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.
By construction, the marginal distribution over is an inverse 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
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 .
Generating normal-inverse-Wishart random variates
Generation of random variates is straightforward:
- Sample from an inverse Wishart distribution with parameters and
- Sample from a multivariate normal distribution with mean and variance