Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)
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A non-Gaussian posterior can have different MAP and mean

Statement refuted

For every probability density p on R with finite first absolute moment and at least one global maximizer, every maximum a posteriori (MAP) point, meaning every global maximizer of p, equals its mean ∫Rt p(t) dt.

Facts & Assumptions

Given: AC; a prior uniform on [0,1]; independent standard-normal observation noise; and the observation model in the Counterexample.

[A1]

AC is assumed by the standard-normal law. The Axiom of Choice

[F1]

The standard-normal noise has a strictly positive Gaussian density. Standard normal and normal laws, A positive-definite multivariate normal law has its Gaussian density

[F2]

A normalized potential update is the prior reweighted by its likelihood and divided by its evidence when the evidence is positive and finite. Misfit evidence and the potential posterior law

[F3]

The equality of MAP, posterior mean, and support-restricted Tikhonov minimizer proved for the finite Gaussian model uses Gaussian prior and likelihood assumptions. Finite Gaussian MAP, posterior mean, and support-restricted Tikhonov minimizer coincide

Counterexample

Given: Let θ have the uniform probability density 1[0,1] on R. Independently let ϵ∼N(0,1), observe Y=θ+ϵ, and condition on y=0. The likelihood potential is Φ0(θ)=θ2/2 on [0,1] and the posterior density relative to Lebesgue measure is p(θ∣0)=e−θ2/2∫01e−t2/2 dt 1[0,1](θ).

1.1A1F1F2givenalgebra

The evidence in [F2] is Z=∫01e−t2/2 dt. Since e−1/2≤e−t2/2≤1 on [0,1], one has 0<e−1/2≤Z≤1. Thus the displayed normalized posterior density is proper.

2.1F1step 1.1algebra

On (0,1], the derivative of e−θ2/2 is −θe−θ2/2<0. The posterior density vanishes outside [0,1], so its unique global maximizer is θ=0. Its support is the bounded interval [0,1] and it is not a point mass. A nondegenerate one-dimensional Gaussian has a strictly positive density on all of R by [F1], whereas a zero-variance Gaussian is a point mass. Therefore this posterior is non-Gaussian.

3.1F3step 1.1step 2.1algebra∎

The posterior mean is ∫01te−t2/2 dtZ=1−e−1/2Z>0, which differs from the MAP point 0. This refutes the universal claim. It also shows why [F3] is restricted to the Gaussian model.

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