TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 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.

Posterior kernel and dominated formula for a latent Gibbs observation

Statement

Assume the Axiom of Choice. Let the parameter and data spaces $(Θ, T)$ and $(Y, Y)$ of a finite latent Gibbs observation model both be standard Borel, and let $Π$ be its proper prior. Let $J$ be the joint prior-predictive probability determined by $Π$ and the marginal sampling kernel $Q$ , and let $J_{Y}$ be its data marginal. There exists a data-indexed probability kernel $y \mapsto Π_{y}$ satisfying $J (A \times B) = \int_{B} Π_{y} (A) J_{Y} (d y) (A \in T, B \in Y),$ and any two such posterior kernels agree as probability measures for $J_{Y}$ -almost every $y$ .

In the dominated case with data measure $ν$ and joint likelihood density $L (θ, y) = \sum_{x \in X} π_{ϑ (θ)} (x) k (y ∣ θ, x)$ from the Definition, define $m (y) : = \int_{Θ} L (θ, y) Π (d θ) .$ Then $m$ is measurable, $\int_{Y} m d ν = 1$ , and for $ν$ -almost every $y$ with $0 < m (y) < \infty$ the posterior has the version $Π_{y} (A) = \frac{\int_{A} L (θ, y) Π (d θ)}{m (y)} (A \in T) .$ The prior-predictive law gives zero mass to the set where $m = 0$ or $m = \infty$ ; the conditional identities do not determine pointwise posterior values there.

Statement references

Finite latent Gibbs observation model

The Axiom of Choice

Standard Borel spaces

Measure kernel and probability kernel

Standard-Borel Bayesian posterior kernel exists

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Facts & Assumptions

Given: A proper prior $Π$ , the measurable canonical-parameter map and finite latent Gibbs sampling kernel in the Definition, standard-Borel parameter and data spaces, AC, and, for the dominated clause, a sigma-finite measure $ν$ and jointly measurable normalized density $k$ .

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function; AC is an explicit hypothesis of the posterior-kernel result used below.

[F1]

Finite latent Gibbs observation model: summing the finite latent state law with weights $π_{ϑ (θ)} (x)$ against the observation kernel gives a parameter-measurable probability kernel $Q$ ; in the dominated case the finite sum $L (θ, y) = \sum_{x} π_{ϑ (θ)} (x) k (y ∣ θ, x)$ is its joint likelihood density.

[F2]

Standard Borel spaces: a measurable space is standard Borel when it is measurably isomorphic to the Borel space of a Polish topology; both parameter and data measurable structures are assumed to have this property.

[F3]

Standard-Borel Bayesian posterior kernel exists: under AC, a proper prior and parameter-measurable probability kernel on standard-Borel parameter and data spaces give a data-indexed posterior probability kernel satisfying the rectangle identity, unique as a probability measure outside one data-marginal null set.

[F4]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: for a nonnegative product-measurable function on two sigma-finite measure spaces, section integrals are measurable and either iterated integral equals the product integral.

[F5]

Measure kernel and probability kernel: a probability kernel is a measurable family of probability measures, with total mass one at every source point.

Proof

Given: The hypotheses and notation in the Statement.

1.1givenA1F1F2F3

By [F1], the prior and $Q$ form a proper Bayesian model. The parameter and data spaces are standard Borel by [F2], and AC holds by [A1]. Therefore [F3] supplies the posterior probability kernel and its $J_{Y}$ -almost-everywhere uniqueness, with the exact rectangle identity in the Statement.

1.2givenF1F4algebra

In the dominated case, $L$ is nonnegative and jointly measurable by its finite-sum definition. The prior $Π$ is finite and hence sigma-finite; $ν$ is sigma-finite by the Definition. Apply [F4] to $L$ on their product. It gives measurability of $m (y) = \int_{Θ} L (θ, y) Π (d θ)$ and $\int_{Y} m (y) ν (d y) = \int_{Θ} \sum_{x} π_{ϑ (θ)} (x) (\int_{Y} k (y ∣ θ, x) ν (d y)) Π (d θ) = \int_{Θ} \sum_{x} π_{ϑ (θ)} (x) Π (d θ) = 1$ . For each $B \in Y$ , the same calculation with $1_{B} (y) L (θ, y)$ gives $J_{Y} (B) = \int_{B} m (y) ν (d y)$ . Here the finite sum and each normalized observation density are those of [F1].

2.1givenF1F4F5step 1.2

The set ${m = \infty}$ has $ν$ -measure zero because $m \geq 0$ has integral one, and the set ${m = 0}$ has zero data-marginal mass because $J_{Y} (d y) = m (y) ν (d y)$ by step 1.2. For each $A \in T$ , the numerator $n_{A} (y) = \int_{A} L (θ, y) Π (d θ)$ is measurable by [F4]. Define ${\hat{Π}}_{y} (A) = n_{A} (y) / m (y)$ when $0 < m (y) < \infty$ , and set ${\hat{Π}}_{y} = Π$ otherwise. On valid $y$ , the measure induced on $Θ$ by the nonnegative density $L (\cdot, y)$ with respect to $Π$ has total mass $n_{Θ} (y) = m (y)$ , so after division it is a probability measure; on the exceptional set it is the prior. Thus each section is a probability measure, and its evaluations are measurable, so it is a probability kernel by [F5].

3.1F1F3F4step 1.2step 2.1algebra∎

For every $A \in T$ and $B \in Y$ , Tonelli and the data-marginal density established in step 1.2 give $\int_{B} {\hat{Π}}_{y} (A) J_{Y} (d y) = \int_{B} n_{A} (y) ν (d y) = \int_{A \times B} L (θ, y) (Π \times ν) (d θ, d y) = J (A \times B)$ ; the exceptional values contribute zero because their $J_{Y}$ -measure is zero. Thus the displayed ratio is a posterior version on its stated domain, and the general uniqueness assertion remains only $J_{Y}$ -almost everywhere.

Remarks

The prior is part of this mathematical model, and the posterior is conditional on it and on the stated likelihood kernel. This theorem proves neither posterior concentration nor computational efficiency. It does not report data or establish that any Gibbs model describes a physical system.

Depends on

Dependency tree · 2 levels

Sources

Alicia A. Johnson, Miles Q. Ott, and Mine Dogucu, Bayes Rules!, Chapter 2