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

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Multinomial and residual resampling have unbiased offspring counts

Statement

Fix $N \geq 1$ and nonnegative input weights $a_{1}, \dots, a_{N}$ . Let $p$ be the normalized probability vector, with the uniform fallback when all weights vanish, and let $C_{j}$ be the offspring count from Multinomial and residual offspring resampling. Conditional on the input weights, for either scheme and every $1 \leq j \leq N$ , $E [C_{j} ∣ a_{1}, \dots, a_{N}] = N p_{j} .$ This assertion concerns the counts only; it gives no variance comparison for arbitrary weights.

Facts & Assumptions

Given: The finite weights, $N \geq 1$ , and the resampling procedure in the Statement.

[F1]

Multinomial and residual offspring resampling: Multinomial counts are sums of $N$ categorical indicators with probabilities $p_{j}$ ; residual counts add $⌊ N p_{j} ⌋$ deterministic copies to categorical draws with probabilities proportional to the fractional remainders.

Proof

Given: The hypotheses in the Statement.

1.1F1givenalgebra

In the multinomial branch, $C_{j} = \sum_{ℓ = 1}^{N} 1_{{J_{ℓ} = j}}$ and each indicator has conditional expectation $p_{j}$ by the specified product mass. Linearity of a finite sum gives $E [C_{j} ∣ a] = N p_{j}$ , including the uniform fallback.

1.2F1givenalgebra

In the residual branch, write $q_{j} = ⌊ N p_{j} ⌋$ and $r_{j} = N p_{j} - q_{j}$ . The remainder $R = \sum_{j} r_{j} = N - \sum_{j} q_{j}$ is an integer. If $R = 0$ , every $r_{j} = 0$ and $C_{j} = q_{j} = N p_{j}$ ; if $R > 0$ , then $C_{j} = q_{j} + \sum_{ℓ = 1}^{R} 1_{{K_{ℓ} = j}}$ with each indicator having conditional expectation $r_{j} / R$ , so $E [C_{j} ∣ a] = q_{j} + R (r_{j} / R) = N p_{j}$ .

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 establish the same conditional expected-count identity for both schemes in the procedure, completing the assertion.

Remarks

The proof uses only finite sums of categorical indicators and is choice-free. Both schemes satisfy the expected-count interface used by the particle normalizer induction.

Depends on

Multinomial and residual offspring resampling

Used by

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Sources

Arnaud Doucet and Adam M. Johansen, A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later, §3.4 (standard reference, not scraped)
Simo Särkkä and Lennart Svensson, Bayesian Filtering and Smoothing, 2nd ed., §11.4 (standard reference, not scraped)
Nicolas Chopin, Central Limit Theorem for Sequential Monte Carlo Methods and its Application to Bayesian Inference, §2.1 (standard reference, not scraped)