CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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A nonconstant summary can lose an identifiable parameter

Statement refuted

Full-data identifiability and nonconstancy of a summary do not imply that the summary retains information about the identifiable parameter or that the ABC target equals the exact posterior.

Facts & Assumptions

Given: Let $Θ = {0, 1}$ have equal prior mass, let $Y = {0, 1}^{2}$ have the power-set sigma-algebra, and for parameter $θ$ define $P_{θ} {(z, w)} = \frac{1}{2} 1_{{z = θ}} .$ Take $S (z, w) = w$ , observe $y = (0, 0)$ , and use zero tolerance $ϵ = 0$ .

[F1]

For standard Borel parameter/data spaces and a positive acceptance event, the indicator ABC target is the prior-simulator law conditioned on the closed summary-tolerance ball. Approximate Bayesian computation and the tolerance-conditioned target

[F2]

A probability kernel assigns a probability measure at each parameter and measurable event probabilities in the parameter; a statistic is a measurable map. Measure kernel and probability kernel Statistics and estimators

[F3]

Observational equivalence means equality of the full sampling laws, and the identity estimand is identifiable when equivalent parameters must agree. Parametric statistical models Observational equivalence Identifiability of an estimand

[F4]

A fixed measurable likelihood with finite positive evidence defines the exact target by normalizing its prior-weighted integral. Exact inferential targets and approximation chains

[F5]

A finite discrete space is standard Borel when its discrete metric is complete and separable and its Borel sigma-algebra is its power set; $R$ is standard Borel because its Euclidean metric is complete and $Q$ is countable and dense. Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric Complete metric space: every Cauchy sequence converges in the space Separability: the existence of an at most countable dense subset Polish spaces are separable completely metrizable spaces Standard Borel spaces $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ $Q$ is countably infinite The rationals embed densely in the reals

[F6]

The TV distance is the supremum of event-probability differences. Total variation distance for probability laws

Counterexample

Given: The finite prior, model, summary, observation, and tolerance in Facts & Assumptions.

1.1givenF5construct

Put the discrete metric on $Θ$ and $Y$ . A Cauchy sequence is eventually constant because its tail has pairwise distances below $1 / 2$ ; each finite space is separable since the whole finite set is a countable dense subset. Every subset is open, so the Borel sigma-algebra is the power set. Thus both are standard Borel by [F5]. The real target is standard Borel by [F5].

1.2F3given

The event $A = {(z, w) : z = 0}$ has probabilities $P_{0} (A) = 1$ and $P_{1} (A) = 0$ . Thus the two full-data laws are not observationally equivalent, and the identity parameter is identifiable by [F3].

1.3F4givenalgebra

The full-data likelihood at $y = (0, 0)$ is $1 / 2$ under $θ = 0$ and $0$ under $θ = 1$ . Its prior evidence is $(1 / 2) (1 / 2) + (1 / 2) 0 = 1 / 4$ , so [F4] gives exact posterior $Π_{y} = δ_{0}$ .

2.1F2step 1.1givenalgebra

For each $θ$ , exactly two pairs $(z, w)$ have $z = θ$ , each with mass $1 / 2$ , so $P_{θ}$ is a probability law; because the parameter sigma-algebra is also the power set, the displayed family is a probability kernel. The map $S$ is nonconstant, since $S (0, 0) = 0$ and $S (0, 1) = 1$ , and it is measurable because every subset of the finite source is measurable.

2.2F1givenstep 1.1algebra

The summary law is Bernoulli $(1 / 2)$ under each parameter. The zero-tolerance ball in $R$ centered at $S (y) = 0$ is ${0}$ , and each parameter has acceptance probability $1 / 2$ , so prior-predictive acceptance is $1 / 2 > 0$ . Conditioning the equal prior on this event leaves $Π_{A B C, 0} {θ = 0} = \frac{(1 / 2) (1 / 2)}{1 / 2} = 1 / 2$ by [F1].

3.1F6step 1.3step 2.2algebra∎

On event ${θ = 0}$ the exact posterior gives mass $1$ and the zero-tolerance ABC target gives mass $1 / 2$ . By [F6], their total-variation distance is at least $1 / 2$ . The full model is identifiable, but the nonconstant summary has parameter-independent law and its ABC target remains the prior.

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Sources

Christian P. Robert, Jean-Marie Cornuet, Jean-Michel Marin, and Natesh S. Pillai, Lack of Confidence in Approximate Bayesian Computation Model Choice (2011) (standard reference, not scraped)
Jarno Lintusaari et al., Fundamentals and Recent Developments in Approximate Bayesian Computation (2017) (standard reference, not scraped)
Paul Fearnhead and Dennis Prangle, Constructing Summary Statistics for Approximate Bayesian Computation: Semi-automatic ABC (2012) (standard reference, not scraped)
Michael G. B. Blum, Approximate Bayesian Computation: A Nonparametric Perspective (2010) (standard reference, not scraped)