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).Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media. Page 690.
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
Cross Validated, https://stats.stackexchange.com/q/324925
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
Related distributions The normal-inverse Wishart distribution is essentially the same distribution parameterized by variance rather than precision. The normal-gamma distribution is the one-dimensional equivalent. The multivariate normal distribution and Wishart distribution are the component distributions out of which this distribution is made.
Notes
References Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.