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]
Notation | |||
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Parameters |
location (vector of real) (real) scale matrix (pos. def.) (real) | ||
Support | covariance matrix (pos. def.) | ||
Suppose
has a multivariate normal distribution with mean and covariance matrix , where
has a Wishart distribution. Then has a normal-Wishart distribution, denoted as
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.
After making observations , the posterior distribution of the parameters is
where
Generation of random variates is straightforward: