Inverse Gamma Distribution
X ~ InvGamma(α, β)
Bayesian / Conjugate-Prior Family • Published July 22, 2026
The standard conjugate prior for a Normal distribution's variance: defined directly as the reciprocal of the already-published Gamma (confirmed via 3-million-draw simulation), moments that exist only up to order α (confirmed by watching higher moments numerically diverge right at the cutoff), no moment generating function due to a polynomial right tail, a maximum-likelihood fit derived by reusing the Gamma sheet's own digamma equation on reciprocal data, and a Normal-variance posterior conjugacy confirmed to 13 decimal places against a brute-force grid computation.
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About This Reference Sheet
The standard conjugate prior for a Normal distribution's variance: defined directly as the reciprocal of the already-published Gamma (confirmed via 3-million-draw simulation), moments that exist only up to order α (confirmed by watching higher moments numerically diverge right at the cutoff), no moment generating function due to a polynomial right tail, a maximum-likelihood fit derived by reusing the Gamma sheet's own digamma equation on reciprocal data, and a Normal-variance posterior conjugacy confirmed to 13 decimal places against a brute-force grid computation.
Support
x ∈ (0, ∞)
Parameters
α > 0 (shape), β > 0 (scale)