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Uniform SMC backward-operator contraction controls oscillation

Statement

Let $E_{s}$ be the forward operators of Finite-horizon SMC CLT classes and variance recursions, and for bounded real $h$ put $osc (h) = \sup h - \inf h$ . Suppose there is one $β \in [0, 1)$ such that $osc (E_{s} h) \leq β osc (h)$ for every $s$ and every bounded real $h$ with $π_{s} (h) = 0$ , with bounded measurable versions of $E_{s} h$ . The stronger condition on every bounded $h$ is sufficient but is not necessary here. For bounded $ϕ : Θ_{t} \to R$ , put $Δ ϕ = osc (ϕ)$ and $\overset{ˉ}{ϕ} = ϕ - π_{t} (ϕ)$ . Then for every $0 \leq k \leq t$ , $∥ E_{k + 1 : t} \overset{ˉ}{ϕ} ∥_{\infty} \leq β^{t - k} Δ ϕ,$ with $β^{0} = 1$ and $E_{t + 1 : t}$ the identity. This condition concerns the actual SMC backward operators; it is not implied merely by mixing ratios of the proposal densities.

Facts & Assumptions

Given: The SMC flow and operators in the Statement, bounded real $ϕ$ , and the uniform operator contraction hypothesis.

[A1]

The Axiom of Choice: AC is inherited through the finite-horizon path and Radon–Nikodym construction.

[F1]

Finite-horizon SMC CLT classes and variance recursions: $E_{s} h = k_{s} (υ_{s} h)$ , ${\tilde{π}}_{s} = π_{s - 1} k_{s}$ , and ${\tilde{π}}_{s} (υ_{s} h) = π_{s} (h)$ .

Proof

Given: The hypotheses in the Statement.

1.1A1F1given

For any bounded measurable $h$ and $s \geq 1$ , [F1] gives $π_{s - 1} (E_{s} h) = π_{s - 1} (k_{s} (υ_{s} h)) = {\tilde{π}}_{s} (υ_{s} h) = π_{s} (h) .$ Iterating this identity yields $π_{k} (E_{k + 1 : t} \overset{ˉ}{ϕ}) = π_{t} (\overset{ˉ}{ϕ}) = 0$ for every $k < t$ ; for $k = t$ the identity operator also gives mean zero.

2.1F1step 1.1given∎

Repeated application of the assumed oscillation contraction gives $osc (E_{k + 1 : t} \overset{ˉ}{ϕ}) \leq β^{t - k} osc (\overset{ˉ}{ϕ}) = β^{t - k} Δ ϕ$ . By step 1.1 its $π_{k}$ -mean is zero, so its infimum is at most zero and its supremum at least zero; hence its sup norm is at most its oscillation. For $k = t$ this is $∥ \overset{ˉ}{ϕ} ∥_{\infty} \leq Δ ϕ$ .

Depends on

Used by

Time-uniform SMC asymptotic variance under uniform weight and operator bounds Theorem

Dependency tree · two levels

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Sources

Nicolas Chopin, Central Limit Theorem for Sequential Monte Carlo Methods and its Application to Bayesian Inference, §3.3 and Appendix A.4 (standard reference, not scraped)
Arnaud Doucet and Adam M. Johansen, A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later, §3.6 (standard reference, not scraped)