability density for the off-diagonal element is therefore thevariance-gamma distribution

${\displaystyle f(x_{12})=\frac{{\left|x_{12}\right|}^{\frac{n-1}{2}}}{\mathrm{\Gamma }\left(\frac{n}{2}\right)\sqrt{2^{n-1}\pi \left(1-{\rho }^2\right){\left({\sigma }_1{\sigma }_2\right)}^{n+1}}}\cdot K_{\frac{n-1}{2}}\left(\frac{\left|x_{12}\right|}{{\sigma }_1{\sigma }_2\left(1-{\rho }^2\right)}\right)\exp \left(\frac{\rho x_{12}}{{\sigma }_1{\sigma }_2(1-{\rho }^2)}\right)}$

whereK_ν(z) is themodified Bessel function of the second kind.^[22] Similar results may be found for higher dimensions. In general, if${\displaystyle X}$ follows a Wishart distribution with parameters,${\displaystyle \mathrm{\Sigma },n}$, then for${\displaystyle i\ne j}$, the off-diagonal elements

${\displaystyle X_{ij}\sim \text{VG}(n,{\mathrm{\Sigma }}_{ij},({\mathrm{\Sigma }}_{ii}{\mathrm{\Sigma }}_{jj}-{\mathrm{\Sigma }}_{ij}^2{)}^{1/2},0)}$.^[23]

It is also possible to write down themoment-generating function even in thenoncentral case (essentially thenth power of Craig (1936)^[24] equation 10) although the probability density becomes an infinite sum of Bessel functions.

It can be shown^[25] that the Wishart distribution can be defined if and only if the shape parametern belongs to the set

${\displaystyle {\mathrm{\Lambda }}_p:=\{0,\dots ,p-1\}\cup \left(p-1,\mathrm{\infty }\right).}$

This set is named afterSimon Gindikin, who introduced it^[26] in the 1970s in the context of gamma distributions on homogeneous cones. However, for the new parameters in the discrete spectrum of the Gindikin ensemble, namely,

${\displaystyle {\mathrm{\Lambda }}_p^{\ast }:=\{0,\dots ,p-1\},}$

the corresponding Wishart distribution has no Lebesgue density.

In random matrix theory, the Wishart family is often studied through itsLaguerre ensembles. For the real case (orthogonal symmetry,β=1), the joint density of the eigenvalues${\displaystyle {\lambda }_1,\dots ,{\lambda }_p\ge 0}$ of${\displaystyle \mathbf{X}\sim W_p(\mathbf{I},n)}$ is

which is theLaguerre orthogonal ensemble (LOE). The complex and quaternion analogues are theLaguerre unitary (LUE,${\displaystyle \beta =2}$) andLaguerre symplectic (LSE,${\displaystyle \beta =4}$) ensembles, respectively.^[27]

A further generalization, theβ-Laguerre ensemble, allows the Dyson index${\displaystyle \beta >0}$ to vary continuously. Its joint eigenvalue density has theCoulomb gas form

which reduces to LOE/LUE/LSE for${\displaystyle \beta =1,2,4}$. For the classical Gaussian Wishart case (scale${\displaystyle \mathbf{I}}$), one has the identification

${\displaystyle \alpha =\frac{\beta }{2}(n-p+1)-1.}$

A concrete probabilistic construction for any${\displaystyle \beta >0}$ is provided by theDumitriu–Edelman bidiagonal model. One samples a random bidiagonal matrix${\displaystyle \mathbf{B}}$ with independent chi variables and sets${\displaystyle \mathbf{L}=\mathbf{B}{\mathbf{B}}^T}$; the eigenvalues of${\displaystyle \mathbf{L}}$ follow the β-Laguerre law with parameter${\displaystyle \alpha }$.^[28][29]

