ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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.

A finite Bayesian update and a null-data posterior version

Example

Let $Θ = {0, 1}$ with prior $Π (1) = 1 / 3$ , and let $Y = {0, 1, 2}$ have its power-set sigma-algebra. Given $θ = 1$ , assign probabilities $(1 / 4, 3 / 4, 0)$ to $(0, 1, 2)$ ; given $θ = 0$ , assign probabilities $(3 / 4, 1 / 4, 0)$ . The posterior probability of $θ = 1$ is $1 / 7$ after data $0$ and $3 / 5$ after data $1$ . Data value $2$ has prior-predictive probability zero, so two versions may assign different posterior laws there.

Facts & Assumptions

Given: The finite parameter and data spaces and the prior and sampling probabilities specified in the example.

[F1]

A proper Bayesian model consists of a prior probability and a parameter-measurable sampling probability kernel. Bayesian model as a prior and sampling kernel

[F2]

A data-indexed posterior kernel satisfies the rectangle conditional-law identity; its values on prior-predictive null data may vary by version. Bayesian posterior as a regular conditional law

[F3]

A proper prior and sampling kernel define the joint law by $J (A \times B) = \int_{A} K (θ, B) Π (d θ)$ . A prior and model kernel define a joint law

Verification

Given: The assumptions and finite model in the example.

1.1givenF1F3algebra

The joint masses are $J (1, 0) = 1 / 12$ , $J (0, 0) = 1 / 2$ , $J (1, 1) = 1 / 4$ , and $J (0, 1) = 1 / 6$ , while both masses at data $2$ are zero. Therefore $J_{Y} (0) = 7 / 12$ , $J_{Y} (1) = 5 / 12$ , and $J_{Y} (2) = 0$ .

2.1F1F2step 1.1algebraconstruct

Define $Π_{0} (1) = 1 / 7$ , $Π_{1} (1) = 3 / 5$ , and choose any probability law for $Π_{2}$ ; set $Π_{y} (0) = 1 - Π_{y} (1)$ . Each $Π_{y}$ is a probability, and measurability is automatic on the finite power-set data space. The displayed ratios give $Π_{0} (1) J_{Y} (0) = J (1, 0)$ and $Π_{1} (1) J_{Y} (1) = J (1, 1)$ ; subtraction from the marginal gives the same identities for parameter value $0$ .

3.1F2step 1.1step 2.1algebra∎

For any $A \subseteq Θ$ and $B \subseteq Y$ , summing the singleton identities in step 2.1 over $y \in B$ gives $\int_{B} Π_{y} (A) J_{Y} (d y) = J (A \times B)$ ; the zero-mass term at $y = 2$ contributes zero for every choice of $Π_{2}$ . Thus every such kernel is a data-indexed posterior version by [F2]. In particular, $Π_{2} = δ_{0}$ and ${\tilde{Π}}_{2} = δ_{1}$ , with the same values at $0, 1$ , are distinct versions of the same posterior law.

Depends on

Used by

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Dependency tree · two levels

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Sources

Alicia A. Johnson, Miles Q. Ott, and Mine Dogucu, Bayes Rules!, Chapter 2 §§2.1.1–2.1.4 (standard reference, not scraped)
Andrew Gelman, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin, Bayesian Data Analysis, 3rd ed., Chapter 1 §1.3 (standard reference, not scraped)