Normal-Wishart distribution
In probability theory and statistics, the normal-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.
Definition
Supposehas a multivariate normal distribution with mean and covariance matrix, where
has a Wishart distribution. Then
has a normal-Wishart distribution, denoted as
Properties
Marginal distributions
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
After making observations, the posterior distribution of the parameters iswhere
Generating normal-Wishart random variates
Generation of random variates is straightforward:- Sample from a Wishart distribution with parameters and
- Sample from a multivariate normal distribution with mean and variance
Related distributions
- The normal-inverse Wishart distribution is essentially the same distribution parameterized by variance rather than precision.
- The normal-gamma distribution is the one-dimensional equivalent.
- The multivariate normal distribution and Wishart distribution are the component distributions out of which this distribution is made.