Normal-inverse-Wishart distribution

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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[edit]



Marginal distributions[edit]

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[edit]

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[edit]

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[edit]

  • 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." [1]
  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." [2]