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Finite-horizon bootstrap particle filters are weakly consistent

Statement

Assume AC. Let $E_{0}, \dots, E_{T}$ be compact metric spaces, let $μ_{0}$ be a probability on $E_{0}$ , and let $M_{t}$ be a Feller probability kernel from $E_{t - 1}$ to $E_{t}$ . Here Feller means that $x \mapsto M_{t} h (x) : = \int h (x^{'}) M_{t} (x, d x^{'})$ is continuous for every bounded continuous real $h$ on $E_{t}$ . Let each potential $G_{t} : E_{t} \to R$ be continuous and satisfy fixed bounds $0 < c \leq G_{t} (x) \leq C < \infty$ . Let $P_{t}^{M}$ be the Markov path law and $Z_{t} = \int \prod_{s = 0}^{t} G_{s} (x_{s}) P_{t}^{M} (d x_{0 : t})$ the Feynman–Kac normalizer from Feynman–Kac path targets and normalizers. Since $P_{t}^{M}$ is a probability, $c^{t + 1} \leq Z_{t} \leq C^{t + 1}$ , so the normalized path target exists. For the initial law and transitions use the bootstrap proposals $ν_{0} = μ_{0}$ , $R_{t} = M_{t}$ , initial potential $G_{0}$ , and multinomial resampling in Sequential importance resampling for Feynman–Kac path targets. Write $η_{t}$ for the last-state marginal of $γ_{t} / Z_{t}$ and $η_{t}^{N} = \sum_{i} w_{t}^{i} δ_{X_{t}^{i}}$ for the particle filter's current weighted empirical measure. For every fixed $t \leq T$ , $η_{t}^{N} \Rightarrow η_{t} in probability as N \to \infty,$ where convergence in probability means that for every bounded continuous $h : E_{t} \to R$ , $\int h d η_{t}^{N} \to \int h d η_{t}$ in probability. This is a fixed-horizon, positive-bounded-potential result; it asserts no time-uniform bound and no particle central limit theorem.

Facts & Assumptions

Given: The hypotheses in the Statement, a finite horizon, and the multinomial bootstrap particle system.

[A1]

AC is assumed by the finite-horizon Feynman–Kac path-law convention used to define the target. The Axiom of Choice

[F1]

The bootstrap specialization of the sequential importance-resampling algorithm uses IID initial particles, multinomial parent indices, conditionally independent propagation through $M_{t}$ , and incremental weights $G_{t} (X_{t}^{i})$ . Sequential importance resampling for Feynman–Kac path targets

[F2]

Multinomial parent indices are conditionally IID with the normalized input weights. Multinomial and residual offspring resampling

[F3]

The normalized current-state marginals of the Feynman–Kac path measures obey $η_{0} (h) = μ_{0} (G_{0} h) / μ_{0} (G_{0})$ and $η_{t} (h) = η_{t - 1} (M_{t} (G_{t} h)) / η_{t - 1} (M_{t} G_{t})$ . Feynman–Kac path targets and normalizers Feynman–Kac path measures satisfy the forward recursion

[F4]

Weak convergence of probability measures is convergence of integrals against every bounded continuous function. Weak convergence of borel probability measures

[F5]

For nonnegative measurable $F$ and $a > 0$ , $μ (F \geq a) \leq a^{- 1} \int F d μ$ on any measure space. Chebyshev-Markov inequality for the integral

Proof

Given: The hypotheses in the Statement. The empirical measure before potential weighting at time $t$ is denoted $ν_{t}^{N} = N^{- 1} \sum_{i} δ_{X_{t}^{i}}$ .

1.1A1F1F3F5given

At time zero, for any bounded continuous $h$ , put $V_{i} = G_{0} (X_{0}^{i}) h (X_{0}^{i})$ . Expanding the squared centered sample average, the cross terms vanish by independence and the diagonal terms give $E [(ν_{0}^{N} (G_{0} h) - μ_{0} (G_{0} h))^{2}] \leq C^{2} ∥ h ∥_{\infty}^{2} / N$ . The same calculation with $V_{i} = G_{0} (X_{0}^{i})$ gives the denominator bound $C^{2} / N$ . Applying [F5] to each squared error shows both averages converge in probability to their $μ_{0}$ integrals. The denominator limit is at least $c$ , so the ratio $η_{0}^{N} (h) = ν_{0}^{N} (G_{0} h) / ν_{0}^{N} (G_{0})$ converges in probability to $μ_{0} (G_{0} h) / μ_{0} (G_{0}) = η_{0} (h)$ .

1.2F1F2F4F5given

Suppose $η_{t - 1}^{N} (g) \to η_{t - 1} (g)$ in probability for every bounded continuous $g$ . Conditional on the previous particles, [F1]–[F2] make $X_{t}^{1}, \dots, X_{t}^{N}$ IID with common law $η_{t - 1}^{N} M_{t}$ . For bounded continuous $h$ , the squared error of their empirical average under this product law is the sum of $N$ diagonal centered terms divided by $N^{2}$ ; the cross terms vanish by independence, and each diagonal term is at most $∥ h ∥_{\infty}^{2}$ . Thus its integral is at most $∥ h ∥_{\infty}^{2} / N$ . Applying [F5] under the conditional product law and then integrating over the previous particles gives $ν_{t}^{N} (h) - η_{t - 1}^{N} (M_{t} h) \to 0$ in probability. The Feller hypothesis makes $M_{t} h$ bounded continuous, so the induction hypothesis gives $η_{t - 1}^{N} (M_{t} h) \to η_{t - 1} (M_{t} h)$ .

2.1A1F1F3step 1.1step 1.2givenalgebra∎

Apply step 1.2 to $h$ and $G_{t} h$ , both bounded continuous. Then $η_{t}^{N} (h) = ν_{t}^{N} (G_{t} h) / ν_{t}^{N} (G_{t})$ ; the denominator converges to $η_{t - 1} (M_{t} G_{t}) \geq c$ . The ratio therefore converges in probability to $η_{t - 1} (M_{t} (G_{t} h)) / η_{t - 1} (M_{t} G_{t}) = η_{t} (h)$ by [F3]. Induction from step 1.1 proves the assertion for each fixed $t \leq T$ . The lower bound $c$ rules out zero normalizers, while the upper bound $C$ gives the finite variance bounds used above.

Depends on

Used by

Fixed-horizon particle backward smoothing is consistent under stated conditions Theorem

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Simo Särkkä, Bayesian Filtering and Smoothing, Chapter 7, §§7.3–7.4 (standard reference, not scraped)
Arnaud Doucet and Adam M. Johansen, A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later, §§3.4–3.5 (standard reference, not scraped)