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Fixed-level full-resampling SIR splitting is unbiased

Statement

Assume AC. Use the full-resampling SIR model in Full-resampling fixed-level SIR splitting with invariant mutations with $N \geq 1$ , positive level probabilities, and each $K_{ℓ}$ supported on $A_{ℓ}$ and invariant for $π_{ℓ} = P (\cdot ∣ A_{ℓ})$ . Then every prefix Feynman–Kac normalizer is $Z_{ℓ} = P (A_{ℓ})$ , and the product normalizer estimate $\hat{p}$ is unbiased: $E [\hat{p}] = P (A_{L}) .$ This conclusion asserts no finite- $N$ variance bound, accuracy guarantee, or uniform-in-rarity efficiency.

Facts & Assumptions

Given: The nested events, positive level probabilities, mutation kernels, full multinomial resampling, and finite particle count in Full-resampling fixed-level SIR splitting with invariant mutations.

[A1]

The Axiom of Choice: AC is inherited through the Feynman–Kac path-law and SIR unbiasedness suppliers.

[F1]

Nested rare-event levels and conditional laws: $π_{ℓ} = P (\cdot ∣ A_{ℓ})$ and $p_{ℓ} = P (A_{ℓ}) / P (A_{ℓ - 1}) > 0$ .

[F2]

Full-resampling fixed-level SIR splitting with invariant mutations: $G_{1} = 1_{A_{1}}$ , $M_{1}$ is the identity, and $G_{ℓ + 1} = 1_{A_{ℓ + 1}}$ follows mutation $K_{ℓ}$ .

[F3]

Feynman–Kac path measures satisfy the forward recursion: The next normalizer is obtained by integrating the next potential against the preceding normalized target and its transition kernel.

[F4]

The sequential particle estimator is unbiased for Feynman–Kac path integrals: Full-resampling SIR estimates each nonnegative Feynman–Kac normalizer without bias when every prefix normalizer is finite and positive.

Proof

1.1A1F1F2given

The initial law is $P$ , $G_{0} \equiv 1$ , and $M_{1}$ is the identity. Therefore the first weighted normalizer is $Z_{1} = P (A_{1})$ , and the normalized state law after the event potential is $π_{1}$ .

2.1F1F3step 1.1giveninduction

Suppose the normalized law after level $ℓ$ is $π_{ℓ}$ and $Z_{ℓ} = P (A_{ℓ})$ . The mutation $K_{ℓ}$ preserves $π_{ℓ}$ , so [F3] gives $Z_{ℓ + 1} = Z_{ℓ} \int 1_{A_{ℓ + 1}} d π_{ℓ} = P (A_{ℓ}) p_{ℓ + 1} = P (A_{ℓ + 1}) .$ The normalized next law is $P (\cdot ∣ A_{ℓ + 1}) = π_{ℓ + 1}$ ; induction proves all prefix normalizers are finite and positive.

3.1A1F2F4step 2.1given∎

Apply [F4] to the SIR model with all prefix normalizers from step 2.1 and the nonnegative terminal test $1$ . It yields $E [\hat{p}] = Z_{L} = P (A_{L})$ . The conclusion is only unbiasedness; no second moment was estimated.

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Sources

Frédéric Cérou, Pierre Del Moral, and Arnaud Guyader, A nonasymptotic theorem for unnormalized Feynman–Kac particle models, Theorem 1.5 (standard reference, not scraped)
Adam M. Johansen, Pierre Del Moral, and Arnaud Doucet, Sequential Monte Carlo Samplers for Rare Events, §§2–3 (standard reference, not scraped)