DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Bayesian model as a prior and sampling kernel

Definition

Let $E = ((Y, Y), Θ, (P_{θ})_{θ \in Θ})$ be a statistical experiment (Statistical experiments). Equip the parameter set with a specified measurable structure $(Θ, T)$ (Measurable spaces and measurable sets). A proper Bayesian model based on this experiment consists of a prior probability measure $Π$ on $(Θ, T)$ (Probability measures and probability spaces) and a probability kernel $K$ from $(Θ, T)$ to $(Y, Y)$ (Measure kernel and probability kernel) that represents the experiment: $K (θ, B) = P_{θ} (B) (θ \in Θ, B \in Y) .$ Thus $θ \mapsto K (θ, B)$ is $T$ -measurable for every data event $B$ , as required for a parameter-measurable model (Parameter-measurable statistical models as kernels). The family of measures in a parametric model alone does not supply this parameter sigma-algebra or measurability condition (Parametric statistical models).

The word proper means that the prior is a probability measure. A sigma-finite improper prior is not a proper Bayesian model in this definition; fixed-data posterior normalizers for such priors are handled separately.

Depends on

Used by

Posterior RCD exists for a standard-Borel parameter Corollary
An identifiable model can have persistent ABC summary distortion Counterexample
Approximate Bayesian computation and the tolerance-conditioned target Definition
Augmented ABC tolerance target Definition
Bayesian posterior as a regular conditional law Definition
Evidence normalizer and fiberwise posterior update Definition
A Bayesian posterior RCD indexed by the sample point Example
A Bernoulli count is a sufficient summary for a finite-sample posterior Example
A dominated posterior on a nonstandard-Borel observation space Example
A finite Bayesian update and a null-data posterior version Example
A standard-Borel posterior kernel for countable data Example
A sufficient summary preserves the Bayesian posterior Theorem
Beta-Binomial conjugate update Theorem
Conjugate update for an exponential-family likelihood Theorem
Dirichlet-Multinomial conjugate update Theorem
Dirichlet-process posterior update Theorem
Exposure-adjusted Gamma-Poisson conjugate update Theorem
Known-variance Normal-Normal conjugate update Theorem
Standard-Borel Bayesian posterior kernel exists Theorem

Dependency tree · two levels

12 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. (standard reference, not scraped)
Alicia A. Johnson, Miles Q. Ott, and Mine Dogucu, Bayes Rules!, Chapter 2 (standard reference, not scraped)