In probability theory and statistics, the normal-inverse-Wishart distribution (or Gaussian-inverse-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 covariance matrix (the inverse of the precision matrix).
location (vector of real)|
inverse scale matrix (pos. def.)
|Support||covariance matrix (pos. def.)|
has an inverse Wishart distribution. Then has a normal-inverse-Wishart distribution, denoted as
Probability density function
Posterior distribution of the parameters
Suppose the sampling density is a multivariate normal distribution
where is an matrix and (of length ) is row of the matrix .
With the mean and covariance matrix of the sampling distribution is unknown, we can place a Normal-Inverse-Wishart prior on the mean and covariance parameters jointly
The resulting posterior distribution for the mean and covariance matrix will also be a Normal-Inverse-Wishart
Generating normal-inverse-Wishart random variates
Generation of random variates is straightforward:
- The normal-Wishart distribution is essentially the same distribution parameterized by precision rather than variance. If then .
- The normal-inverse-gamma distribution is the one-dimensional equivalent.
- The multivariate normal distribution and inverse Wishart distribution are the component distributions out of which this distribution is made.