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ABC rejection returns the tolerance-conditioned posterior

Statement

Use the model and notation of Approximate Bayesian computation and the tolerance-conditioned target and the procedure ABC rejection sampling. Let $B = B_{ϵ} (s_{0})$ and $p = M (Θ \times B) > 0$ . Suppose the proposals and simulator outputs form an IID sequence $(Θ_{j}, Y_{j})_{j \geq 1}$ with the prior-simulator joint law. Then the first accepted index $N : = \inf {j \geq 1 : S (Y_{j}) \in B}$ is finite almost surely, and the accepted parameter has law $P (Θ_{N} \in A) = \frac{M (A \times B)}{p} = Π_{ϵ} (A ∣ s_{0}) (A \in T) .$ Thus the accepted parameter has exactly the tolerance-conditioned target law. This statement does not identify that target with the full-data posterior or assert an expected proposal count.

Facts & Assumptions

Given: A proper prior-simulator model, $s_{0}$ , $ϵ \geq 0$ , $p = M (Θ \times B_{ϵ} (s_{0})) > 0$ , and an IID sequence of proposal-simulator pairs with the model's prior-simulator joint law.

[F1]

The summary-predictive law $M$ is the law of $(θ, S (Y))$ , and the indicator ABC target is $M (A \times B) / M (Θ \times B)$ when the denominator is positive. Approximate Bayesian computation and the tolerance-conditioned target

[F2]

ABC rejection draws independent prior-simulator pairs and accepts precisely when the simulated summary lies in $B_{ϵ} (s_{0})$ . ABC rejection sampling

[F3]

IID means identically distributed and mutually independent, not merely pairwise independent. Identical distribution and IID families

[F4]

If $0 \leq r < 1$ , then $\sum_{k = 0}^{\infty} r^{k} = 1 / (1 - r)$ . For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges

Proof

Given: The hypotheses in the Statement.

1.1givenF1F2

For each $j$ , let $E_{j} = {S (Y_{j}) \in B}$ . By [F1] and [F2], $P (E_{j}) = p$ and, for every $A \in T$ , $P (Θ_{j} \in A, E_{j}) = M (A \times B)$ .

2.1givenF3F4step 1.1algebra

By [F3], for each $k \geq 1$ , $P (N = k) = P (E_{1}^{c} \cap \dots \cap E_{k - 1}^{c} \cap E_{k}) = (1 - p)^{k - 1} p .$ Since $0 \leq 1 - p < 1$ , summing these disjoint events and applying [F4] gives $P (N < \infty) = \sum_{k = 1}^{\infty} (1 - p)^{k - 1} p = p \frac{1}{1 - (1 - p)} = 1.$ Thus the algorithm terminates almost surely.

3.1givenF1F3step 1.1F4algebra∎

For any $A \in T$ , partition the accepted-output event by the first accepted index. Independence in [F3] and step 1.1 give $P (Θ_{N} \in A) = \sum_{k = 1}^{\infty} (1 - p)^{k - 1} M (A \times B) = \frac{M (A \times B)}{p} .$ By [F1] this is $Π_{ϵ} (A ∣ s_{0})$ , proving the asserted law.

Depends on

Used by

An identifiable model can have persistent ABC summary distortion Counterexample
Positive ABC tolerance changes an identity-summary target Counterexample
A Bernoulli count is a sufficient summary for a finite-sample posterior Example
Vanishing acceptance probability with a stable zero-tolerance target Example

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Sources

Jarno Lintusaari, Michael U. Gutmann, Ritabrata Dutta, Samuel Kaski, and Jukka Corander, Fundamentals and Recent Developments in Approximate Bayesian Computation (standard reference, not scraped)
Aki Honkela, Computational Statistics I, Chapter 9 (standard reference, not scraped)