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
has a multivariate normal distribution with mean
has a Wishart distribution. Then
Probability density function
Marginal distributions
By construction, the marginal distribution over
Generating normal-Wishart random variates
Generation of random variates is straightforward:
- Sample
Λ from a Wishart distribution with parametersW andν - Sample
μ from a multivariate normal distribution with meanμ 0 ( λ Λ ) − 1