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Feynman–Kac path measures satisfy the forward recursion

Statement

Assume AC and the finite-horizon setup of Feynman–Kac path targets and normalizers. For $1 \leq k \leq T$ and every nonnegative product-measurable $φ : X_{0 : k} \to [0, \infty]$ , $γ_{k} (φ) = \int_{X_{0 : k - 1}} γ_{k - 1} (d x_{0 : k - 1}) \int_{X_{k}} G_{k} (x_{k}) φ (x_{0 : k}) M_{k} (x_{k - 1}, d x_{k}),$ with equality in $[0, \infty]$ .

Facts & Assumptions

Given: AC, the finite-horizon path measures and potentials in the Statement, and $k \geq 1$ .

[F1]

Feynman–Kac path targets and normalizers: $γ_{k}$ is the path law $μ_{0} \prod_{j = 1}^{k} M_{j}$ weighted by the nonnegative product $\prod_{j = 0}^{k} G_{j}$ .

[F2]

Measurability of integration against a kernel: A nonnegative product-measurable integrand has a measurable section integral against a probability kernel.

[F3]

The product sigma-algebra and its finite iterates: Coordinate projections from finite product measurable spaces are measurable.

[F4]

Composition with a Borel measurable outer map preserves measurability: The composition of measurable maps is measurable.

[F5]

Arithmetic and lattice operations preserve measurability whenever they are defined: Finite products of nonnegative measurable functions are measurable.

[F6]

Integrating against a density agrees with integrating the product: If $f, g \geq 0$ are measurable, then $\int g d (f d μ) = \int g f d μ$ , including infinite values.

Proof

Given: The hypotheses in the Statement.

1.1F2F3F4F5given

The coordinate projection $x_{0 : k} \mapsto x_{k}$ is measurable by [F3], so $G_{k} (x_{k})$ is measurable by [F4]. Its product with the nonnegative product-measurable $φ$ is measurable by [F5]; hence $H (x_{0 : k - 1}) = \int G_{k} (x_{k}) φ (x_{0 : k}) M_{k} (x_{k - 1}, d x_{k})$ is measurable by [F2].

2.1F1F6step 1.1∎

By [F1], the finite path law at time $k$ is obtained by appending $M_{k}$ to the time- $(k - 1)$ path law, and $γ_{k - 1} = V_{k - 1} d P_{k - 1}^{M}$ . Applying [F6] to $H$ and then expanding the ordered kernel integral gives $\int H d γ_{k - 1} = \int H V_{k - 1} d P_{k - 1}^{M} = γ_{k} (φ)$ . These are nonnegative integrals, so the identity holds in $[0, \infty]$ and is the stated recursion.

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Sources

Arnaud Doucet and Adam M. Johansen, A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later, §§3.3–3.5 (standard reference, not scraped)
Simo Särkkä and Lennart Svensson, Bayesian Filtering and Smoothing, 2nd ed., §§11.2–11.5 (standard reference, not scraped)
Nicolas Chopin, Central Limit Theorem for Sequential Monte Carlo Methods and its Application to Bayesian Inference, §§2.1–2.2 (standard reference, not scraped)