16.5.1: 計算： $\Big( F_{s} (\Xi_{s}) \Phi^{s,s+1}{\widehat F}_{s+1}({{{\times}}}_{t=s+1}^n \Xi_{t}) \Big)$ in (16.9) (in $\S$16.3)

さて,次のように考えて, \begin{align} \widetilde{f}_{x_n}(\omega_n) & = \frac{1}{\sqrt{2 \pi} {q}_n} \exp[ -\frac{(x_n -c_n \omega_n - d_n)^2}{2{q}_n^2} ] \nonumber \\ & \approx \exp[ -\frac{(c_n \omega_n -( x_n-d_n))^2}{2{q}_n^2} ] \equiv \exp[ - \frac {1}{2} \Big(\widetilde{u}_n \omega_n -\widetilde{v}_n \Big)^2 ] \tag{16.19} \end{align}

ここで, $c_n$, $d_n$と$q_n$ $(t \in T)$ は既知とする. さて,次のようにおく.

\begin{align} {\widetilde{u}_n} =\frac{c_n}{q_n} ,\quad {\widetilde{v}_n} = \frac{x_n - d_n}{q_n} \tag{16.20} \end{align}

さらに, 補題16.3から,因果作用素 $\Phi^{t-1.t}:L^\infty (\Omega_t) \to L^\infty ( \Omega_{t-1})$ を次のように計算できる.

\begin{align} & f_{t-1}(\omega_{t-1}) = [\Phi^{t-1,t} \widetilde{f}_{x_t}](\omega_{t-1} ) \nonumber \\ \approx & \int_{- \infty}^{\infty} \frac{1}{\sqrt{2 \pi} r_t} \exp[ -\frac{(\omega_t - a_t \omega_{t-1} -b_t )^2}{2r_t^2} ] \exp[ -\frac{({\widetilde{u}}_t \omega_t-{\widetilde{v}_t})^2}{2} ] d \omega_{t} \nonumber \\ \approx & \exp[ -\frac{1}{2} \Big( \frac{\widetilde{v}_t}{\sqrt{1 +r_t^2 {\widetilde{u}}_t^2 }} - \frac{{\widetilde{u}}_t(a_t \omega_{t-1} + b_t)}{\sqrt{1 +r_t^2 {\widetilde{u}}_t^2}} \Big)^2 ] \approx \exp[ -\frac{1}{2} \Big({{u}}_{t-1} \omega_{t-1}-{{v}_{t-1}} \Big)^2 ] \tag{16.21} \end{align} ここで, \begin{align} & u_{t-1}=-\frac{a_t {\widetilde u}_t}{\sqrt{1 +r_t^2 {\widetilde{u}}_t^2 }}, \quad v_{t-1}=\frac{ b_t {\widetilde u}_t - {\widetilde v}_t }{\sqrt{1 +r_t^2 {\widetilde{u}}_t^2 }} \tag{16.22} \end{align}

さらに, 補題16.3より, \begin{align} \widetilde{f}_{x_{t-1}}(\omega_{t-1}) & = \exp[ -\frac{(c_{t-1} \omega_{t-1} + d_{t-1}-x_{t-1})^2}{2{q}_{t-1}^2} ] \exp[ -\frac{({{u}}_{t-1} \omega_{t-1}-{{v}_{t-1}})^2}{2} ] \nonumber \\ & \approx \exp [ -\frac{1}{2} ( \frac{c_{t-1}^2 + u_{t-1}^2 q_{t-1}^2}{q_{t-1}^2 } ) \Big(\omega_{t-1} - \frac{ {c_{t-1}(d_{t-1}-t_{t-1})} + {u_{t-1}v_{t-1}}{q_{t-1}^2} }{{c_{t-1}^2} + {u_{t-1}^2}{q_{t-1}^2}} \Big)^2 ] \nonumber \\ & \approx \exp[ -\frac{1}{2} \Big({\widetilde{u}}_{t-1} \omega_{t-1}-{\widetilde{v}_{t-1}} \Big)^2 ] \tag{16.23} \end{align} ここで, \begin{align} {{\widetilde{u}}_{t-1}} = \frac{\sqrt{c_{t-1}^2 + u_{t-1}^2 q_{t-1}^2}}{q_{t-1} }, \;\; {{\widetilde{v}}_{t-1}} = \frac{ {c_{t-1}(d_{t-1}-t_{t-1})} + {u_{t-1}v_{t-1}}{q_{t-1}^2} }{q_{t-1} \sqrt{{c_{t-1}^2} + {u_{t-1}^2}{q_{t-1}^2}}} \tag{16.24} \end{align} 上の(16.19)-(16.24)をまとめると, \begin{align*} {\scriptsize { \overset{\widetilde{u}_s, \widetilde{v}_s}{ \underset{\widetilde{w}_s}{\fbox{$\widetilde{f}_{x_s}$}} }} \xleftarrow[]{\mbox{$x_s$}} \cdots \xleftarrow[]{\Phi^{t-2,t-1}} { \overset{\widetilde{u}_{t-1}, \widetilde{v}_{t-1}}{ \underset{\widetilde{w}_{t-1}}{ {{\fbox{$\widetilde{f}_{x_{t-1}}$}}}}}} \xleftarrow[(16.24)]{{x_{t-1}}} { \overset{{u}_{t-1}, {v}_{t-1}}{ \underset{w_{t-1}}{ \fbox{${f}_{t-1}$}}} } \xleftarrow[(16.22)]{\Phi^{t-1,t}} { \overset{\widetilde{u}_t, \widetilde{v}_t}{ \underset{{\widetilde{w}_t}}{ \fbox{$\widetilde{f}_{x_t}$}}} } \xleftarrow[]{{x_{t}}} \cdots \xleftarrow[]{{x_{n-1}}} { \overset{{u}_{n-1}, {v}_{n-1}}{ \underset{{w}_{n-1}}{ \fbox{${f}_{n-1}$}}} } \xleftarrow[]{\Phi^{n-1,n}} { \overset{{\widetilde{u}_n \widetilde{v}_n}}{\underset{{\widetilde{w}_n}}{\fbox{$\widetilde{f}_{x_n}$=(16.20)}}} } } \end{align*} したがって,次を得る. \begin{align} \widetilde{f}_{x_s} \approx \lim_{\Xi_t \to x_t \;( t \in \{s.s+1, \cdots, n\})} \frac{ \Big( F_{s} (\Xi_{s}) \Phi^{s,s+1}{\widehat F}_{s+1}({{{\times}}}_{t=s+1}^n \Xi_{t}) \Big)} { \| F_{s} (\Xi_{s}) \Phi^{s,s+1}{\widehat F}_{s+1}({{{\times}}}_{t=s+1}^n \Xi_{t}) \Big) \|_{L^\infty (\Omega_s )} } \tag{16.25} \end{align} in (16.9)(in $\S$16.3)

すなわち,

 $(A_1):$ 平滑化: $0 \le s < n$の場合
 $(A_2):$ フィルタ: $s= n$の場合
 $(A_3):$ 予測: $s= n$の場合で,しかも, $n_0 \le m < n$を満たす任意の$m$に対して,, 観測量 $(X_m, {\mathcal F}_m, F_m )= (\{1\}, \{\emptyset ,\{1 \} \}, F_m )$ が $F_m(\emptyset )\equiv 0$, $F_m(\{ 1 \} )\equiv 1$ を満たす場合