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Exact state-space filtering recursion under dominated observations

Statement

Assume AC and a finite-horizon model as in General finite-horizon state-space models by kernels. For each $1 \leq t \leq T$ , let $ν_{t}$ be a sigma-finite measure on $(F_{t}, F_{t})$ and suppose the observation kernel has a specified product-measurable density $g_{t} : E_{t} \times F_{t} \to [0, \infty]$ with respect to $E_{t} \otimes F_{t}$ : $G_{t} (x, B) = \int_{B} g_{t} (x, y) ν_{t} (d y), \int_{F_{t}} g_{t} (x, y) ν_{t} (d y) = 1$ for every state $x \in E_{t}$ and measurable $B \subseteq F_{t}$ . For $h \in F_{1 : t - 1}$ , let $π_{t - 1}$ be the filtering kernel at time $t - 1$ from Filtering, predictive, and smoothing laws in a finite-horizon state-space model when $t > 1$ , and for $t = 1$ let $π_{0}$ be the initial kernel $μ_{0}$ on the singleton empty-history space. For $A \in E_{t}$ , define the predictive kernel ${\hat{π}}_{t} (h, A) : = \int_{E_{t - 1}} K_{t} (x, A) π_{t - 1} (h, d x),$ the evidence density $z_{t} (h, y) : = \int_{E_{t}} g_{t} (x, y) {\hat{π}}_{t} (h, d x),$ and for $h \in F_{1 : t - 1}$ , $y \in F_{t}$ , and $A \in E_{t}$ define the update $π_{t} (h, y, A) : = {\begin{cases} \frac{\int_{A} g_{t} (x, y) {\hat{π}}_{t} (h, d x)}{z_{t} (h, y)}, & 0 < z_{t} (h, y) < \infty, \\ {\hat{π}}_{t} (h, A), & z_{t} (h, y) = 0 or z_{t} (h, y) = \infty . \end{cases}$ Then ${\hat{π}}_{t}$ is a predictive kernel for $X_{t}$ given $Y_{1 : t - 1}$ , and $π_{t}$ is a filtering kernel for $X_{t}$ given $Y_{1 : t}$ . For each fixed history $h$ , the observation marginal under the conditional version determined by ${\hat{π}}_{t} (h, \cdot)$ has density $z_{t} (h, \cdot)$ with respect to $ν_{t}$ and assigns zero mass to both ${y : z_{t} (h, y) = 0}$ and ${y : z_{t} (h, y) = \infty}$ . Thus the formula never divides by zero or infinity; values on those fibers are conditional-law fillers. The statement is exact and finite-horizon and gives no numerical approximation or long-time stability claim.

Facts & Assumptions

Given: AC; the finite-horizon state-space model; for every $1 \leq t \leq T$ , a sigma-finite $ν_{t}$ and product-measurable $g_{t} \geq 0$ satisfying the displayed kernel-density identity and pointwise normalization; and the stated time- $(t - 1)$ filtering kernel.

[A1]

AC states that every family of nonempty sets has a choice function. The Axiom of Choice

[F1]

The model path law has the ordered factorization $μ_{0} (d x_{0}) \prod_{s = 1}^{T} K_{s} (x_{s - 1}, d x_{s}) G_{s} (x_{s}, d y_{s})$ . Finite-horizon state-space kernel law exists and is unique

[F2]

A filtering kernel composed with the observation history is an RCD of the current state given that history; the initial filtering kernel is $μ_{0}$ on the singleton empty history. Filtering, predictive, and smoothing laws in a finite-horizon state-space model

[F3]

Integrating a nonnegative measurable function through an RCD gives the conditional integral identities for every conditioning event. Conditional integration through a regular conditional law

[F4]

Integrating a nonnegative product-measurable function against a probability kernel gives a measurable function of the source variable. Measurability of integration against a kernel

[F5]

A probability kernel has pointwise probability-measure sections and measurable evaluations. Measure kernel and probability kernel

[F6]

For a proper prior probability and a jointly measurable normalized kernel density, the dominated Bayes kernel has density-normalizer marginal; it uses the quotient only at finite positive evidence and fills both zero and infinite evidence fibers with the supplied prior probability. Bayes formula for dominated kernels

[F7]

Measurable rectangles generate the finite product sigma-algebra. The product sigma-algebra and its finite iterates

[F8]

A pi-system generating a sigma-algebra determines finite measures and extends equality of finite measures from the pi-system. Dynkin's pi-lambda theorem

[F9]

Expectations of measurable functions of a random element equal their integrals against its law. Change of variables for expectation

[F10]

Increasing limits pass through nonnegative integrals. Monotone convergence for the integral

[F11]

Under AC, the required data-indexed filtering kernels and smoothing kernel exist on the standard-Borel state and observation spaces. Filtering and finite-horizon smoothing laws exist

[F12]

The recursive filtering procedure defines prediction, evidence, and finite-positive updates with the predictive measure as the zero/infinite evidence filler. Recursive Bayesian filtering under dominated observations

Proof

Given: The hypotheses in the Statement.

