Normal-Wishart distribution

In probability theory and statistics, the normal-Wishart distribution (or Gaussian-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 precision matrix (the inverse of the covariance matrix).[1]

Normal-Wishart
Notation
Parameters location (vector of real)
(real)
scale matrix (pos. def.)
(real)
Support covariance matrix (pos. def.)
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Definition

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Suppose

 

has a multivariate normal distribution with mean   and covariance matrix  , where

 

has a Wishart distribution. Then   has a normal-Wishart distribution, denoted as

 

Characterization

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Probability density function

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Properties

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Scaling

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Marginal distributions

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By construction, the marginal distribution over   is a 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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After making   observations  , the posterior distribution of the parameters is

 

where

 
 
 
 [2]

Generating normal-Wishart random variates

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Generation of random variates is straightforward:

  1. Sample   from a Wishart distribution with parameters   and  
  2. Sample   from a multivariate normal distribution with mean   and variance  
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Notes

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  1. ^ Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media. Page 690.
  2. ^ Cross Validated, https://stats.stackexchange.com/q/324925

References

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  • Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.