Sampling

• Classical${\displaystyle \beta =1,2,4}$. Bartlett’s decomposition gives${\displaystyle \mathbf{X}=\mathbf{L}\mathbf{A}{\mathbf{A}}^T{\mathbf{L}}^T}$ with${\displaystyle \mathbf{A}}$ lower-triangular having chi/normal entries, yielding LOE/LUE/LSE exactly.^[30]
• General${\displaystyle \beta >0}$. In the Dumitriu–Edelman model, for matrix size${\displaystyle p}$ and parameter${\displaystyle \alpha >-1}$, one samples independently

${\displaystyle B_{ii}\sim \frac{1}{\sqrt{\beta }}{\chi }_{\beta (\alpha +p-i+1)},B_{i,i-1}\sim \frac{1}{\sqrt{\beta }}{\chi }_{\beta (p-i+1)}(i=2,\dots ,p),}$

and sets${\displaystyle \mathbf{L}=\mathbf{B}{\mathbf{B}}^T}$. Then the eigenvalues of${\displaystyle \mathbf{L}}$ have the β-Laguerre joint density above.^[28]

Rectangular data matrices

If one wishes to realize these spectra as the singular values of a rectangular${\displaystyle M\times N}$ matrix (with${\displaystyle M\ge N=p}$), draw independent Haar-distributed${\displaystyle \mathbf{U}\in O(M)}$ (or${\displaystyle U(M)}$) and${\displaystyle \mathbf{V}\in O(N)}$, and set

${\displaystyle X=U\left[\begin{array}{c}\mathrm{diag}(\sqrt{{\lambda }_1},\dots ,\sqrt{{\lambda }_p})\\ 0\end{array}\right]V^{\ast },}$

so that${\displaystyle X^{\ast }X}$ has eigenvalues${\displaystyle \{{\lambda }_i\}}$. For i.i.d. Gaussian entries (true Wishart),${\displaystyle \beta }$ is fixed by the field (real, complex, quaternion) and${\displaystyle \alpha }$ reduces to${\displaystyle \frac{\beta }{2}(n-p+1)-1}$.^[29][31]

At the “hard edge” (near${\displaystyle \lambda =0}$), β-Laguerre ensembles exhibitBessel-kernel correlations and level repulsion of order${\displaystyle \beta }$. In particular, with${\displaystyle p\to \mathrm{\infty }}$ and fixed${\displaystyle \alpha }$, the distribution of the smallest eigenvalue converges to a universal hard-edge (Bessel) law that depends only on${\displaystyle \alpha }$ and${\displaystyle \beta }$.^[32][29]

Under proportional growth${\displaystyle p/n\to \gamma \in (0,\mathrm{\infty })}$, the empirical spectral distribution of${\displaystyle n^{-1}\mathbf{G}{\mathbf{G}}^T}$ (with i.i.d. entries of variance${\displaystyle {\sigma }^2}$) converges almost surely to theMarchenko–Pastur law, supported on${\displaystyle {\sigma }^2(1\pm \sqrt{\gamma }{)}^2}$. When${\displaystyle \gamma =1}$, the density diverges as${\displaystyle x^{-1/2}}$ at the origin (a hard-edge singularity).^[33]

• The Wishart distribution is related to theinverse-Wishart distribution, denoted by${\displaystyle W_p^{-1}}$, as follows: IfX ~W_p(V,n) and if we do the change of variablesC =X⁻¹, then${\displaystyle \mathbf{C}\sim W_p^{-1}({\mathbf{V}}^{-1},n)}$. This relationship may be derived by noting that the absolute value of theJacobian determinant of this change of variables is|C|^p+1, see for example equation (15.15) in.^[34]
• InBayesian statistics, the Wishart distribution is aconjugate prior for theprecision parameter of themultivariate normal distribution, when the mean parameter is known.^[12]
• A generalization is themultivariate gamma distribution.
• A different type of generalization is thenormal-Wishart distribution, essentially the product of amultivariate normal distribution with a Wishart distribution.

• A C++ library for random matrix generator

• Continuous distributions
• Multivariate continuous distributions
• Covariance and correlation
• Random matrices
• Conjugate prior distributions
• Exponential family distributions

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