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] Suppose has a multivariate normal distribution with mean

and covariance matrix

has a normal-Wishart distribution, denoted as By construction, the marginal distribution over

is a multivariate normal distribution.

is a multivariate t-distribution.

, the posterior distribution of the parameters is where Generation of random variates is straightforward: