CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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A particle Gibbs proposal that misses posterior path support

Statement refuted

A particle Gibbs exactness construction remains well-defined with its usual finite importance weights even if its proposal assigns zero probability to a positive-mass posterior path.

Facts & Assumptions

Given: The support requirements in Conditional particle filter for particle Gibbs and Feynman–Kac path Radon–Nikodym weight.

[F1]

Conditional particle filter for particle Gibbs: its reference path and proposal weights must be defined under the full target-path support condition.

[F2]

Feynman–Kac path Radon–Nikodym weight: the target path measure is absolutely continuous with respect to the proposal path measure and the displayed importance weight is its Radon–Nikodym density.

Counterexample

Use the one-time path space ${0, 1}$ with posterior $π = (δ_{0} + δ_{1}) / 2$ and proposal $q = δ_{0}$ . The posterior assigns positive mass to state $1$ , but the proposal assigns it zero mass.

1.1F1F2given

Since $q ({1}) = 0$ while $π ({1}) = 1 / 2$ , $π ≪̸ q$ . Therefore the Radon–Nikodym weight required by [F2] does not exist on the full target support, so the stated conditional particle filter hypotheses fail.

2.1F1F2step 1.1given∎

Starting from reference state $0$ , every free particle is also $0$ because it is drawn from $q = δ_{0}$ . The particle filter therefore cannot represent the positive posterior mass at state $1$ by a weighted proposal path. The exact conditional particle filter in [F1] is not defined with the required target-to-proposal weight on the full posterior support. Thus the support-free exactness claim is false as stated.

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Sources

Andrieu, Doucet, and Holenstein, Particle Markov chain Monte Carlo methods, Assumption 5 and Theorem 5 (standard reference, not scraped)
Lindsten and Schön, On the use of backward simulation in particle Markov chain Monte Carlo methods (arXiv 1110.2873v2), Assumption 1 and §4.1 (standard reference, not scraped)