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Dirichlet-process Pólya-urn predictive law

Statement

Assume AC. In the conditionally IID model $G \sim D P (α)$ , $X_{i} ∣ G \sim G$ , the posterior version after $x_{1 : n} = (x_{1}, \dots, x_{n})$ has one-step predictive probability $\Pr (X_{n + 1} \in A ∣ X_{1 : n} = x_{1 : n}) = \frac{α (A) + \sum_{i = 1}^{n} 1_{A} (x_{i})}{α (X) + n}$ for every measurable $A$ and every data vector. The formula is a data-indexed version on prior-predictive null data sets; those pointwise values are not uniquely determined by the disintegration identity.

Facts & Assumptions

Given: Full AC; a standard Borel space; finite nonzero $α$ ; and conditionally IID sampling from $G \sim D P (α)$ .

[A1]

Full AC is assumed. The Axiom of Choice.

[A2]

AC implies Countable Choice, as required by the Dirichlet-multinomial update. AC implies DC implies countable choice The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

The data-indexed posterior version is $D P (α + \sum_{i} δ_{x_{i}})$ . Dirichlet-process posterior update.

[F2]

Given the random measure and the observed past, the next observation has conditional probability $G (A)$ of falling in $A$ , by the conditionally IID product sampling kernel. Dirichlet process Probability measures and probability spaces.

[F3]

On every finite partition, positive-base cells have the Dirichlet law and zero-base cells have mass zero. Dirichlet process Existence of a Dirichlet process by normalized gamma Poisson random measure.

[F4]

Under Countable Choice, the one-trial Dirichlet-multinomial evidence for the first cell of a two-cell partition with positive parameters $(b, c)$ is $b / (b + c)$ . Dirichlet-Multinomial conjugate update.

Proof

Given: The hypotheses in the Statement and $n \geq 0$ .

1.1givenF1construct

Put $β_{x} = α + \sum_{i = 1}^{n} δ_{x_{i}}$ . For $n \geq 1$ , [F1] makes the posterior law $D P (β_{x})$ ; for $n = 0$ it is the prior $D P (α)$ . In either case, $β_{x} (X) = α (X) + n > 0$ .

2.1A1A2F2F3F4step 1.1casesalgebra∎

Let $b = β_{x} (A)$ and $c = β_{x} (A^{c})$ . If $b = 0$ , [F3] gives $G (A) = 0$ almost surely under $D P (β_{x})$ ; if $c = 0$ , it gives $G (A) = 1$ almost surely. If both are positive, apply [F4] to one Bernoulli draw on the partition $(A, A^{c})$ : its probability of the first cell is the Dirichlet-multinomial evidence $b / (b + c)$ , which equals $E_{D P (β_{x})} G (A)$ by the sampling kernel [F2]. Thus in all cases the predictive probability is $b / (b + c) = (α (A) + \sum_{i} 1_{A} (x_{i})) / (α (X) + n)$ .

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)