Budan's theorem

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Template:Probability 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]

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

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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

Notes

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References

  • Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.
  • Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [1]

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  1. Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [2]