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

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The particle Gibbs extended target has the exact smoothing marginal

Statement

Assume the hypotheses of Particle Gibbs extended target and Sequential importance resampling for Feynman–Kac path targets, with multinomial resampling in sampled order, including the finite-horizon support conditions and $0 < L_{θ} < \infty$ for prior-almost every $θ$ , with $0 < C = \int L_{θ} π_{0} (d θ) < \infty$ . Then $φ_{N}$ in Particle Gibbs extended target is a probability measure and its selected $(θ, X_{0 : T}^{K})$ marginal is $C^{- 1} π_{0} (d θ) γ_{θ, T} (d x_{0 : T}),$ the exact joint parameter and smoothing posterior, for every $N \geq 2$ .

Facts & Assumptions

Given: The finite-horizon path model, full-support proposal conditions, SIR particle system, and $N \geq 2$ .

[A1]

The Axiom of Choice: AC is assumed through the path Radon–Nikodym and probability-kernel suppliers.

[F1]

Particle Gibbs extended target: $φ_{N} = C^{- 1} π_{0} (d θ) ψ_{θ} (d u) {\hat{Z}}_{T} (u) W_{T}^{k} (u)$ and $X_{0 : T}^{k}$ is the ancestry-traced path.

[F2]

Sequential importance resampling for Feynman–Kac path targets: the terminal weights sum to one, ${\hat{Z}}_{T}$ is the product of average incremental weights, and the output weighted path measure is ${\hat{π}}_{T}^{N} = \sum_{k} W_{T}^{k} δ_{X_{0 : T}^{k}}$ .

[F3]

Feynman–Kac path Radon–Nikodym weight: the initial and incremental SIR weights convert each proposal path factor into the corresponding Feynman–Kac path factor under the stated support conditions.

[F4]

Feynman–Kac path targets and normalizers: the normalized smoothing path target is $γ_{θ, T} / L_{θ}$ when $0 < L_{θ} < \infty$ .

Proof

Given: [A1] and the hypotheses in the Statement.

1.1A1F1F2F3givenalgebra

Fix a terminal index $k$ and let $b_{0}, \dots, b_{T}$ be its ancestry, so $A_{t}^{b_{t}} = b_{t - 1}$ . On a positive-weight particle system, [F1] has ${\hat{Z}}_{T} W_{T}^{k} = (\prod_{t = 0}^{T - 1} {\overset{ˉ}{w}}_{t}) w_{T}^{b_{T}} / N$ . In the forward SIR law [F2], the selected lineage contributes $\prod_{t = 1}^{T} W_{t - 1}^{b_{t - 1}} R_{t} (x_{t - 1}^{b_{t - 1}}, d x_{t}^{b_{t}})$ and its initial proposal $ν_{0} (d x_{0}^{b_{0}})$ . Substituting $W_{t - 1}^{b_{t - 1}} = w_{t - 1}^{b_{t - 1}} / (N {\overset{ˉ}{w}}_{t - 1})$ cancels all average-weight denominators, giving $φ_{N} (d θ, d u, k) = \frac{π_{0} (d θ) γ_{θ, T} (d x_{0 : T}^{b})}{C N^{T + 1}} \prod_{i \neq b_{0}} ν_{0} (d x_{0}^{i}) \prod_{t = 1}^{T} \prod_{i \neq b_{t}} W_{t - 1}^{A_{t}^{i}} R_{t} (x_{t - 1}^{A_{t}^{i}}, d x_{t}^{i})$ . The selected path factors are $ν_{0} w_{0} = G_{0} μ_{0}$ and $R_{t} w_{t} = G_{t} M_{t}$ by [F3]; all off-path factors are probability proposal/resampling kernels by [F2]. If any average weight is zero, the extended-target mass and selected target path measure are both zero at that step.

2.1F1F2F3F4step 1.1algebra∎

For each fixed lineage $b$ , the right side of step 1.1 integrates over the off-path variables to $C^{- 1} π_{0} (d θ) γ_{θ, T} (d x^{b}) / N^{T + 1}$ because those variables have the normalized SIR conditional law. There are exactly $N^{T + 1}$ index sequences $b$ , and summing over them gives total mass one and selected $(θ, X_{0 : T}^{K})$ marginal $C^{- 1} π_{0} (d θ) γ_{θ, T} (d x_{0 : T})$ , which is the posterior by [F4].

Depends on

Used by

A particle Gibbs proposal that misses posterior path support Counterexample
An invariant particle Gibbs chain can be sticky Example
The conditional particle Gibbs sweep is a Gibbs update Theorem

Cited to discharge well-definedness by Particle Gibbs extended target.

Dependency tree · two levels

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Sources

Andrieu, Doucet, and Holenstein, Particle Markov chain Monte Carlo methods, §4.5, equations (36)–(37), 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), §4.1 and Proposition 1 (standard reference, not scraped)