Normal-inverse-Wishart distribution


In probability theory and statistics, the normal-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.

Definition

Suppose
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

Characterization

Probability density function

Properties

Scaling

Marginal distributions

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 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
where
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:
  1. Sample from an inverse Wishart distribution with parameters and
  2. Sample from a multivariate normal distribution with mean and variance

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