Normal-Wishart distribution

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Normal-Wishart
Notation
Parameters location (vector of real)
(real)
scale matrix (pos. def.)
(real)
Support covariance matrix (pos. def.)
PDF

In probability theory and statistics, the normal-Wishart distribution (or Gaussian-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix (the inverse of the covariance matrix).[1]

Definition[edit]

Suppose

has a multivariate normal distribution with mean and covariance matrix , where

has a Wishart distribution. Then has a normal-Wishart distribution, denoted as

Characterization[edit]

Probability density function[edit]

Properties[edit]

Scaling[edit]

Marginal distributions[edit]

By construction, the marginal distribution over is a Wishart distribution, and the conditional distribution over given is a multivariate normal distribution. The marginal distribution over is a multivariate t-distribution.

Posterior distribution of the parameters[edit]

Generating normal-Wishart random variates[edit]

Generation of random variates is straightforward:

  1. Sample from a Wishart distribution with parameters and
  2. Sample from a multivariate normal distribution with mean and variance

Related distributions[edit]

Notes[edit]

  1. ^ Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media. Page 690.

References[edit]

  • Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.