Introduction to the Wishart Distribution for Random Matrix Theory

Understanding the Wishart Distribution

The Wishart distribution, named after John Wishart, is a multivariate generalization of the chi-squared distribution and plays a crucial role in multivariate statistical analysis. It is commonly used as a conjugate prior for the precision matrix in Bayesian statistics.

Conjugate Prior for Bayesian Analysis

In a multivariate normal distribution, the Wishart distribution serves as the conjugate prior for the precision matrix. This means that if the prior distribution of Ω (the precision matrix) is a Wishart distribution, then the posterior distribution (after observing data) is also a Wishart distribution.

Definition

A Wishart random matrix with parameters and can be expressed as the sum of outer products of independent multivariate normal random vectors (each with mean 0). The probability density function of the Wishart distribution is given by a complicated expression involving the number of degrees of freedom, the scale matrix, and the argument matrix.

Properties

The Wishart distribution possesses several important properties, including: * It is widely used in random matrix theory and multivariate statistics for its ability to model the distribution of positive definite matrices. * It is the multivariate generalization of the univariate chi-squared distribution, serving an analogous role in multivariate analysis. * The moments of the Wishart distribution can provide insights into the behavior of random matrices.