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

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Beta-Binomial conjugate update

Statement

Assume Countable Choice. Let $P \sim Beta (α, β)$ with $α, β > 0$ , and, conditional on $P = p$ , let $Y$ have the Binomial $(n, p)$ law for $n \in N_{0}$ . The data space ${0, \dots, n}$ is equipped with finite counting measure. For every observed $y \in {0, \dots, n}$ , the evidence is $m (y) = (\binom{n}{y}) \frac{B (α + y, β + n - y)}{B (α, β)} \in (0, \infty),$ and the posterior is $Beta (α + y, β + n - y)$ . The endpoints $y = 0$ and $y = n$ are included; all posterior shape parameters remain positive.

Facts & Assumptions

Given: Countable Choice, positive $α, β$ , integer $n \geq 0$ , and observed $y \in {0, \dots, n}$ .

[A1]

Countable Choice is used by the Beta density normalization supplier. The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

The Beta prior has density $p^{α - 1} (1 - p)^{β - 1} / B (α, β)$ on $(0, 1)$ . Beta distribution.

[F2]

The Binomial $(n, p)$ mass at $y$ is $(\binom{n}{y}) p^{y} (1 - p)^{n - y}$ . The Binomial count has its usual probability mass function.

[F3]

The integral of the Beta kernel with positive parameters is finite and equals $B$ ; this value is strictly positive. The Beta density has finite positive Lebesgue integral.

[F4]

A proper prior paired with a parameter-measurable sampling probability kernel is a Bayesian model. Bayesian model as a prior and sampling kernel.

[F5]

The evidence is the prior integral of the specified likelihood section, and the quotient update is formed only on positive finite evidence. Evidence normalizer and fiberwise posterior update.

[F6]

A finite-positive evidence is exactly the condition for probability normalization of the likelihood-weighted prior. Fiberwise posterior propriety is equivalent to finite positive evidence.

[F7]

The normalized likelihood quotient for a jointly measurable dominated model is a posterior kernel on its valid fibers. Bayes formula for dominated kernels.

Proof

Given: Countable Choice, the hypotheses and observed count in the Statement.

1.1A1F1F2F4F5given

By [F2], the likelihood section is $L_{y} (p) = (\binom{n}{y}) p^{y} (1 - p)^{n - y}$ and the Binomial law is a probability kernel. Its dependence on $p$ is measurable, and [F1] supplies a proper prior, so [F4] gives the Bayesian model. The finite counting measure on the data space is sigma-finite, and [F5] identifies its evidence as $\int_{0}^{1} L_{y} (p) f_{α, β} (p) d p$ .

2.1A1F1F3step 1.1algebra

Multiplying the likelihood and prior densities gives $\frac{(\binom{n}{y})}{B (α, β)} p^{α + y - 1} (1 - p)^{β + n - y - 1} .$ Both new shapes are positive because $y \geq 0$ and $n - y \geq 0$ . Integrating with [F3] yields the stated evidence, which is finite and strictly positive.

3.1A1F1F5F6F7step 2.1givenalgebra∎

Divide the product density in step 2.1 by its evidence. The coefficient becomes $1 / B (α + y, β + n - y)$ , giving exactly the Beta $(α + y, β + n - y)$ density. The evidence is in the valid domain of [F5] and [F6], so [F7] identifies this normalized law as the posterior conditional kernel. At $y = 0$ or $y = n$ , the unchanged positive shape on one side and the increased positive shape on the other keep the same proof valid.

Depends on

Used by

A Beta(2,3) prior updated by five Bernoulli trials Example

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Sources

Gelman et al., Bayesian Data Analysis, 3rd ed., §2.1 (standard reference, not scraped)
Johnson, Ott & Dogucu, Bayes Rules!, Chapter 3, §3.1 (standard reference, not scraped)