State-space models and exact Bayesian filtering

Assume AC and fix a finite horizon $T\ge 0$ with a state-space model by probability kernels as in General finite-horizon state-space models by kernels. If $T=0$, put ${\mathrm{\Omega }}_T=E_0$; otherwise put ${\mathrm{\Omega }}_T=E_0\times E_1\times F_1\times \cdots \times E_T\times F_T$. Let ${\mathcal{Z}}_T$ be the corresponding finite-product sigma-algebra and write $({\mathrm{\Omega }}_T,{\mathcal{Z}}_T,P_T)$ for its finite-horizon probability space. The law $P_T$ is the unique law supplied by Finite-horizon state-space kernel law exists and is unique. Let $X_t$ and $Y_t$ be the coordinate random elements taking values in $E_t$ and $F_t$. For $t\ge 1$, write $Y_{1:t}:=(Y_1,\dots ,Y_t)$ and ${\mathsf{F}}_{1:t}:=F_1\times \cdots \times F_t$ with its finite-product sigma-algebra; let ${\mathsf{F}}_{1:0}$ be a singleton and $Y_{1:0}$ its constant map. The factors are standard Borel, meaning measurably isomorphic to Polish spaces (Standard Borel spaces), and their finite products are standard Borel (Finite products of standard Borel spaces are standard Borel).

The initial filtering kernel ${\pi }_0:{\mathsf{F}}_{1:0}\to E_0$ assigns ${\mu }_0$ to the unique empty history; it is the conditional law of $X_0$ given the trivial history. For $1\le t\le T$, a filtering kernel at time $t$ is a probability kernel ${\pi }_t:{\mathsf{F}}_{1:t}\to E_t$ such that $\omega \mapsto {\pi }_t(Y_{1:t}(\omega ),\cdot )$ is a regular conditional distribution of $X_t$ given $\sigma (Y_{1:t})$. For $1\le t\le T$, a predictive kernel at time $t$ is a probability kernel ${\pi }_{t\mathrm{\mid }t-1}:{\mathsf{F}}_{1:t-1}\to E_t$ whose composition with $Y_{1:t-1}$ is a regular conditional distribution of $X_t$ given $\sigma (Y_{1:t-1})$. A finite-horizon smoothing kernel is a probability kernel ${\psi }_T:{\mathsf{F}}_{1:T}\to E_0\times \cdots \times E_T$ whose composition with $Y_{1:T}$ is a regular conditional distribution of the full state path $(X_0,\dots ,X_T)$ given $\sigma (Y_{1:T})$. Here a regular conditional distribution is an everywhere probability kernel satisfying the conditional event identities in Regular conditional distribution, and a conditional law given a random element is its data-indexed kernel form in Conditional law given a random element.

These kernels are versions: their values can be changed on a measurable set of observation histories of probability zero without changing the conditional identities. No density is assumed in these definitions. The state-space model here is one fixed law. A parametric statistical model is instead a map $\theta \mapsto P_{\theta }$ from a parameter set to probability laws; its measurability is not asserted without a parameter sigma-algebra (Parametric statistical models).

Assume AC and a finite-horizon state-space model by standard-Borel state and observation spaces, an initial probability measure, and transition and observation probability kernels, as in General finite-horizon state-space models by kernels. Let $P_T$ be its finite-horizon path law. The initial filtering kernel is ${\pi }_0={\mu }_0$ on the singleton empty-history space. For each $1\le t\le T$, data-indexed filtering and predictive kernels from Filtering, predictive, and smoothing laws in a finite-horizon state-space model exist. A data-indexed finite-horizon smoothing kernel for the full state path also exists, including when $T=0$. The claim is existence of conditional versions only; it asserts no density, filtering recursion, uniqueness on null observation histories, or numerical approximation.

Assume AC and a finite-horizon model as in General finite-horizon state-space models by kernels. For each $1\le t\le T$, suppose the observation kernel has a supplied product-measurable density $g_t:E_t\times F_t\to [0,\mathrm{\infty }]$ with respect to a sigma-finite measure ${\nu }_t$ on $F_t$, normalized so that $G_t(x,B)={\int }_B{g}_t(x,y){\nu }_t(dy),{\int }_{F_t}{g}_t(x,y){\nu }_t(dy)=1$ for every state $x$ and measurable $B\subseteq F_t$.

