Bayes Disintegration and Filtering Formula for the Observation Record

Adopt the setting of Conditional Density of the Observation Record Given the Initial States and Transition Clocks: the controlled $N$-agent dynamics with control set $\mathcal{A}$, rate families $\beta$ and $\tilde{\beta}$, horizon $T>0$, driving system $(\Omega,\mathcal{F},P)$, $\mathcal{A}$-valued policy $h$, and a solution on $[0,T]$ with regular event $\Omega_0$, empirical state measure $\Sigma_t$, observation filtration $(\mathcal{G}_s)_{s\in[0,T]}$, and observation record $W$; the observation record space $(\mathbf{R},\mathcal{R},\rho)$; the $\sigma$-algebra $\mathcal{T}$ and reconstruction data with reconstructed empirical state measures $\Sigma^r$; the record density kernel $f$; and the marginal record density $p(r)=\mathbb{E}[f(r,\cdot)]$ of claim 5 of Conditional Density of the Observation Record Given the Initial States and Transition Clocks. Write $P|_{\mathcal{T}}$ for $P$ considered on $\mathcal{T}$ only.

1. (Joint law) The map $\omega\mapsto(W(\omega),\omega)$ is measurable from $(\Omega,\mathcal{F})$ to $(\mathbf{R}\times\Omega,\mathcal{R}\otimes\mathcal{T})$, and its image measure is the measure with density $f$ with respect to the product measure $\rho\otimes P|_{\mathcal{T}}$. In particular, for every $\mathcal{R}\otimes\mathcal{T}$-measurable $\Psi:\mathbf{R}\times\Omega\to[0,\infty]$, $\mathbb{E}\bigl[\Psi(W(\cdot),\cdot)\bigr]=\int_{\mathbf{R}\times\Omega}\Psi\,f\,d(\rho\otimes P|_{\mathcal{T}})\qquad\text{in }[0,\infty].$

2. (The record generates the observations) $\mathcal{G}_T$ is the $\sigma$-algebra generated by the events $W^{-1}(A)$, $A\in\mathcal{R}$, together with every event of probability zero.

3. (Bayes ratio for conditional expectations) Almost surely $p(W)>0$. Let $\Psi:\mathbf{R}\times\Omega\to\mathbb{R}$ be $\mathcal{R}\otimes\mathcal{T}$-measurable and bounded. Then the map $\varphi_\Psi(r)=\frac{\mathbb{E}\bigl[\Psi(r,\cdot)\,f(r,\cdot)\bigr]}{p(r)}\quad\text{where }p(r)>0,\qquad \varphi_\Psi(r)=0\quad\text{where }p(r)=0,$ is $\mathcal{R}$-measurable and bounded, and $\varphi_\Psi(W)$ is a version of the conditional expectation of the square-integrable random variable $\Psi(W(\cdot),\cdot)$ given $\mathcal{G}_T$.

4. (Filtering formula) Let $t\in(0,T]$. The restrictions to $[0,t]$ of the state, observation, and control processes, together with the regular event $\Omega_0$ and the policy whose members are the restrictions of the $h_j$ to $[0,t]\times R_j(t)\times\{1,\dots,\tilde{l}\}^j$ (which is $\mathcal{A}$-valued), form a solution of the controlled $N$-agent dynamics on $[0,t]$, whose observation filtration at time $t$ is $\mathcal{G}_t$ and whose observation record $W_t$ is $\mathcal{G}_t$-measurable. Writing $f^{(t)}$, $p^{(t)}$, and $\Sigma^{r,(t)}$ for the record density kernel, marginal record density, and reconstructed empirical state measures of this horizon-$t$ solution (for any fixed choice of horizon-$t$ reconstruction data), the following holds for every bounded Borel-measurable $g:\mathbb{R}^l\to\mathbb{R}$: almost surely, $\mathbb{E}\bigl[g(\Sigma_t)\bigm|\mathcal{G}_t\bigr]=\frac{\mathbb{E}\bigl[g\bigl(\Sigma^{r,(t)}_t\bigr)\,f^{(t)}(r,\cdot)\bigr]}{p^{(t)}(r)}\Bigg|_{r=W_t},$ the right side read as $0$ where $p^{(t)}(W_t)=0$, an event of probability zero.