ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Bootstrap and informed proposals on a two-state particle filter

Example

Let $X_{0} = {*}$ and $X_{1} = {a, b}$ with counting sigma-algebras. Set $μ_{0} = ν_{0} = δ_{*}$ , $M_{1} (*, {a}) = M_{1} (*, {b}) = 1 / 2$ , $G_{0} (*) = 1$ , $G_{1} (a) = 1 / 5$ , and $G_{1} (b) = 4 / 5$ . Compare the bootstrap proposal $R_{1} = M_{1}$ with the informed proposal $R_{1} (*, {a}) = 1 / 5$ , $R_{1} (*, {b}) = 4 / 5$ . With one particle, the bootstrap normalizer estimate is random, while the informed proposal gives the exact normalizer at this horizon.

Facts & Assumptions

Given: The finite model in the Example and the one-particle algorithm.

[F1]

Feynman–Kac path targets and normalizers: The path target weights the initial/transition path law by the product of the nonnegative potentials, and normalizes only for a finite positive normalizer.

[F2]

Sequential importance resampling for Feynman–Kac path targets: The initial and incremental particle weights are $G_{0} ρ_{0}$ and $G_{k} r_{k}$ , and the normalizer estimate is the product of their sample averages.

[F3]

Ordinary importance sampling: For a target density $\tilde{p}$ and proposal density $q$ , the ordinary importance weight is proportional to $\tilde{p} / q$ .

[F4]

Ordinary importance sampling is unbiased: Under positive proposal support on the target integrand and integrability, the ordinary importance-sampling estimator has expectation equal to the target expectation.

Verification

Given: Counting measure on ${a, b}$ , one particle, and the model specified above.

1.1F1givenalgebra

The unnormalized path masses are $γ_{1} (a) = M_{1} (*, {a}) G_{1} (a) = 1 / 10$ and $γ_{1} (b) = M_{1} (*, {b}) G_{1} (b) = 2 / 5$ . Thus $Z_{1} = 1 / 2$ and the normalized target masses are $(1 / 5, 4 / 5)$ .

2.1F2step 1.1algebra

For the bootstrap proposal, $ρ_{0} = 1$ and $r_{1} (a) = r_{1} (b) = 1$ . The one-particle incremental weight is therefore $1 / 5$ with probability $1 / 2$ and $4 / 5$ with probability $1 / 2$ , so its expectation is $1 / 2$ but it is not constant.

2.2F2step 1.1algebra

For the informed proposal, $r_{1} (a) = (1 / 2) / (1 / 5) = 5 / 2$ and $r_{1} (b) = (1 / 2) / (4 / 5) = 5 / 8$ . Multiplication by $G_{1}$ gives $G_{1} (a) r_{1} (a) = G_{1} (b) r_{1} (b) = 1 / 2$ , so the one-particle normalizer estimate equals $Z_{1}$ for either sampled state.

3.1F3F4step 1.1step 2.1step 2.2algebra∎

Against counting measure, the unnormalized target density is $\tilde{p} (a) = 1 / 10$ , $\tilde{p} (b) = 2 / 5$ , and both proposal densities are strictly positive, so the support and finite-integrability hypotheses of [F4] hold. Setting $f = 1$ in [F4] gives $E_{Q} [\tilde{p} (X) / q (X)] = Z_{1}$ after multiplying the unbiased normalized estimate by $Z_{1}$ ; the explicit bootstrap and informed weights above are exactly these ordinary importance-sampling ratios.

Remarks

The informed proposal is tailored to this single fixed potential and horizon. This finite example makes no claim that such a proposal is generally available or computationally cheaper.

Depends on

Used by

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Sources

Arnaud Doucet and Adam M. Johansen, A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later, §§3.3–3.5 (standard reference, not scraped)
Simo Särkkä and Lennart Svensson, Bayesian Filtering and Smoothing, 2nd ed., §§11.2–11.5 (standard reference, not scraped)
Nicolas Chopin, Central Limit Theorem for Sequential Monte Carlo Methods and its Application to Bayesian Inference, §2.2 (standard reference, not scraped)