TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Hierarchical posterior normalization on positive finite evidence

Statement

Assume the finite-group hierarchical model of Finite-group hierarchical Bayesian model. Write $P_{0}$ for its proper prior on $Ξ = H \times Θ^{m}$ , let $L_{y} (ξ)$ be its specified product likelihood density with respect to the finite product measure $ν$ , and define $Z (y) = \int_{Ξ} L_{y} (ξ) P_{0} (d ξ) .$ For every observation $y$ with $0 < Z (y) < \infty$ , the measure $P_{y} (A) = \frac{\int_{A} L_{y} (ξ) P_{0} (d ξ)}{Z (y)} (A \in H \otimes T^{\otimes m})$ is the joint posterior law of the latent tuple. Defining $P_{y} = P_{0}$ when $Z (y) = 0$ or $Z (y) = \infty$ gives a regular conditional distribution under the prior-predictive law; no quotient is formed on either exceptional fiber.

Facts & Assumptions

Given: A proper hyperprior, finite product group and observation kernels, and their specified jointly measurable likelihood densities.

[F1]

The finite-group model defines a proper latent prior $P_{0}$ and the product observation kernel with likelihood section $L_{y}$ . Finite-group hierarchical Bayesian model

[F2]

The finite product of the group observation kernels has jointly measurable product density $L (ξ, y) = \prod_{j} f_{j} (θ_{j}, y_{j})$ relative to the sigma-finite product measure $ν$ . Finite products of dominated kernels have the product density

[F3]

A proper prior and probability model kernel define a joint probability law with the corresponding iterated-integral identity. A prior and model kernel define a joint law

[F4]

For a proper prior and a dominated probability kernel with a specified jointly measurable density, the posterior kernel is the normalized density on $0 < m (y) < \infty$ fibers and uses the prior as filler on zero or infinite normalizer fibers. Bayes formula for dominated kernels

Proof

technique · identify the finite product likelihood density and apply the dominated-kernel Bayes formula

Given: The hypotheses in the Statement.

1.1F1F2F3F4given

By [F1] and [F3], the latent prior $P_{0}$ is a probability measure and the model kernel gives a joint probability law for the latent tuple and the observations. By [F2], that observation kernel is dominated by the sigma-finite product measure $ν$ with the specified jointly measurable density $L (ξ, y) = \prod_{j} f_{j} (θ_{j}, y_{j})$ . Thus all hypotheses of [F4] apply with prior $P_{0}$ and likelihood density $L$ .

2.1F4step 1.1∎

The resulting evidence is $m (y) = \int_{Ξ} L_{y} (ξ) P_{0} (d ξ) = Z (y)$ . On every fiber with $0 < Z (y) < \infty$ , [F4] gives the probability measure $P_{y} (A) = Z (y)^{- 1} \int_{A} L_{y} d P_{0}$ and the conditional-event identities. On fibers with zero or infinite evidence [F4] uses the probability filler $P_{0}$ , so its kernel remains a regular conditional distribution there without dividing by zero or infinity. The same theorem says these filled fibers have prior-predictive measure zero.

Depends on

Used by

Dependency tree · two levels

27 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

Andrew Gelman, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin, Bayesian Data Analysis, 3rd ed., §§5.2–5.3 (standard reference, not scraped)
Alicia A. Johnson, Miles Q. Ott, and Mine Dogucu, Bayes Rules!, Chapter 16 (standard reference, not scraped)