Normal-Inverse-Wishart Distribution
(μ, Σ) ~ NIW(m, λ, ν, Ψ)
Bayesian / Conjugate-Prior Family • Published July 22, 2026
The multivariate capstone tying together four already-published sheets at once: the complete joint conjugate prior for a Normal's mean vector and covariance matrix together, generalizing the already-published Normal-Inverse-Gamma one dimension higher via the already-published Wishart and Inverse Wishart, with closed-form InvWishart and multivariate-t marginals, exact machine-precision confirmation of the closed-form posterior-update conjugacy, and simulation via its own hierarchical construction -- the standard prior underneath Bayesian multivariate regression and Gaussian mixture models.
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About This Reference Sheet
The multivariate capstone tying together four already-published sheets at once: the complete joint conjugate prior for a Normal's mean vector and covariance matrix together, generalizing the already-published Normal-Inverse-Gamma one dimension higher via the already-published Wishart and Inverse Wishart, with closed-form InvWishart and multivariate-t marginals, exact machine-precision confirmation of the closed-form posterior-update conjugacy, and simulation via its own hierarchical construction -- the standard prior underneath Bayesian multivariate regression and Gaussian mixture models.
Support
μ ∈ ℝᵖ, Σ a p×p positive-definite matrix -- joint over a mean vector and a covariance matrix
Parameters
m ∈ ℝᵖ (prior mean), λ > 0 (precision scale), ν > p-1 (degrees of freedom), Ψ a p×p positive-definite scale matrix