TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Sequential ABC Radon–Nikodym weights target each tolerance law

Statement

Assume the setup of Sequential ABC importance-resampling with tolerance populations and put $γ_{t} (d θ, d y) = Π (d θ) K (θ, d y) 1_{A_{ϵ_{t}}} (y), π_{t} = γ_{t} / Z_{ϵ_{t}} = {\tilde{Π}}_{ϵ_{t}} .$ For a measure $μ$ , write $μ (f) = \int f d μ$ for bounded nonnegative measurable $f$ . For every bounded nonnegative product-measurable $f : Θ \times Y \to [0, \infty)$ , let $X_{0}^{j} = (Θ_{0}^{j}, Y_{0}^{j})$ , $1 \leq j \leq N$ , be the initial $N$ independent prior-simulator draws with law $H$ and weights $a_{0}^{j} = 1_{A_{ϵ_{0}}} (Y_{0}^{j})$ satisfy $E ⁣ [\frac{1}{N} \sum_{j = 1}^{N} a_{0}^{j} f (X_{0}^{j})] = γ_{0} (f) .$ For each $1 \leq t \leq T$ reached with positive probability, let $Z_{t - 1} = ((Θ_{t - 1}^{i}, Y_{t - 1}^{i})_{i = 1}^{N}, (W_{t - 1}^{i})_{i = 1}^{N})$ be the previous population and let $λ_{t - 1}$ be its law conditional on reaching stage $t$ . Use the standard-Borel population space and probability kernel $ϕ_{t}^{N} (z, \cdot)$ from Sequential ABC importance-resampling with tolerance populations, and assume $Π ≪ ϕ_{t}^{N} (z, \cdot)$ for $λ_{t - 1}$ -almost every $z$ . The kernel Radon–Nikodym lemma Jointly measurable Radon–Nikodym derivatives for probability kernels then supplies a product-measurable derivative; use the finite-valued version specified in Sequential ABC importance-resampling with tolerance populations after changing infinite values to zero. Write this version as $r_{t}^{N} (z, θ) = d Π / d ϕ_{t}^{N} (z, \cdot) (θ)$ for almost every $z$ . Put $a_{t}^{j} = 1_{A_{ϵ_{t}}} (Y_{t}^{j}) r_{t}^{N} (Z_{t - 1}, Θ_{t}^{j})$ for the candidates produced by multinomial ancestor resampling, perturbation, and simulation. For every bounded nonnegative product-measurable $f : Θ \times Y \to [0, \infty)$ , $E ⁣ [\frac{1}{N} \sum_{j = 1}^{N} a_{t}^{j} f (Θ_{t}^{j}, Y_{t}^{j}) | previous population] = γ_{t} (f) almost surely on reaching stage t .$ In particular, the average weight is conditionally unbiased for $Z_{ϵ_{t}}$ . If candidates with positive total weight are resampled by the multinomial or residual scheme from normalized weights, then for every bounded measurable $f$ their conditional empirical mean equals the weighted candidate empirical mean in expectation. No unbiasedness is asserted for the self-normalized finite- $N$ estimate, and a finite sample may have zero total weight or fail to represent positive target mass.

Facts & Assumptions

Given: AC, a prior $Π$ , simulator probability kernel $K$ , tolerance event $A_{ϵ_{t}}$ , a previous weighted population, mixture $ϕ_{t}^{N}$ , $Π ≪ ϕ_{t}^{N}$ , and $Z_{ϵ_{t}} > 0$ .

[A1]

The Axiom of Choice: AC states that every family of nonempty sets has a choice function.

[F1]

Augmented ABC tolerance target: the unnormalized augmented target is $γ_{t} = Π (d θ) K (θ, d y) 1_{A_{ϵ_{t}}} (y)$ with normalizer $Z_{ϵ_{t}} > 0$ and normalized target $π_{t} = γ_{t} / Z_{ϵ_{t}}$ .

[F2]

Sequential ABC importance-resampling with tolerance populations: the initial draws use $H$ with indicator weights; later stages use multinomial ancestor-resampling, perturbation, simulation, the tolerance indicator, and a product-measurable prior-to-mixture derivative evaluated at the previous population and candidate parameter. Its stage-specific support condition holds almost surely under the population law.

[F3]

Jointly measurable Radon–Nikodym derivatives for probability kernels: for probability kernels $M, R$ on standard-Borel spaces, if $M (z, \cdot) ≪ R (z, \cdot)$ for $λ$ -almost every $z$ , there is a product-measurable density that represents $d M (z, \cdot) / d R (z, \cdot)$ for almost every $z$ .

[F4]

For this joint density $h$ , the section identity at $B = Θ$ gives $\int h (z, θ) ϕ_{t}^{N} (z, d θ) = 1$ for almost every $z$ . Thus the set where $h (z, \cdot) = + \infty$ is $ϕ_{t}^{N} (z, \cdot)$ -null there, and setting $h$ to zero on its product-measurable infinite-value set gives a finite-valued product-measurable density version.

