TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck pass

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The event-conditioned proposal is the zero-variance benchmark

Statement

Let $(X, X, μ)$ be a measure space, let $P$ have density $f \geq 0$ with $\int f d μ = 1$ , and let $A \in X$ have $p = P (A) > 0$ . For $q^{*} (x) = 1_{A} (x) f (x) / p$ and $X \sim Q^{*}$ with this density, the ordinary importance-sampling contribution $Z = 1_{A} (X) f (X) / q^{*} (X)$ , defined on the $Q^{*}$ -support, equals $p$ almost surely. Thus the estimator from $n \geq 1$ IID $Q^{*}$ -draws is unbiased and has variance zero. This proposal depends on the unknown $p$ , so it is a benchmark and not a generally available algorithm.

Facts & Assumptions

Given: The probability density $f$ , event $A$ , and $p = P (A) > 0$ in the Statement.

[F1]

Rare-event efficiency and work criteria: The zero-variance event proposal has density $q^{*} (x) = 1_{A} (x) f (x) / p$ .

[F2]

Ordinary importance sampling: Ordinary importance sampling averages the contributions $1_{A} (X_{i}) f (X_{i}) / q (X_{i})$ over IID proposal draws where $q > 0$ .

Proof

1.1F1givenalgebra

The density $q^{*}$ is nonnegative and $\int q^{*} d μ = P (A) / p = 1$ , so it defines a proposal probability measure $Q^{*}$ supported on ${x \in A : f (x) > 0}$ up to a $Q^{*}$ -null set.

2.1F1F2step 1.1givenalgebra∎

For $Q^{*}$ -almost every $X$ , we have $X \in A$ and $f (X) > 0$ . Substitution of $q^{*} (X) = f (X) / p$ into [F2] gives $Z = p$ almost surely. Every term in the IID sample average is therefore the same constant $p$ ; its expectation is $p$ and its variance is zero.

Depends on

Used by

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Dependency tree · two levels

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Sources

Gino Biondini, An Introduction to Rare Event Simulation and Importance Sampling, Handbook of Statistics 33, Chapter 2, §3.3 (standard reference, not scraped)
Patrick Laub, Algorithms for Rare Event Estimation, Short Course, Lecture 1 Part 2, Optimal IS Proposal Density (standard reference, not scraped)