Normal-Inverse-Gamma Distribution
(μ, σ²) ~ NIG(m, λ, a, b)
Bayesian / Conjugate-Prior Family • Published July 22, 2026
The first fully joint conjugate prior on this site: the complete Bayesian prior for a Normal's mean and variance together (extending the already-published Inverse Gamma's known-mean-only case), built hierarchically from the already-published Inverse Gamma and Normal sheets, with closed-form InvGamma and non-standardized Student's-t marginals verified against scipy, exact machine-precision confirmation of the closed-form posterior-update conjugacy, and simulation via its own two-stage hierarchical construction.
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About This Reference Sheet
The first fully joint conjugate prior on this site: the complete Bayesian prior for a Normal's mean and variance together (extending the already-published Inverse Gamma's known-mean-only case), built hierarchically from the already-published Inverse Gamma and Normal sheets, with closed-form InvGamma and non-standardized Student's-t marginals verified against scipy, exact machine-precision confirmation of the closed-form posterior-update conjugacy, and simulation via its own two-stage hierarchical construction.
Support
μ ∈ ℝ, σ² ∈ (0, ∞) -- joint over a mean and a variance
Parameters
m ∈ ℝ (prior mean), λ > 0 (precision scale), a > 0 (shape), b > 0 (scale)