[F5]

Integrating against a Radon-Nikodym derivative recovers integration against the measure: if $r = d Π / d ϕ$ , then $Π (B) = \int_{B} r d ϕ$ for every measurable $B$ ; finite simple-function integration follows by finite addition.

[F6]

Every nonnegative measurable function is the increasing limit of simple measurable functions and Monotone convergence for the integral: every nonnegative measurable function is an increasing limit of simple functions, and integrals pass to that limit.

[F7]

Multinomial and residual offspring resampling: the resampling procedure returns a length- $N$ ancestor list encoded by its offspring counts.

[F8]

Multinomial and residual resampling have unbiased offspring counts: conditional on input weights, each offspring count has expectation $N$ times its normalized input weight.

[F9]

A prior and model kernel define a joint law: the prior and simulator kernel define a probability measure on $Θ \times Y$ and integrate nonnegative product-measurable functions by iterated integration.

[F10]

Measurability of integration against a kernel: integrating a nonnegative product-measurable function against a probability kernel gives a measurable section integral.

[F11]

Closure properties of measurable functions used by the integral: finite sums, scalar multiples, and products of measurable functions with measurable indicators remain measurable.

Proof

Given: The hypotheses in the Statement.

1.1F1givenalgebra

Each initial draw $X_{0}^{j}$ has law $H$ and weight $a_{0}^{j} = 1_{A_{ϵ_{0}}} (Y_{0}^{j})$ , so definition of $γ_{0}$ gives $E [a_{0}^{j} f (X_{0}^{j})] = \int f 1_{A_{ϵ_{0}}} d H = γ_{0} (f)$ ; linearity of the finite average proves the identity.

1.2A1F1F3F4F5F6F9F10F11given

Fix $t$ with positive probability of reaching stage $t$ , and let $λ_{t - 1}$ be the conditional law of the previous population. By [F3], the constant kernel $z \mapsto Π$ and the proposal-mixture kernel $ϕ_{t}^{N}$ have a product-measurable density $h (z, θ)$ ; by [F4], its finite version $r_{t}^{N} (z, θ)$ is a derivative for $λ_{t - 1}$ -almost every $z$ . By [F10], $g_{f} (θ) = \int_{Y} 1_{A_{ϵ_{t}}} (y) f (θ, y) K (θ, d y)$ is measurable and bounded. Choose simple functions $s_{n} ↑ g_{f}$ by [F6]; each $s_{n}$ is a finite sum of measurable indicators, so $s_{n} (θ) r_{t}^{N} (z, θ)$ is measurable by [F11]. For almost every $z$ , applying the setwise RN identity [F5] to the level sets of $s_{n}$ gives $\int s_{n} (θ) r_{t}^{N} (z, θ) ϕ_{t}^{N} (z, d θ) = \int s_{n} d Π .$ Since $r_{t}^{N}$ is finite and nonnegative, [F6] and monotone convergence give $\int g_{f} (θ) r_{t}^{N} (z, θ) ϕ_{t}^{N} (z, d θ) = \int g_{f} d Π .$ The right side is $\int f 1_{A_{ϵ_{t}}} d H = γ_{t} (f)$ by the iterated-integral formula [F9] and target definition [F1]. Thus the one-candidate conditional proposal expectation equals $γ_{t} (f)$ for $λ_{t - 1}$ -almost every previous population.

2.1F2F8step 1.2algebra

Given parent $i$ , each child has law $Q_{t} (θ_{t - 1}^{i}, d θ) K (θ, d y)$ ; the multinomial expected count is $N W_{t - 1}^{i}$ by [F8]. Conditional on the previous population $z$ , summing contributions over parents and dividing by $N$ gives the one-candidate expectation under $ϕ_{t}^{N} (z, d θ) K (θ, d y)$ with weight $1_{A_{ϵ_{t}}} (y) r_{t}^{N} (z, θ)$ . Step 1.2 makes this $γ_{t} (f)$ for $λ_{t - 1}$ -almost every $z$ , proving the conditional identity.

3.1F2F7F8algebra∎

For candidate weights $a_{j} = 1_{A_{ϵ_{t}}} (y_{j}) r_{t}^{N} (Z_{t - 1}, θ_{j})$ with $\sum_{j} a_{j} > 0$ , [F8] gives conditional offspring-count means $N a_{j} / \sum_{k} a_{k}$ ; hence for bounded measurable $f$ , $E [\frac{1}{N} \sum_{j} C_{j} f (θ_{t}^{j}, y_{t}^{j}) ∣ (θ_{t}^{i}, y_{t}^{i}, a_{i})_{i = 1}^{N}] = \sum_{j} a_{j} f (θ_{t}^{j}, y_{t}^{j}) / \sum_{k} a_{k}$ , the weighted empirical mean. If the total weight is zero, [F2] reports degeneracy and makes no normalized-target claim.

Depends on

Used by

Cited to discharge well-definedness by Sequential ABC importance-resampling with tolerance populations.

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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, §5.2 and Appendix, Algorithm 4 (standard reference, not scraped)
Benjamin M. Turner and Thomas Van Zandt, A tutorial on approximate Bayesian computation, §3.2 (standard reference, not scraped)