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 an unknown mean and covariance matrix (the inverse of the precision matrix).
Definition Suppose
μ | μ 0 , λ , Σ ∼ N ( μ | μ 0 , 1 λ Σ ) {\displaystyle {\boldsymbol {\mu }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Sigma }}\sim {\mathcal {N}}\left({\boldsymbol {\mu }}{\Big |}{\boldsymbol {\mu }}_{0},{\frac {1}{\lambda }}{\boldsymbol {\Sigma }}\right)}
has a multivariate normal distribution with mean μ 0 {\displaystyle {\boldsymbol {\mu }}_{0}} and covariance matrix 1 λ Σ {\displaystyle {\tfrac {1}{\lambda }}{\boldsymbol {\Sigma }}} , where
Σ | Ψ , ν ∼ W − 1 ( Σ | Ψ , ν ) {\displaystyle {\boldsymbol {\Sigma }}|{\boldsymbol {\Psi }},\nu \sim {\mathcal {W}}^{-1}({\boldsymbol {\Sigma }}|{\boldsymbol {\Psi }},\nu )}
has an inverse Wishart distribution. Then ( μ , Σ ) {\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}
has a normal-inverse-Wishart distribution, denoted as
( μ , Σ ) ∼ N I W ( μ 0 , λ , Ψ , ν ) . {\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})\sim \mathrm {NIW} ({\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu ).}
Characterization
Probability density function
f ( μ , Σ | μ 0 , λ , Ψ , ν ) = N ( μ | μ 0 , 1 λ Σ ) W − 1 ( Σ | Ψ , ν ) {\displaystyle f({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu )={\mathcal {N}}\left({\boldsymbol {\mu }}{\Big |}{\boldsymbol {\mu }}_{0},{\frac {1}{\lambda }}{\boldsymbol {\Sigma }}\right){\mathcal {W}}^{-1}({\boldsymbol {\Sigma }}|{\boldsymbol {\Psi }},\nu )}
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