Parameters location (vector of real)
inverse scale matrix (pos. def.)
Support 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).[1]



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:[2]

Here is the multivariate gamma function and is the Trace of the given matrix.



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 (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 .[3]

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

Related distributions


  1. ^ Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [1]
  2. ^ Simon J.D. Prince(June 2012). Computer Vision: Models, Learning, and Inference. Cambridge University Press. 3.8: "Normal inverse Wishart distribution".
  3. ^ Gelman, Andrew, et al. Bayesian data analysis. Vol. 2, p.73. Boca Raton, FL, USA: Chapman & Hall/CRC, 2014.