The procedure takes the initial law ${\mu }_0$, transition kernels $K_t$, observation densities $g_t$ and measures ${\nu }_t$ as inputs. It defines data-indexed prediction, evidence, and filtering functions on every observation history as follows.

1. Initialize ${\pi }_0$ on the singleton empty-history space by ${\pi }_0(\ast ,A)={\mu }_0(A)$.

2. For $t=1,\dots ,T$, each history $h\in {\mathsf{F}}_{1:t-1}$ and each measurable state set $A\in {\mathcal{E}}_t$, form the predictive measure ${\widehat{\pi }}_t(h,A):={\int }_{E_{t-1}}{K}_t(x,A){\pi }_{t-1}(h,dx)\mathrm{.}$

3. For each $y\in F_t$, calculate the evidence density $z_t(h,y):={\int }_{E_t}{g}_t(x,y){\widehat{\pi }}_t(h,dx)\mathrm{.}$

4. Define the filtering measure at $(h,y)$ by ${\pi }_t(h,y,A):=\{\begin{array}{ll}{\displaystyle \frac{{\int }_A{g}_t(x,y){\widehat{\pi }}_t(h,dx)}{z_t(h,y)},}& 0<z_t(h,y)<\mathrm{\infty },\\ {\widehat{\pi }}_t(h,A),& z_t(h,y)=0\text{ or }{z}_t(h,y)=\mathrm{\infty }\mathrm{.}\end{array}$ Use ${\pi }_t$ as the input filtering function at the next time step.

The procedure returns the initialized filter ${\pi }_0={\mu }_0$ and, for every $1\le t\le T$, ${\widehat{\pi }}_t$, $z_t$, and ${\pi }_t$. It is an exact measure-valued recursion; it makes no claim that the integrals have closed forms or can be evaluated with finite computational cost. Exact state-space filtering recursion under dominated observations ↗ proves that the returned time-$t$ functions are predictive and filtering kernels for the specified finite-horizon model.

Assume AC and a finite-horizon model as in General finite-horizon state-space models by kernels. For each $1\le t\le T$, let ${\nu }_t$ be a sigma-finite measure on $(F_t,{\mathcal{F}}_t)$ and suppose the observation kernel has a specified product-measurable density $g_t:E_t\times F_t\to [0,\mathrm{\infty }]$ with respect to ${\mathcal{E}}_t\otimes {\mathcal{F}}_t$: $G_t(x,B)={\int }_B{g}_t(x,y){\nu }_t(dy),{\int }_{F_t}{g}_t(x,y){\nu }_t(dy)=1$ for every state $x\in E_t$ and measurable $B\subseteq F_t$. For $h\in {\mathsf{F}}_{1:t-1}$, let ${\pi }_{t-1}$ 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 ${\pi }_0$ be the initial kernel ${\mu }_0$ on the singleton empty-history space. For $A\in {\mathcal{E}}_t$, define the predictive kernel ${\widehat{\pi }}_t(h,A):={\int }_{E_{t-1}}{K}_t(x,A){\pi }_{t-1}(h,dx),$ the evidence density $z_t(h,y):={\int }_{E_t}{g}_t(x,y){\widehat{\pi }}_t(h,dx),$ and for $h\in {\mathsf{F}}_{1:t-1}$, $y\in F_t$, and $A\in {\mathcal{E}}_t$ define the update ${\pi }_t(h,y,A):=\{\begin{array}{ll}{\displaystyle \frac{{\int }_A{g}_t(x,y){\widehat{\pi }}_t(h,dx)}{z_t(h,y)},}& 0<z_t(h,y)<\mathrm{\infty },\\ {\widehat{\pi }}_t(h,A),& z_t(h,y)=0\text{ or }{z}_t(h,y)=\mathrm{\infty }\mathrm{.}\end{array}$ Then ${\widehat{\pi }}_t$ is a predictive kernel for $X_t$ given $Y_{1:t-1}$, and ${\pi }_t$ 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 ${\widehat{\pi }}_t(h,\cdot )$ has density $z_t(h,\cdot )$ with respect to ${\nu }_t$ and assigns zero mass to both $\{y:z_t(h,y)=0\}$ and $\{y:z_t(h,y)=\mathrm{\infty }\}$. 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.