Supriya Ghosh (Editor)

Normal Wishart distribution

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Notation
  
( μ , Λ ) ∼ N W ( μ 0 , λ , W , ν ) {\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Lambda }})\sim \mathrm {NW} ({\boldsymbol {\mu }}_{0},\lambda ,\mathbf {W} ,\nu )}

Parameters
  
μ 0 ∈ R D {\displaystyle {\boldsymbol {\mu }}_{0}\in \mathbb {R} ^{D}\,} location (vector of real) λ > 0 {\displaystyle \lambda >0\,} (real) W ∈ R D × D {\displaystyle \mathbf {W} \in \mathbb {R} ^{D\times D}} scale matrix (pos. def.) ν > D − 1 {\displaystyle \nu >D-1\,} (real)

Support
  
μ ∈ R D ; Λ ∈ R D × D {\displaystyle {\boldsymbol {\mu }}\in \mathbb {R} ^{D};{\boldsymbol {\Lambda }}\in \mathbb {R} ^{D\times D}} covariance matrix (pos. def.)

PDF
  
f ( μ , Λ | μ 0 , λ , W , ν ) = N ( μ | μ 0 , ( λ Λ ) − 1 )   W ( Λ | W , ν ) {\displaystyle f({\boldsymbol {\mu }},{\boldsymbol {\Lambda }}|{\boldsymbol {\mu }}_{0},\lambda ,\mathbf {W} ,\nu )={\mathcal {N}}({\boldsymbol {\mu }}|{\boldsymbol {\mu }}_{0},(\lambda {\boldsymbol {\Lambda }})^{-1})\ {\mathcal {W}}({\boldsymbol {\Lambda }}|\mathbf {W} ,\nu )}

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).

Contents

Definition

Suppose

μ | μ 0 , λ , Λ ∼ N ( μ | μ 0 , ( λ Λ ) − 1 )

has a multivariate normal distribution with mean μ 0 and covariance matrix ( λ Λ ) − 1 , where

Λ | W , ν ∼ W ( Λ | W , ν )

has a Wishart distribution. Then ( μ , Λ ) has a normal-Wishart distribution, denoted as

( μ , Λ ) ∼ N W ( μ 0 , λ , W , ν ) .

Probability density function

f ( μ , Λ | μ 0 , λ , W , ν ) = N ( μ | μ 0 , ( λ Λ ) − 1 )   W ( Λ | W , ν )

Marginal distributions

By construction, the marginal distribution over Λ is a Wishart distribution, and the conditional distribution over μ given Λ is a multivariate normal distribution. The marginal distribution over μ is a multivariate t-distribution.

Generating normal-Wishart random variates

Generation of random variates is straightforward:

  1. Sample Λ from a Wishart distribution with parameters W and ν
  2. Sample μ from a multivariate normal distribution with mean μ 0 and variance ( λ Λ ) − 1
  • The normal-inverse Wishart distribution is essentially the same distribution parameterized by variance rather than precision.
  • The normal-gamma distribution is the one-dimensional equivalent.
  • The multivariate normal distribution and Wishart distribution are the component distributions out of which this distribution is made.

    • References

    Normal-Wishart distribution Wikipedia
    (Text) CC BY-SA