ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generated

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Delayed information creates fixed-lag smoothing bias

Example

Let $X_{0} \sim Bernoulli (1 / 2)$ and set $X_{1} = X_{0}$ and $X_{2} = X_{1}$ . Let $Y_{1} = 0$ deterministically and $Y_{2} = X_{2}$ . For target time $t = 0$ and lag $L = 1$ , the fixed-lag procedure conditions only on $Y_{1}$ , whereas the full smoother also uses $Y_{2}$ . In the exact-kernel mode of Fixed-lag particle smoother, the lagged law is $Bernoulli (1 / 2)$ while, for either observed value $Y_{2} = b$ , the full smoothing marginal at time $0$ is $δ_{b}$ . Their total-variation distance under the convention $\sup_{A} ∣ μ (A) - ν (A) ∣$ is $1 / 2$ . Increasing the number of particles does not remove this truncation bias at fixed lag.

Facts & Assumptions

Given: The finite model, $t = 0$ , and $L = 1$ in the Example.

[A1]

AC is inherited through the finite-horizon conditional-law convention for the supplied smoothing kernels. The Axiom of Choice

[F1]

A smoothing law is a conditional law of the full state path given the specified observation history. Filtering, predictive, and smoothing laws in a finite-horizon state-space model

[F2]

The backward conditional-kernel mode returns the exact smoother for the observations it is given, and fixed lag uses only $Y_{1 : \min (t + L, T)}$ . Fixed-lag particle smoother

[F3]

Exact backward conditional kernels factor the finite-horizon smoother. Bayes factorization of the smoothing backward kernel

[F4]

The exact-kernel fixed-lag branch targets the smoothing marginal with the truncated observation history. Exact-kernel fixed-lag smoothing targets the truncated conditional law

Verification

Given: $P (X_{0} = 0) = P (X_{0} = 1) = 1 / 2$ , $X_{1} = X_{2} = X_{0}$ , $Y_{1} = 0$ , and $Y_{2} = X_{2}$ .

1.1A1F1F2F4givenalgebra

Since $Y_{1}$ is constant, conditioning on $Y_{1}$ leaves the prior law of $X_{0}$ unchanged. The time- $0$ target for lag $1$ uses $s_{0} = \min (0 + 1, 2) = 1$ , so [F2] and [F4] give the lagged law $Bernoulli (1 / 2)$ .

1.2A1F1F3givenalgebra

For each $b \in {0, 1}$ , the event $Y_{2} = b$ has probability $1 / 2$ . On this event, $X_{0} = X_{1} = X_{2} = b$ surely. Thus the full smoothing path law is $δ_{(b, b, b)}$ , and its time- $0$ marginal is $L (X_{0} ∣ Y_{1} = 0, Y_{2} = b) = δ_{b}$ .

2.1step 1.1step 1.2algebra∎

For $A = {b}$ , the probabilities under the lagged law and the full smoothing marginal at time $0$ are $1 / 2$ and $1$ , respectively; for any subset $A \subseteq {0, 1}$ their probability difference has absolute value at most $1 / 2$ . Hence $\sup_{A} ∣ Bernoulli (1 / 2) (A) - δ_{b} (A) ∣ = 1 / 2$ . This difference is between the truncated and full targets at time $0$ , so particle-count growth alone cannot remove it.

Depends on

Used by

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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)