DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Finite latent Gibbs observation model

Definition

Let $(Θ, T)$ be a nonempty measurable parameter space and let $ϑ : (Θ, T) \to (R^{k}, B (R^{k}))$ be measurable. Let $X$ be the finite state set of the finite exponential Gibbs family from the A-page Definition, evaluated at canonical parameter $ϑ (θ)$ ; write its mass as $π_{ϑ (θ)} (x)$ . A finite latent Gibbs observation model consists of a prior probability measure $Π$ on $(Θ, T)$ and a probability kernel $K$ from $(Θ \times X, T \otimes 2^{X})$ to a nonempty measurable data space $(Y, Y)$ , where $2^{X}$ is the full power-set sigma-algebra on $X$ and $T \otimes 2^{X}$ is the product sigma-algebra. The kernel $K ((θ, x), B)$ is the observation law for data event $B$ when parameter $θ$ and latent configuration $x$ are fixed. Define the marginal sampling kernel $Q (θ, B) : = \sum_{x \in X} π_{ϑ (θ)} (x) K ((θ, x), B) (B \in Y) .$ Thus the latent configuration is summed out before the data likelihood is evaluated.

In a dominated observation model, a specified measure $ν$ on $(Y, Y)$ is sigma-finite, meaning that $Y$ is a countable union of measurable sets of finite $ν$ -measure. A density is given by a map $k : Θ \times X \times Y \to [0, \infty]$ measurable from $(Θ \times X \times Y, (T \otimes 2^{X}) \otimes Y)$ to $[0, \infty]$ with its Borel sigma-algebra. Writing its value as $k (y ∣ θ, x)$ , for every $(θ, x) \in Θ \times X$ and $B \in Y$ it satisfies $K ((θ, x), B) = \int_{B} k (y ∣ θ, x) ν (d y), \int_{Y} k (y ∣ θ, x) ν (d y) = 1.$ The marginal data density, or likelihood density at $y$ , is then $L (θ, y) : = \sum_{x \in X} π_{ϑ (θ)} (x) k (y ∣ θ, x) .$ The general kernel model does not require a pointwise likelihood density.

Definition references

Finite exponential Gibbs family for inverse inference

Bayesian model as a prior and sampling kernel

The product sigma-algebra and its finite iterates

Measure kernel and probability kernel

Facts & Assumptions

Given: The finite exponential family, measurable parameter and data spaces, the proper prior, and the observation kernel in the Definition.

[F1]

Finite exponential Gibbs family for inverse inference: $X$ is finite and, for each $x \in X$ , $π_{u} (x) = w (x) \exp (u \cdot T (x) - A (u))$ with $w (x) > 0$ ; its proof establishes $A \in C^{2} (R^{k})$ , positivity, and normalization.

[F2]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: $∣ (u - v) \cdot T (x) ∣ \leq ∥ u - v ∥_{2} ∥ T (x) ∥_{2}$ for $u, v \in R^{k}$ , and the Euclidean norm is a norm.

[F3]

The exponential function is strictly increasing: $\exp : R \to (0, \infty)$ is continuous.

[F4]

The product sigma-algebra and its finite iterates: $T \otimes 2^{X}$ is generated by measurable rectangles; for each fixed $x \in X$ , the section map $θ \mapsto (θ, x)$ is measurable because the preimage of a rectangle $A \times D$ is $A$ if $x \in D$ and empty otherwise.

[F5]

Measure kernel and probability kernel: a probability kernel is a pointwise probability measure in its data argument and is measurable in its source argument for each fixed data event.

[F6]

Bayesian model as a prior and sampling kernel: a proper Bayesian model consists of a prior probability measure and a parameter-measurable probability kernel from parameter space to data space.

Proof

Given: The ingredients of the Definition.

1.1givenF1F2F3algebra

Fix $x \in X$ and define $f_{x} (u) = u \cdot T (x)$ and $g_{x} (u) = f_{x} (u) - A (u)$ . By [F2], $∣ f_{x} (u) - f_{x} (v) ∣ \leq ∥ u - v ∥_{2} ∥ T (x) ∥_{2}$ , so $f_{x}$ is continuous. The function $A$ is continuous by [F1], hence $g_{x}$ is continuous: at any $u_{0}$ and for any $ε > 0$ , intersect neighborhoods on which $∣ f_{x} (u) - f_{x} (u_{0}) ∣ < ε / 2$ and $∣ A (u) - A (u_{0}) ∣ < ε / 2$ . By continuity of $\exp$ [F3], its composition with $g_{x}$ is continuous (choose the continuity neighborhood of $g_{x} (u_{0})$ first, then use continuity of $g_{x}$ ); multiplying by the fixed scalar $w (x)$ preserves continuity by rescaling $ε$ by $w (x) > 0$ . Thus the displayed formula makes $u \mapsto π_{u} (x)$ continuous. A continuous real-valued function is Borel measurable because preimages of open sets are open and open sets generate the Borel sigma-algebra. Therefore composing $u \mapsto π_{u} (x)$ with the measurable $ϑ$ makes $θ \mapsto π_{ϑ (θ)} (x)$ measurable.

1.2givenF1F5algebra

For each fixed $θ$ , every summand in $Q (θ, B)$ is nonnegative. For pairwise disjoint events $B_{j}$ , countable additivity of each $K ((θ, x), \cdot)$ and the finiteness of $X$ let the finite sum over $x$ commute with the nonnegative series over $j$ , so $Q (θ, \cdot)$ is countably additive. Also $Q (θ, Y) = \sum_{x} π_{ϑ (θ)} (x) = 1$ .

2.1givenF1F4F5F6step 1.1step 1.2algebra

For each $B \in Y$ , $(θ, x) \mapsto K ((θ, x), B)$ is measurable by [F5]; composing with each measurable section from [F4] makes $θ \mapsto K ((θ, x), B)$ measurable. Also $θ \mapsto π_{ϑ (θ)} (x)$ is measurable by step 1.1. Therefore $Q (θ, B)$ is a finite sum of products of measurable functions and is measurable in $θ$ . Thus $Q$ is a probability kernel and, with the proper prior $Π$ , is a Bayesian model in the sense of [F6].

3.1givenF1step 1.1step 1.2algebra∎

In the dominated case, each term $π_{ϑ (θ)} (x) k (y ∣ θ, x)$ is jointly measurable and nonnegative by the Definition and step 1.1. Its finite sum $L (θ, y)$ is jointly measurable; integrating it over $B$ gives $\int_{B} L (θ, y) ν (d y) = \sum_{x} π_{ϑ (θ)} (x) K ((θ, x), B) = Q (θ, B)$ . For $B = Y$ , this integral is $1$ by step 1.2, so $L$ is exactly a density for the marginal sampling kernel, with latent states already summed out.

Remarks

A product of one-site conditional probabilities is a pseudo-likelihood, as defined separately on this page. It is a different inferential input from the joint-data kernel $Q$ above. The B-page finite counterexample shows that treating it as the exact likelihood can give a different posterior; the pseudo-likelihood product is therefore not substituted for $Q$ in this definition.

Finite latent Gibbs observation model

Definition

Definition references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources