{"document_url": "https://en.wikipedia.org/wiki/Normal-Wishart_distribution"}
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).

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:
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