|Notation||X ~ Wp(V, n)|
n > p − 1 degrees of freedom (real)|
V > 0 scale matrix (p × p pos. def)
|Support||X(p × p) positive definite matrix|
|Mode||(n − p − 1)V for n ≥ p + 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.
Then the Wishart distribution is the probability distribution of the p × p random matrix
known as the scatter matrix. One indicates that S has that probability distribution by writing
The positive integer n is the number of degrees of freedom. Sometimes this is written W(V, p, n). For n ≥ 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.
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
Let X be a p × p symmetric matrix of random variables that is positive definite. Let V be a (fixed) symmetric positive definite matrix of size p × p.
Then, if n ≥ p, X has a Wishart distribution with n degrees of freedom if it has the probability density function
The density above is not the joint density of all the elements of the random matrix X (such -dimensional density does not exists because of the symmetry constrains ), it is rather the joint density of elements for (, page 38). Also, the density formula above applies only to positive definite matrices for other matrices the density is equal to zero.
where is a constant.
In fact the above definition can be extended to any real n > p − 1. If n ≤ 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 × p matrices.
Use in Bayesian statistics
Choice of parameters
The least informative, proper Wishart prior is obtained by setting n = p.
The prior mean of Wp(V, n) is nV, suggesting that a reasonable choice for V would be n−1Σ0, where Σ0 is some prior guess for the covariance matrix.
where is the multivariate digamma function (the derivative of the log of the multivariate gamma function).
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.
The information entropy of the distribution has the following formula::693
where B(V, n) is the normalizing constant of the distribution:
This can be expanded as follows:
The cross entropy of two Wishart distributions with parameters and with parameters is
Note that when we recover the entropy.
The Kullback–Leibler divergence of from is
The characteristic function of the Wishart distribution is
In other words,
where E[⋅] 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).
Since the determinant's range contains a closed line through the origin for matrix dimensions greater than two, the above formula is only correct for small values of the Fourier variable. (see https://arxiv.org/pdf/1901.09347.pdf)
If a p × p random matrix X has a Wishart distribution with m degrees of freedom and variance matrix V — write — and C is a q × p matrix of rank q, then
In this case, is the chi-squared distribution and (note that is a constant; it is positive because V is positive definite).
Consider the case where zT = (0, ..., 0, 1, 0, ..., 0) (that is, the j-th element is one and all others zero). 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 (MLE) of the covariance matrix of a multivariate normal distribution. A derivation of the MLE uses the spectral theorem.
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:
where L is the Cholesky factor of V, and:
Marginal distribution of matrix elements
Let V be a 2 × 2 variance matrix characterized by correlation coefficient −1 < ρ < 1 and L its lower Cholesky factor:
Multiplying through the Bartlett decomposition above, we find that a random sample from the 2 × 2 Wishart distribution is
The diagonal elements, most evidently in the first element, follow the χ2 distribution with n degrees of freedom (scaled by σ2) 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 χ2 distribution. The corresponding marginal probability density for the off-diagonal element is therefore the variance-gamma distribution
where Kν(z) is the modified Bessel function of the second kind. 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) equation 10) although the probability density becomes an infinite sum of Bessel functions.
The range of the shape parameter
This set is named after Gindikin, who introduced it 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,
the corresponding Wishart distribution has no Lebesgue density.
Relationships to other distributions
- The Wishart distribution is related to the inverse-Wishart distribution, denoted by , as follows: If X ~ Wp(V, n) and if we do the change of variables C = X−1, then . 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.
- 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.
- 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.
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