LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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A bounded-potential SIR update has a uniform unnormalized variance bound

Statement

Let $ν$ be a probability measure on $(E, E)$ , $M$ a probability kernel on $E$ , $G, f : E \to [0, 1]$ measurable, $z \in [0, 1]$ , and $N \geq 1$ . Draw $Y_{1}, \dots, Y_{N}$ IID from the probability law $ν M$ . Put $a = N^{- 1} \sum_{i = 1}^{N} G (Y_{i})$ . If $a > 0$ , set $η^{N} = \frac{\sum_{i = 1}^{N} G (Y_{i}) δ_{Y_{i}}}{N a};$ if $a = 0$ , let $η^{N}$ be any probability measure. Define the one-step unnormalized estimate $Γ^{N} (f) = z a η^{N} (f)$ . Then $E [Γ^{N} (f)] = z ν M (G f), Var (Γ^{N} (f)) \leq \frac{z^{2}}{4 N} .$ The estimate is the normalizer update $z a$ multiplied by the weighted empirical mean of $f$ . The zero-weight fallback does not affect $Γ^{N}$ .

Facts & Assumptions

Given: The probability measure $ν$ , probability kernel $M$ , measurable functions $G, f$ valued in $[0, 1]$ , $z \in [0, 1]$ , and $N$ independent proposals from $ν M$ .

[F1]

Measure kernel and probability kernel: Integrating a probability measure against a probability kernel defines a probability law $ν M$ .

[F2]

Unbiasedness and exact variance of the IID Monte Carlo mean: The mean of an IID sample average is the common mean, and its variance is the common variance divided by $N$ when the variables are square-integrable.

[F3]

Moments, variance, and covariance on a probability space: For square-integrable $H$ , $Var (H) = E [H^{2}] - (E H)^{2}$ .

Proof

1.1givenalgebra

If $a > 0$ , then $z a η^{N} (f) = \frac{z}{N} \sum_{i = 1}^{N} G (Y_{i}) f (Y_{i})$ by the definition of $η^{N}$ . If $a = 0$ , each nonnegative $G (Y_{i}) = 0$ , so both sides equal zero for any fallback $η^{N}$ . Thus the identity holds on every proposal outcome.

2.1F1F2F3step 1.1givenalgebra∎

The variables $H_{i} = G (Y_{i}) f (Y_{i})$ are IID and lie in $[0, 1]$ . For $H = H_{1}$ , $H^{2} \leq H$ , so [F3] gives $Var (H) \leq E H - (E H)^{2} \leq 1 / 4$ , since $u (1 - u) \leq 1 / 4$ for $u \in [0, 1]$ . By [F2], the average $N^{- 1} \sum_{i} H_{i}$ has mean $ν M (G f)$ and variance at most $1 / (4 N)$ . Multiplication by $z$ and step 1.1 give both claims.

Source notes

The cited rare-event SMC papers motivate the particle update; this exact one-step bound is proved locally from the canonical IID sample-mean moments, including the zero-weight branch.

Depends on

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Sources

Adam M. Johansen, Pierre Del Moral, and Arnaud Doucet, Sequential Monte Carlo Samplers for Rare Events, §§2–3 (standard reference, not scraped)
Frédéric Cérou, Pierre Del Moral, and Arnaud Guyader, A nonasymptotic theorem for unnormalized Feynman–Kac particle models, §§1–2 (standard reference, not scraped)