16.3: Bayes=Kalman operator $B_{\widehat{\mathsf{O}}_{0} }^s (\times_{t \in T} \{x_t \})$

In what follows, we solve Problem 16.2. For this, it suffices to find the ${{z}}_s \in L^1_{+1} (\Omega_s )$ such that

\begin{align} \lim_{ \Xi_t \to x_t \;\;(t\in T)} \frac{\int_{\Omega_0} [{\widehat F}_{0}((\times_{t=0}^n \Xi_{t}) \times \Gamma_s)](\omega_0) \;\; {{z}}_0 (\omega_0 ) d\omega_0}{\int_{\Omega_0} [{\widehat F}_{0}(\times_{t=0}^n \Xi_{t}) ](\omega_0)\;\; {{z}}_0 (\omega_0 ) d\omega_0} = \int_{\Omega_s} [G_s(\Gamma_s )](\omega_s) \;\; {{z}}_s (\omega_s ) d \omega_s \quad (\forall \Gamma_s \in {\mathcal F}_s ) \nonumber \end{align}

Let us calculate ${{z}}_s= [B_{\widehat{\mathsf{O}}_{0} }^s (\times_{t \in T} \{x_t \})]({{{z}}}_0)$ as follows.

\begin{align} & \int_{\Omega_0} [{\widehat F}_{0}((\times_{t=0}^n \Xi_{t}) \times \Gamma_s)](\omega_0) \;\; {{z}}_0 (\omega_0 ) d\omega_0 \nonumber \\ = & {}_{{}_{L^1(\Omega_0)}} \langle {{z}}_0, {\widehat F}_{0}((\times_{t=0}^n \Xi_{t}) \times \Gamma_s) \rangle {}_{{}_{L^\infty(\Omega_0)}} \nonumber \\ = & {}_{{}_{L^1(\Omega_1)}} \langle \Phi^{0,1}_* (F_0(\Xi_0) {{z}}_0), {\widehat F}_{1}((\times_{t=1}^n \Xi_{t}) \times \Gamma_s) \rangle {}_{{}_{L^\infty(\Omega_1)}} \tag{16.7} \end{align}
$(A):$ and, putting $\widetilde{{{z}}}_0=F_0(\Xi_0) {{z}}_0$ (or, exactly, its normalization, i.e., $\widetilde{{{z}}}_0=\lim_{\Xi_0 \to x_0} \frac{F_0(\Xi_0) {{z}}_0}{\int_{\Omega_0}{F_0(\Xi_0) {{z}}_0} d\omega_0}$) , $\widetilde{{{z}}}_1=F_1(\Xi_1) \Phi^{0,1}_* (\widetilde{{{z}}}_0)$, $\widetilde{{{z}}}_2=F_2(\Xi_2) \Phi^{1,2}_* ( \widetilde{{{z}}}_1)$, $\cdots$ , $\widetilde{{{z}}}_{s-1}=F_{s-1}(\Xi_{s-1}) \Phi^{s-2,s-1}_* (\widetilde{{{z}}}_{s-2})$, we see that
\begin{align} (16.7) = & {}_{{}_{L^1(\Omega_1)}} \langle \Phi^{0,1}_* (\widetilde{{{z}}}_0), {\widehat F}_{1}((\times_{t=1}^n \Xi_{t}) \times \Gamma_s) \rangle {}_{{}_{L^\infty(\Omega_1)}} \nonumber \\ = & {}_{{}_{L^1(\Omega_2)}} \langle \Phi^{1,2}_* ( \widetilde{{{z}}}_1), {\widehat F}_{2}((\times_{t=2}^n \Xi_{t}) \times \Gamma_s) \rangle {}_{{}_{L^\infty(\Omega_2)}} \nonumber \\ & \cdots \cdots \nonumber \\ = & {}_{{}_{L^1(\Omega_{s+1})}} \langle \Phi^{s,s+1}_* ( \widetilde{{{z}}}_{s}), {\widehat F}_{s+1}((\times_{t=s+1}^n \Xi_{t}) \times \Gamma_s) \rangle {}_{{}_{L^\infty(\Omega_{s+1})}} \nonumber \\ = & {}_{{}_{L^1(\Omega_{s})}} \langle \Phi^{s-1,s}_* ( \widetilde{{{z}}}_{s-1}), {\widehat F}_{s}((\times_{t=s}^n \Xi_{t}) \times \Gamma_s) \rangle {}_{{}_{L^\infty(\Omega_{s})}} \nonumber \\ = & {}_{{}_{L^1(\Omega_{s})}} \langle \Phi^{s-1,s}_* (\widetilde{{{z}}}_{s-1}), F_{s} (\Xi_{s})G_{s} (\Gamma_{s}) \Phi^{s,s+1}{\widehat F}_{s+1}(\times_{t=s+1}^n \Xi_{t}) \rangle {}_{{}_{L^\infty(\Omega_{s})}} \nonumber \\ = & {}_{{}_{L^1(\Omega_{s})}} \langle \Big( F_{s} (\Xi_{s}) \Phi^{s,s+1}{\widehat F}_{s+1}(\times_{t=s+1}^n \Xi_{t}) \Big) \Big( \Phi^{s-1,s}_* (\widetilde{{{z}}}_{s-1}) \Big), G_{s} (\Gamma_{s}) \rangle {}_{{}_{L^\infty(\Omega_{s})}} \tag{16.8} \end{align} Thus, we see \begin{align} [B_{\widehat{\mathsf{O}}_{0} }^s (\times_{t \in T} \{x_t \})]({{z}}_0) = \lim_{ \Xi_t \to x_t \;\;(t\in T)} \frac{ \Big( F_{s} (\Xi_{s}) \Phi^{s,s+1}{\widehat F}_{s+1}(\times_{t=s+1}^n \Xi_{t}) \Big) \times \Big( \Phi^{s-1,s}_* \widetilde{{{z}}}_{s-1}) \Big) } {\int_{\Omega_0} [{\widehat F}_{0}(\times_{t=0}^n \Xi_{t}) ](\omega_0)\;\; {{z}}_0 (\omega_0 ) d\omega_0} \tag{16.9} \end{align}