InverseWishart

Inverse-Wishart is a probability distribution over real symmetric positive definite p×p matrices. It is the distribution of the inverse of a Wishart matrix. If X follows an inverse-Wishart distribution with degrees of freedom ν and scale matrix Ψ (X ~ IW_p(ν, Ψ)) where Ψ ≻ 0 and ν > p−1, then the inverse X−1 has a Wishart distribution with the same degrees of freedom and with scale Ψ−1: X−1 ~ W_p(ν, Ψ−1).

The density of X is given, for X ≻ 0, by

f(X) = |Ψ|^{ν/2} / (2^{ν p/2} Γ_p(ν/2)) · |X|^{-(ν+p+1)/2} · exp(-1/2 tr(Ψ X−1)),

where Γ_p is the multivariate gamma function. Key parameter constraints are ν > p−1 for the density to be proper; a finite mean exists when ν > p+1, with E[X] = Ψ / (ν − p − 1).

Common usage and properties: the inverse-Wishart serves as a conjugate prior for the covariance matrix of a multivariate normal distribution. It encodes beliefs about scale (through Ψ) and uncertainty (through ν). In sampling terms, if W ~ W_p(ν, Ψ−1) then X = W−1 ~ IW_p(ν, Ψ).

Notes on parameterization: different texts interchangeably parameterize with scale Ψ or with a precision-like matrix; the core relationship remains X ~ IW_p(ν, Ψ) ↔ X−1 ~ W_p(ν, Ψ−1). The distribution is defined on the space of positive definite matrices and is widely used in Bayesian multivariate analysis.