Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Equal posterior means do not control the full posterior

Statement refuted

For any proper prior probability measure on a measurable real-valued parameter space and any two finite likelihood potentials with positive finite evidence and finite posterior means, equality of the two posterior means implies equality of the posterior laws and hence zero Hellinger distance.

Facts & Assumptions

Given: The false statement above, the fixed-data potential update convention, and the library root-L2 Hellinger convention.

[F1]

For a finite potential Φ on a prior space, the evidence is ZΦ=∫e−Φ dμ0 and the posterior density relative to the prior is e−Φ/ZΦ whenever 0<ZΦ<∞. Misfit evidence and the potential posterior law

[F2]

If P,Q have supplied densities p,q with respect to a common measure ν, then h(P,Q)2=∫(p−q)2 dν. Root-L2 Hellinger distance for dominated laws

[F3]

The natural logarithm is the inverse of the real exponential on (0,∞). The natural logarithm as the inverse of the exponential function

[F4]

A measurable space is a set equipped with a sigma-algebra, and a probability measure is a countably additive measure with total mass one. Measurable spaces and measurable sets, Probability measures and probability spaces

[F5]

A posterior mean of an integrable real parameter is its expectation; on this finite space it is the finite probability-weighted sum. Expectation of a nonnegative or integrable random variable, Expectation of a real random variable on a finite probability space

[F6]

For any a∈R, the intervals (x−∣x−a∣/2,x+∣x−a∣/2), indexed by x≠a, cover R∖{a} and omit a; hence R∖{a} is open and {a} is closed. Thus D is Borel, every subset of the finite set D is Borel, and its trace Borel sigma-algebra is the power set; every map from D to a measurable target is consequently measurable. The Borel sigma-algebra of a topological space

Counterexample

Take X=R, D={−1,0,1} with its Borel subspace structure, and the prior μ0 that assigns mass 1/3 to each point. Put ε=1/100 and define two probability mass functions on D by p(−1)=p(1)=ε/2,p(0)=1−ε, q(−1)=q(1)=(1−ε)/2,q(0)=ε. Define finite potentials ΦP(u)=−log⁡(3p(u)) and ΦQ(u)=−log⁡(3q(u)). Their evidences are both one, so their posterior updates are precisely the laws with mass functions p and q.

1.1F1F3F4F6givenalgebra

The uniform prior is the finite atomic measure μ0(A):=∣A∣/3 for A⊆D; it is countably additive on this finite space and has μ0(D)=1, so [F4] makes it a probability measure. The masses of p and q are positive and each sums to one. By [F6], D is Borel and both finite potentials are measurable. By [F3], e−ΦP(u)=3p(u) and e−ΦQ(u)=3q(u) for every u∈D. Thus ZΦP=13∑u∈D3p(u)=1 and ZΦQ=13∑u∈D3q(u)=1. By [F1], the two posterior updates have masses p and q; they are proper and have finite means.

2.1F5step 1.1algebra

By [F5], their means are (−1)(ε/2)+(1)(ε/2)=0 and (−1)((1−ε)/2)+(1)((1−ε)/2)=0, respectively. Hence their posterior means agree.

3.1F2step 1.1algebra∎

Using μ0 as the common dominating measure, the corresponding densities are 3p and 3q. By [F2], h(P,Q)2=2(1/200−99/200)2+(99/100−1/10)2. Since 99/100>(9/10)2, the central term is greater than (4/5)2=16/25. Since 99/200>(7/10)2 and 1/200<(1/14)2, each endpoint squared difference is greater than (7/10−1/14)2=(22/35)2. Therefore h(P,Q)2>16/25+2(22/35)2=1752/1225>1. Equal posterior means thus coexist with Hellinger distance greater than one.

Remarks

This finite example only refutes a mean-only conclusion. It makes no claim that the two potentials satisfy the Hellinger stability or discretization hypotheses on the A page.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

46 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources