Inverse-Wishart distribution

In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution. In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution. We say X {displaystyle mathbf {X} } follows an inverse Wishart distribution, denoted as X ∼ W − 1 ( Ψ , ν ) {displaystyle mathbf {X} sim {mathcal {W}}^{-1}(mathbf {Psi } , u )} , if its inverse X − 1 {displaystyle mathbf {X} ^{-1}} has a Wishart distribution W ( Ψ − 1 , ν ) {displaystyle {mathcal {W}}(mathbf {Psi } ^{-1}, u )} . Important identities have been derived for the inverse-Wishart distribution. The probability density function of the inverse Wishart is: where x {displaystyle mathbf {x} } and Ψ {displaystyle {mathbf {Psi } }} are p × p {displaystyle p imes p} positive definite matrices, and Γp(·) is the multivariate gamma function. If A ∼ W ( Σ , ν ) {displaystyle {mathbf {A} }sim {mathcal {W}}({mathbf {Sigma } }, u )} and Σ {displaystyle {mathbf {Sigma } }} is of size p × p {displaystyle p imes p} , then X = A − 1 {displaystyle mathbf {X} ={mathbf {A} }^{-1}} has an inverse Wishart distribution X ∼ W − 1 ( Σ − 1 , ν ) {displaystyle mathbf {X} sim {mathcal {W}}^{-1}({mathbf {Sigma } }^{-1}, u )} . Suppose A ∼ W − 1 ( Ψ , ν ) {displaystyle {mathbf {A} }sim {mathcal {W}}^{-1}({mathbf {Psi } }, u )} has an inverse Wishart distribution. Partition the matrices A {displaystyle {mathbf {A} }} and Ψ {displaystyle {mathbf {Psi } }} conformably with each other where A i j {displaystyle {mathbf {A} _{ij}}} and Ψ i j {displaystyle {mathbf {Psi } _{ij}}} are p i × p j {displaystyle p_{i} imes p_{j}} matrices, then we have i) A 11 {displaystyle mathbf {A} _{11}} is independent of A 11 − 1 A 12 {displaystyle mathbf {A} _{11}^{-1}mathbf {A} _{12}} and A 22 ⋅ 1 {displaystyle {mathbf {A} }_{22cdot 1}} , where A 22 ⋅ 1 = A 22 − A 21 A 11 − 1 A 12 {displaystyle {mathbf {A} _{22cdot 1}}={mathbf {A} }_{22}-{mathbf {A} }_{21}{mathbf {A} }_{11}^{-1}{mathbf {A} }_{12}} is the Schur complement of A 11 {displaystyle {mathbf {A} _{11}}} in A {displaystyle {mathbf {A} }} ; ii) A 11 ∼ W − 1 ( Ψ 11 , ν − p 2 ) {displaystyle {mathbf {A} _{11}}sim {mathcal {W}}^{-1}({mathbf {Psi } _{11}}, u -p_{2})} ;