Filtering, predictive, and smoothing laws in a finite-horizon state-space model

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).