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

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

Probability density function

Properties

Scaling

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 is
where

Generating normal-Wishart random variates

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

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