TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Independent cross-entropy production is conditionally unbiased

Statement

Use Independent-pilot cross-entropy proposal fitting and production. Let $G = σ (Y_{1}, \dots, Y_{m})$ and let $H \in G$ be the event that the pilot succeeds and the algorithm's exact admissibility checker accepts the selected parameter $\hat{θ}$ . On $H$ , the checker guarantees $P ∣_{A} ≪ Q_{\hat{θ}}$ and integrability under $Q_{\hat{θ}}$ of the piecewise contribution $w_{\hat{θ}}$ defined in the algorithm. Define ${\tilde{p}}_{C E}$ to equal the returned production estimate on $H$ and zero off $H$ . Then

E [{\tilde{p}}_{C E} ∣ G] = 1_{H} P (A) almost surely .

In particular, the production estimate is conditionally unbiased almost surely on the successful, admissible pilot event. This gives no probability-of-success, proposal-optimization, finite-variance, logarithmic-efficiency, or total-work guarantee.

Facts & Assumptions

Given: The pilot sigma-algebra $G$ , the successful and admissible event $H$ , the frozen parameter $\hat{θ}$ , the independent production sample, and the support and integrability conditions for $w_{\hat{θ}}$ certified by the algorithm's checker.

[F1]

Independent-pilot cross-entropy proposal fitting and production: After a successful pilot the selected $\hat{θ}$ is frozen, and the production sample is IID from $Q_{\hat{θ}}$ , independent of the pilot conditional on that parameter.

[F2]

Ordinary importance sampling is unbiased: For a fixed proposal with support on the target event and an integrable weighted contribution, the ordinary importance-sampling mean equals the target expectation.

Proof

1.1F1given

The pilot outcome space need not be finite. For each $θ \in Θ$ , the score $S (θ)$ is $G$ -measurable, and $Θ$ is finite. Therefore the success event, the exact-checker acceptance event, and the lexicographically least maximizing selector $\hat{θ}$ on success are $G$ -measurable. Partition $H$ into the finitely many events $H_{θ} = H \cap {\hat{θ} = θ}$ . Conditional independence of the production sample from the pilot given $\hat{θ}$ , as specified in [F1], means that on each $H_{θ}$ its conditional law given $G$ is IID $Q_{θ}$ .

2.1F2step 1.1givenalgebra∎

On each $H_{θ}$ , the checker gives the support and integrability hypotheses for [F2]. That theorem gives conditional expectation $P (A)$ for each production contribution given $G$ , and linearity gives the same conditional expectation for their average. Since the finitely many $H_{θ}$ partition $H$ and ${\tilde{p}}_{C E} = 0$ off $H$ , this proves $E [{\tilde{p}}_{C E} ∣ G] = 1_{H} P (A)$ almost surely.

Depends on

Used by

A pilot with no rare-event hit returns failure Example

Dependency tree · two levels

9 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

Patrick Laub, Algorithms for Rare Event Estimation, Short Course, Lecture 2, Cross-Entropy Method (standard reference, not scraped)
Gino Biondini, An Introduction to Rare Event Simulation and Importance Sampling, Handbook of Statistics 33, Chapter 2, `5 (standard reference, not scraped)