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Dirichlet-Multinomial conjugate update

Statement

Assume Countable Choice. Let $p \sim Dirichlet (α_{1}, \dots, α_{k})$ , with each $α_{j} > 0$ . Conditional on $p$ , observe a Multinomial $(n, p)$ count vector $y \in N_{0}^{k}$ with $\sum_{j} y_{j} = n$ . For $k \geq 2$ , the data space is equipped with counting measure on its finite set of admissible counts. The evidence is $m (y) = \frac{n!}{\prod_{j} y_{j}!} \frac{Γ (α_{+})}{\prod_{j} Γ (α_{j})} \frac{\prod_{j} Γ (α_{j} + y_{j})}{Γ (α_{+} + n)} \in (0, \infty),$ where $α_{+} = \sum_{j} α_{j}$ , and the posterior is $Dirichlet (α_{1} + y_{1}, \dots, α_{k} + y_{k})$ . For $k = 1$ , the parameter and count are both deterministic and the posterior equals the prior point mass.

Facts & Assumptions

Given: Countable Choice, positive $α_{j}$ , $n \geq 0$ , and admissible counts $y_{j} \geq 0$ summing to $n$ .

[A1]

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

[F1]

The Dirichlet law has density $Γ (α_{+}) / \prod_{j} Γ (α_{j}) \prod_{j} p_{j}^{α_{j} - 1}$ on the open simplex for $k \geq 2$ , and is a point mass at $1$ for $k = 1$ . Dirichlet distribution, including the one-category case.

[F2]

The Dirichlet simplex density is normalized by iterated Beta integrals under Countable Choice. Dirichlet density normalization by iterated Beta integrals.

[F3]

The Multinomial $(n, p)$ likelihood is $n! \prod_{j} p_{j}^{y_{j}} / \prod_{j} y_{j}!$ and is normalized. Multinomial distribution.

[F4]

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

[F5]

The evidence is the integral of the likelihood against the prior; no quotient is defined unless it is finite and positive. Evidence normalizer and fiberwise posterior update.

[F6]

A likelihood-weighted prior is a probability after normalization exactly for finite positive evidence. Fiberwise posterior propriety is equivalent to finite positive evidence.

[F7]

The dominated Bayes formula identifies the normalized likelihood quotient as a posterior conditional kernel. Bayes formula for dominated kernels.

Proof

Given: The hypotheses in the Statement and observed count vector $y$ .

1.1A1F1F3F4givencases

If $k = 1$ , then $p_{1} = 1$ and the only admissible count is $y_{1} = n$ . The likelihood and evidence are one, so the point-mass prior is unchanged. Assume henceforth $k \geq 2$ . Multiplying the prior and likelihood densities [F1, F3] gives a constant times $\prod_{j = 1}^{k} p_{j}^{α_{j} + y_{j} - 1},$ with every updated coordinate positive. The finite count space, proper prior, and normalized measurable likelihood give the model in [F4].

2.1A1F2F3F5F6step 1.1algebra

By [F2], the integral of this updated simplex kernel is $\prod_{j} Γ (α_{j} + y_{j}) / Γ (α_{+} + n)$ , finite and positive. Multiplying by the prior normalizing constant and the Multinomial coefficient from [F3] yields exactly the displayed evidence. It is finite positive, so [F5] and [F6] place this fiber in the valid update domain.

3.1A1F2F5F7step 2.1cases∎

Divide the product density from step 1.1 by its evidence from step 2.1. The resulting density is $\frac{Γ (α_{+} + n)}{\prod_{j} Γ (α_{j} + y_{j})} \prod_{j = 1}^{k} p_{j}^{α_{j} + y_{j} - 1},$ the normalized Dirichlet $(α_{1} + y_{1}, \dots, α_{k} + y_{k})$ density by [F2]. The evidence is positive finite, so [F7] makes this normalized law the posterior conditional kernel. Zero counts leave their coordinates positive and unchanged; no boundary parameter points are introduced.

Depends on

Used by

A three-category Dirichlet-Multinomial update Example
Dirichlet-process Pólya-urn predictive law Theorem
Dirichlet-process posterior update Theorem
Type-H histogram posterior weak Bernstein–von Mises theorem Theorem

Dependency tree · two levels

46 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)