location (vector of real)|
scale matrix (pos. def.)
covariance matrix (pos. def.)|
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).
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.
Posterior distribution of the parameters
After making observations , the posterior distribution of the parameters is
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