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

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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.

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Finite-partition posterior and Pólya-urn draws

Example

Assume AC. Let $X = {a, b, c}$ with its discrete sigma-algebra and base masses $α ({a}) = 2$ , $α ({b}) = 1$ , and $α ({c}) = 1$ . For $G \sim D P (α)$ , consider two conditionally IID draws with observed values $X_{1} = a$ , $X_{2} = b$ .

Facts & Assumptions

Given: The finite space and base measure in the Example.

[A1]

Assume AC, as required by the normalized-gamma and posterior results. The Axiom of Choice.

[F1]

The finite-partition masses under $D P (α)$ have the Dirichlet law. Dirichlet process Existence of a Dirichlet process by normalized gamma Poisson random measure.

[F2]

After data with cell counts $y_{j}$ , the finite-partition posterior is Dirichlet with parameters $α_{j} + y_{j}$ . Dirichlet-process posterior update.

[F3]

The next-draw probability in a measurable cell is its base mass plus its observed count, divided by total base mass plus sample size. Dirichlet-process Pólya-urn predictive law.

[F4]

Every nonempty finite discrete space is standard Borel. Dirac set functions are probability measures; finite nonnegative weighted sums of them are measures. A countable discrete measurable space is standard Borel A Dirac set function is a probability measure Nonnegative scalar multiples and countable weighted sums of measures are measures.

Verification

Given: The specified three-cell partition, its base masses, and the observed cell sequence $(a, b)$ .

1.1A1F1F2F4given

The nonempty finite discrete space $X$ is standard Borel by [F4]. Define $α = 2 δ_{a} + δ_{b} + δ_{c}$ , equivalently $α (B) = 21_{B} (a) + 1_{B} (b) + 1_{B} (c)$ . By [F4] it is a measure with total mass $4$ , hence finite and nonzero, so [F1] and [F2] apply. The prior vector $(G ({a}), G ({b}), G ({c}))$ has law $Dirichlet (2, 1, 1)$ by [F1]. The observed counts are $(1, 1, 0)$ , so [F2] gives posterior $Dirichlet (3, 2, 1)$ , whose parameters sum to $6$ .

2.1F3step 1.1algebra

Applying [F3] after the two observations gives next-draw probabilities $3 / 6 = 1 / 2$ for $a$ , $2 / 6 = 1 / 3$ for $b$ , and $1 / 6$ for $c$ ; they sum to one.

3.1F3algebra∎

Before any draw, [F3] gives $\Pr (X_{1} = a) = 2 / 4 = 1 / 2$ . After observing $X_{1} = a$ , the same rule gives $\Pr (X_{2} = a ∣ X_{1} = a) = 3 / 5$ . Therefore $\Pr (X_{1} = a, X_{2} = a) = (1 / 2) (3 / 5) = 3 / 10$ : the first draw increases the next draw's mass for its cell from $2 / 4$ to $3 / 5$ .

Depends on

Used by

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Sources

Thomas S. Ferguson, A Bayesian Analysis of Some Nonparametric Problems, 1973, Sections 2 and 3 (standard reference, not scraped)
Günter Last and Mathew Penrose, Lectures on the Poisson Process, 2017, Chapter 1 (standard reference, not scraped)