ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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A three-category Dirichlet-Multinomial update

Statement

Assume Countable Choice. With prior $p \sim Dirichlet (1, 2, 1)$ , sample size $n = 3$ , and observed counts $y = (2, 1, 0)$ , the evidence is $1 / 10$ and the posterior is $Dirichlet (3, 3, 1)$ .

Facts & Assumptions

Given: Countable Choice, the prior parameters $(1, 2, 1)$ , sample size $3$ , and counts $(2, 1, 0)$ .

[A1]

Countable Choice is used by the Dirichlet simplex normalization theorem. The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

For positive Dirichlet parameters and admissible counts, the posterior adds the counts coordinatewise and the evidence is the Multinomial coefficient times the ratio of Dirichlet normalizers. Dirichlet-Multinomial conjugate update.

[F2]

For positive integers $r$ , $Γ (r) = (r - 1)!$ . $Γ (n + 1) = n!$ for every natural number $n$ .

Verification

Given: The hypotheses in the Statement and facts [A1], [F1], [F2].

1.1A1F1F2givenalgebra

Here $α_{+} = 4$ , $n = 3$ , and $3! / (2! 1! 0!) = 3$ . By [F2], $Γ (4) = 6$ , $Γ (1) = Γ (2) = 1$ , $Γ (3) = 2$ , and $Γ (7) = 720$ . Substitution into [F1] gives $m (y) = 3 \cdot 6 \cdot 4 / 720 = 1 / 10$ .

2.1A1F1givenalgebra∎

The updated coordinates are $(1 + 2, 2 + 1, 1 + 0) = (3, 3, 1)$ , all strictly positive, so [F1] gives the proper posterior $Dirichlet (3, 3, 1)$ . The zero count in the third category leaves its positive prior coordinate unchanged.

Depends on

Used by

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Sources

Gelman et al., Bayesian Data Analysis, 3rd ed., §3.4 (standard reference, not scraped)
Gero Walter, A Technical Note on the Dirichlet-Multinomial Model, 2013 (standard reference, not scraped)