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

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A zero backward-simulation denominator is inadmissible

Example

Take $E_{0} = E_{1} = {0, 1}$ with counting measure, initial law $δ_{0}$ , and deterministic transition $X_{1} = X_{0}$ . Its transition density is $f_{1} (x^{'} ∣ x) = 1$ when $x^{'} = x$ and $0$ otherwise. A supplied pair of particle clouds with time-zero particle $x_{0}^{1} = 0, w_{0}^{1} = 1$ and terminal particle $x_{1}^{1} = 1, w_{1}^{1} = 1$ has backward denominator $\sum_{j = 1}^{1} w_{0}^{j} f_{1} (x_{1}^{1} ∣ x_{0}^{j}) = 0.$ This terminal cloud is inconsistent with the transition and filtering law; the particle backward algorithm must report its density-mode input as inadmissible rather than inventing uniform backward weights.

Facts & Assumptions

Given: The two-state deterministic model and the supplied particle clouds in the Example.

[A1]

AC is inherited through the finite-horizon conditional-law convention used for the impossible conditioning fiber. The Axiom of Choice

[F1]

Conditional kernels and smoothing laws are versions on standard-Borel spaces; values on impossible conditioning fibers are not fixed by the conditional identities. Filtering, predictive, and smoothing laws in a finite-horizon state-space model

[F2]

A density-based backward kernel is normalized only when its predictive denominator is finite and positive. Bayes factorization of the smoothing backward kernel

[F3]

The particle backward algorithm requires finite positive denominators and declares other particle-mode inputs inadmissible. Particle backward simulator

Verification

Given: $η_{0} = δ_{0}$ , $f_{1} (x^{'} ∣ x) = 1 {x^{'} = x}$ , and the particle weights and states in the Example.

1.1F2F3givenalgebra

Substitution gives $f_{1} (1 ∣ 0) = 0$ and $w_{0}^{1} = 1$ , so the unique backward score is zero and its normalizing denominator is zero. No division by this quantity defines a probability vector.

2.1A1F1givenstep 1.1

The exact predictive law is $η_{0} M_{1} = δ_{0}$ , hence the terminal state $1$ has predictive probability zero. By [A1] and [F1] a conditional version on this null fiber is not determined by the model's conditional identities, and the supplied cloud at $1$ is not a possible output of the stated filter.

3.1A1F1F3step 1.1step 2.1∎

Because [F3] restricts the density-based backward draw to a finite positive denominator, the procedure rejects this input; a uniform choice would be an unsupported choice of conditional version, not a consequence of Bayes' formula.

Depends on

Used by

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Sources

Simo Särkkä, Bayesian Filtering and Smoothing, Chapter 11, §11.2 (standard reference, not scraped)
Arnaud Doucet and Adam M. Johansen, A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later, §5.2 (standard reference, not scraped)