location (vector of real)|
inverse scale matrix (pos. def.)
covariance matrix (pos. def.)|
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).
has a multivariate normal distribution with mean and covariance matrix , where
has an inverse Wishart distribution. Then
has a normal-inverse-Wishart distribution, denoted as
Probability density function
The full version of the PDF is as follows:
Here is the multivariate gamma function and is the Trace of the given matrix.
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
To sample from the joint posterior of , one simply draws samples from , then draw . To draw from the posterior predictive of a new observation, draw , given the already drawn values of and .