1.1A1F2F4F5F10F11given

For $t = 1$ use $π_{0} = μ_{0}$ ; for $t > 1$ use the filtering kernel supplied by [F11] under [A1], and define ${\hat{π}}_{t}$ by the displayed mixture. Each section has mass $\int K_{t} (x, E_{t}) π_{t - 1} (h, d x) = 1$ ; countable additivity follows by applying [F10] to increasing partial sums for disjoint target sets, and measurability in $h$ follows from [F4].

2.1F2F4F5step 1.1given

Regard $(h, y) \mapsto {\hat{π}}_{t} (h, d x)$ as a probability kernel constant in $y$ . Product measurability of $(h, y, x) \mapsto g_{t} (x, y)$ and [F4] make $z_{t}$ and each numerator measurable. On $0 < z_{t} < \infty$ the numerator is a finite measure of total mass $z_{t}$ , so division gives a probability; off that set the predictive measure is a probability. Thus $π_{t}$ is an everywhere probability kernel with measurable evaluations, including zero and infinite fibers.

2.2A1F1F2F3F7F8F11step 1.1

For a history rectangle $H$ and measurable $A \subseteq E_{t}$ , inserting $K_{t}$ at the transition factor of [F1] gives $P_{T} (H \cap {X_{t} \in A}) = E_{P_{T}} [1_{H} K_{t} (X_{t - 1}, A)]$ , since later kernels integrate to one; at $t = 1$ this is the one-step transition integral. These are finite measures of $H$ , so [F7]–[F8] extend equality from history rectangles to all $H \in σ (Y_{1 : t - 1})$ . Then [F3] for the filtering kernel from [F11] turns the right side into $E_{P_{T}} [1_{H} {\hat{π}}_{t} (Y_{1 : t - 1}, A)]$ , proving that ${\hat{π}}_{t}$ is predictive.

3.1F6step 1.1step 2.1

Fix $h$ . The prior ${\hat{π}}_{t} (h, \cdot)$ is a probability and $G_{t}$ has the specified jointly measurable normalized density, so [F6] gives observation marginal $z_{t} (h, y) ν_{t} (d y)$ and conditional kernel $π_{t} (h, y, \cdot)$ ; the zero and infinite evidence fibers have zero marginal. In particular, $\int_{C} π_{t} (h, y, A) z_{t} (h, y) ν_{t} (d y) = \int_{A} G_{t} (x, C) {\hat{π}}_{t} (h, d x)$ .

4.1A1F1F2F3F6F7F8step 2.1step 2.2step 3.1

For a history rectangle $B$ and an observation set $C \in F_{t}$ , the path law [F1], predictive identity from step 2.2, and conditional integration [F3] applied to $x \mapsto 1_{A} (x) G_{t} (x, C)$ give $P_{T} (Y_{1 : t - 1} \in B, X_{t} \in A, Y_{t} \in C) = \int_{B} \int_{A} G_{t} (x, C) {\hat{π}}_{t} (h, d x) P_{Y_{1 : t - 1}} (d h)$ . By step 3.1 this is $\int_{B} \int_{C} π_{t} (h, y, A) z_{t} (h, y) ν_{t} (d y) P_{Y_{1 : t - 1}} (d h)$ . Taking $A = E_{t}$ and using $π_{t} (h, y, E_{t}) = 1$ from step 2.1 identifies the law of $(Y_{1 : t - 1}, Y_{t})$ on history-observation rectangles with $P_{Y_{1 : t - 1}} (d h) z_{t} (h, y) ν_{t} (d y)$ ; [F7]–[F8] extend this to the product sigma-algebra.

5.1A1F2F3F7F8F9F11F12step 2.1step 2.2step 4.1∎

Step 2.1 makes $π_{t} (\cdot, \cdot, A)$ measurable and bounded. The joint-observation law just established and [F9] therefore give, on each history-observation rectangle $B \times C$ , $E_{P_{T}} [1_{{Y_{1 : t - 1} \in B, Y_{t} \in C}} π_{t} (Y_{1 : t - 1}, Y_{t}, A)] = \int_{B} \int_{C} π_{t} (h, y, A) z_{t} (h, y) ν_{t} (d y) P_{Y_{1 : t - 1}} (d h) .$ By step 4.1 this is $P_{T} (Y_{1 : t - 1} \in B, X_{t} \in A, Y_{t} \in C)$ . Such rectangles generate $σ (Y_{1 : t})$ by [F7], and for fixed $A$ both sides are finite measures of the conditioning event, so [F8] extends the identity to every event in $σ (Y_{1 : t})$ . Thus $π_{t}$ is a filtering kernel; step 2.2 already proves prediction, and the procedure [F12] returns the claimed kernels.

Depends on

Used by

Cited to discharge well-definedness by Recursive Bayesian filtering under dominated observations.

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Sources

Simo Särkkä and Lennart Svensson, Bayesian Filtering and Smoothing, 2nd ed., Chapter 6 (standard reference, not scraped)
Arnaud Doucet and Adam M. Johansen, A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later, §§2.1–2.3 (standard reference, not scraped)