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Exact-kernel fixed-lag smoothing targets the truncated conditional law

Statement

Assume AC and a finite-horizon Markov state-space model with standard-Borel state and observation spaces. Supply the exact filtering and backward conditional kernels of Bayes factorization of the smoothing backward kernel. Fix an observation history in the full-measure set on which their factorization holds for every prefix horizon $s \leq T$ . In the exact-kernel mode of Fixed-lag particle smoother, fix an integer lag $L \geq 0$ and $t \in {0, \dots, T}$ , and set $s = \min (t + L, T)$ . Each returned path's time- $t$ coordinate has conditional law $L (X_{t} ∣ Y_{1 : s})$ (for the supplied conditional versions), so, for any requested $S \geq 1$ , the empirical measure of its $S$ returned time- $t$ coordinates is a Monte Carlo estimate of that marginal. The target equals the full smoothing marginal only when later observations do not change the conditional law; increasing the particle count cannot remove a difference caused by truncating the observation history.

Facts & Assumptions

Given: A finite-horizon state-space model with supplied exact filtering and backward conditional kernels, integer lag $L \geq 0$ , and target time $t \in {0, \dots, T}$ .

[A1]

AC is inherited through the finite-horizon conditional-law construction that supplies the filtering and backward kernels. The Axiom of Choice

[F1]

The fixed-lag algorithm sets $s = \min (t + L, T)$ , requests $S \geq 1$ paths using observations only through time $s$ , and returns their time- $t$ empirical measure. Fixed-lag particle smoother

[F2]

The exact terminal filter and backward conditional kernels factor the finite-horizon smoothing law. Bayes factorization of the smoothing backward kernel

Proof

Given: The hypotheses in the Statement.

1.1A1F2givenalgebra

Since $L \geq 0$ and $0 \leq t \leq T$ , the integer $s = \min (t + L, T)$ satisfies $0 \leq t \leq s \leq T$ . Apply [F2] at horizon $s$ to the observation history $Y_{1 : s}$ . Its terminal filtering kernel followed by the supplied backward conditional kernels gives a path with law $L (X_{0 : s} ∣ Y_{1 : s})$ .

2.1A1F1F2step 1.1given∎

Each exact-kernel draw in [F1] has the path law from step 1.1 and returns its coordinate projection $X_{t}$ . A coordinate projection of a random element with the path conditional law has the corresponding marginal conditional law, hence each draw has law $L (X_{t} ∣ Y_{1 : s})$ ; averaging the $S$ point masses gives the stated Monte Carlo estimate. If conditioning on additional observations changes this marginal, the exact branch still conditions only on $Y_{1 : s}$ , so increasing either the particle count or the number of backward draws cannot change its target.

Depends on

Used by

Delayed information creates fixed-lag smoothing bias Example

Dependency tree · two levels

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Sources

Arnaud Doucet and Adam M. Johansen, A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later, §5.1 (standard reference, not scraped)