How is the Wishart related to the chi-squared distribution?

In one dimension the Wishart reduces to a scaled chi-squared distribution; the Wishart extends this to the joint distribution of variances and covariances in several dimensions.

What is the inverse-Wishart used for?

It serves as the conjugate prior for a covariance matrix in Bayesian multivariate models, yielding tractable posterior updates for the covariance.

Wishart Distribution

The Wishart distribution is the multivariate generalization of the chi-squared distribution, describing the sampling behavior of covariance matrices from multivariate normal data.

Definition

The Wishart distribution is the probability distribution of the matrix of sums of squares and cross-products formed from a sample of independent mean-zero multivariate normal vectors, parameterized by a scale matrix and degrees of freedom.

Scope

This topic covers the definition of the Wishart distribution as the distribution of a sum of outer products of independent normal vectors, its degrees of freedom and scale matrix, its role as the sampling distribution of the sample covariance matrix, the inverse-Wishart distribution as a conjugate prior for covariance, and its use in deriving multivariate test statistics.

Core questions

What is the sampling distribution of a sample covariance matrix?
How do the scale matrix and degrees of freedom parameterize the Wishart?
How does the Wishart generalize the chi-squared distribution?
Where does the inverse-Wishart distribution arise?

Key theories

Sampling distribution of covariance: For a sample from a multivariate normal population, the matrix of sums of squares and cross-products follows a Wishart distribution, generalizing the result that scaled sample variance from normal data is chi-squared.
Conjugacy of the inverse-Wishart: The inverse-Wishart distribution is the conjugate prior for the covariance matrix of a multivariate normal likelihood, making it central to Bayesian multivariate analysis.

Clinical relevance

The Wishart distribution underlies the null distributions of classical multivariate test statistics and provides the conjugate prior used in Bayesian estimation of covariance matrices.

History

John Wishart derived the distribution of the sample covariance matrix from multivariate normal data in 1928, providing the sampling theory needed for multivariate inference and giving the distribution its name.

Key figures

John Wishart
T. W. Anderson

Seminal works

anderson2003
muirhead1982
mardia1979

Frequently asked questions

How is the Wishart related to the chi-squared distribution?: In one dimension the Wishart reduces to a scaled chi-squared distribution; the Wishart extends this to the joint distribution of variances and covariances in several dimensions.
What is the inverse-Wishart used for?: It serves as the conjugate prior for a covariance matrix in Bayesian multivariate models, yielding tractable posterior updates for the covariance.