Wishart distribution


In statistics, the Wishart distribution is a generalization of the gamma distribution to multiple dimensions. It is named in honor of John Wishart, who first formulated the distribution in 1928. Other names include Wishart ensemble, or Wishart–Laguerre ensemble, or LOE, LUE, LSE.
It is a family of probability distributions defined over symmetric, positive-definite 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 is a matrix, each column of which is independently drawn from a -variate normal distribution with zero mean:
It means :
Then the Wishart distribution is the probability distribution of the random matrix
known as the scatter matrix. One indicates that has that probability distribution by writing
The positive integer is the number of degrees of freedom. Sometimes this is written. For the matrix is invertible with probability if is invertible.
If then this distribution is a chi-squared distribution with 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. It is also encountered in wireless communications, while analyzing the performance of Rayleigh fading MIMO wireless channels.

Probability density function

The Wishart distribution can be characterized by its probability density function as follows:
Let be a symmetric matrix of random variables that is positive semi-definite. Let be a symmetric positive definite matrix of size.
Then, if, has a Wishart distribution with degrees of freedom if it has the probability density function
where is the determinant of and is the multivariate gamma function defined as
The density above is not the joint density of all the elements of the random matrix , it is rather the joint density of elements for . Also, the density formula above applies only to positive definite matrices for other matrices the density is equal to zero.
In fact the above definition can be extended to any real. If, 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 matrices.

Spectral density

The joint-eigenvalue density for the eigenvalues of a random matrix is,
where is a constant. The spectral density can be marginalized to yield the density of a single eigenvalue, by evaluating a Selberg integral.
The spectral density can be integrated to give the probability that all eigenvalues of a Wishart random matrix lie within an interval.

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, where is the covariance matrix. The use of Normal-Wishart conjugate priors is particularly common for vector autoregression models.

Choice of parameters

The least informative, proper Wishart prior is obtained by setting.
A common choice for V leverages the fact that the mean of X ~Wp is nV. Then V' is chosen so that nV equals an initial guess for X'. For instance, when estimating a precision matrix Σ−1 ~ Wp a reasonable choice for V would be n−1Σ0−1, where Σ0 is some prior estimate for the covariance matrix Σ.

Properties

Log-expectation

The following formula plays a role in variational Bayes derivations for Bayes networks
involving the Wishart distribution. From equation,
where is the multivariate digamma function.

Log-variance

The following variance computation could be of help in Bayesian statistics:
where 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:
where is the normalizing constant of the distribution:
This can be expanded as follows:

Cross-entropy

The cross-entropy of two Wishart distributions with parameters and with parameters is
Note that when and we recover the entropy.

KL-divergence

The Kullback–Leibler divergence of from is

Characteristic function

The characteristic function of the Wishart distribution is
where denotes expectation.. Properly interpreting this formula requires a little care, because noninteger complex powers are multivalued; when is noninteger, the correct branch must be determined via analytic continuation.

Theorem

If a random matrix has a Wishart distribution with degrees of freedom and variance matrix — write — and is a matrix of rank, then

Corollary 1

If is a nonzero constant vector, then:
In this case, is the chi-squared distribution and .

Corollary 2

Consider the case where . Then corollary 1 above shows that
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.

Estimator of the multivariate normal distribution

The Wishart distribution is the sampling distribution of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution. A derivation of the MLE uses the spectral theorem.

Bartlett decomposition

The Bartlett decomposition of a matrix from a -variate Wishart distribution with scale matrix and degrees of freedom is the factorization:
where is the Cholesky factor of, and:
where and independently. This provides a useful method for obtaining random samples from a Wishart distribution.

Marginal distribution of matrix elements

Let be a variance matrix characterized by Pearson [product-moment correlation coefficient|correlation coefficient] and its lower Cholesky factor:
Multiplying through the Bartlett decomposition above, we find that a random sample from the Wishart distribution is
The diagonal elements, most evidently in the first element, follow the distribution with degrees of freedom 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 distribution. The corresponding marginal probability density for the off-diagonal element is therefore the variance-gamma distribution
where is the modified Bessel function of the second kind. Similar results may be found for higher dimensions. In general, if follows a Wishart distribution with parameters,, then for, the off-diagonal elements
It is also possible to write down the moment-generating function even in the noncentral case although the probability density becomes an infinite sum of Bessel functions.

The range of the shape parameter

It can be shown that the Wishart distribution can be defined if and only if the shape parameter belongs to the set
This set is named after Simon Gindikin, who introduced it 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,
the corresponding Wishart distribution has no Lebesgue density.

Wishart–Laguerre ensembles and β-extensions

In random matrix theory, the Wishart family is often studied through its Laguerre ensembles. For the real case, the joint density of the eigenvalues of is
which is the Laguerre orthogonal ensemble. The complex and quaternion analogues are the Laguerre unitary and Laguerre symplectic ensembles, respectively.

β-Laguerre ensemble (general \beta>0)

A further generalization, the β-Laguerre ensemble, allows the Dyson index to vary continuously. Its joint eigenvalue density has the Coulomb gas form
which reduces to LOE/LUE/LSE for. For the classical Gaussian Wishart case, one has the identification
A concrete probabilistic construction for any is provided by the Dumitriu–Edelman bidiagonal model. One samples a random bidiagonal matrix with independent chi variables and sets ; the eigenvalues of follow the β-Laguerre law with parameter.
; Sampling
  • Classical. Bartlett’s decomposition gives with lower-triangular having chi/normal entries, yielding LOE/LUE/LSE exactly.
  • General. In the Dumitriu–Edelman model, for matrix size and parameter, one samples independently
and sets. Then the eigenvalues of have the β-Laguerre joint density above.
; Rectangular data matrices
If one wishes to realize these spectra as the singular values of a rectangular matrix, draw independent Haar-distributed and, and set
so that has eigenvalues. For i.i.d. Gaussian entries, is fixed by the field and reduces to.

Hard-edge behavior and universality

At the “hard edge”, β-Laguerre ensembles exhibit Bessel-kernel correlations and level repulsion of order. In particular, with and fixed, the distribution of the smallest eigenvalue converges to a universal hard-edge law that depends only on and.

Macroscopic limit (Marchenko–Pastur law)

Under proportional growth, the empirical spectral distribution of converges almost surely to the Marchenko–Pastur law, supported on. When, the density diverges as at the origin.

Relationships to other distributions