Normal-inverse-Wishart distribution

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]

Parameters location (vector of real)
inverse scale matrix (pos. def.)
Support covariance matrix (pos. def.)



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



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 .[2]

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
  • 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.


  1. Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution."
  2. Gelman, Andrew, et al. Bayesian data analysis. Vol. 2, p.73. Boca Raton, FL, USA: Chapman & Hall/CRC, 2014.


  • Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.
  • Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution."
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