8.2: 擬積観測量の制約条件

\begin{align*} & {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1 \times \Xi_2{}) {}\big) {}_{\overline{\mathcal A} } + {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1 \times \Xi_2^c{}) {}\big) {}_{\overline{\mathcal A} } = {}_{{\mathcal A}^*} \big(\rho, {}F_{12}^{(1)}] (\Xi_1 {}) {}\big) {}_{\overline{\mathcal A} } \\ & {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1^c \times \Xi_2^c{}) {}\big) {}_{\overline{\mathcal A} } + {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1^c \times \Xi_2{}) {}\big) {}_{\overline{\mathcal A} } = {}_{{\mathcal A}^*} \big(\rho, {}F_{12}^{(1)} (\Xi_1^c {}) {}\big) {}_{\overline{\mathcal A} } \\ & {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1^c \times \Xi_2^c{}) {}\big) {}_{\overline{\mathcal A} } + {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1 \times \Xi_2^c{}) {}\big) {}_{\overline{\mathcal A} } = {}_{{\mathcal A}^*} \big(\rho, {}F_{12}^{(2)} (\Xi_2^c {}) {}\big) {}_{\overline{\mathcal A} } \\ & {}_{{\mathcal A}^*}\big(\rho, {}F_{12} (\Xi_1 \times \Xi_2^c{}) {}\big) {}_{\overline{\mathcal A} } + {}_{{\mathcal A}^*} \big(\rho, {}F_{12} (\Xi_1^c \times \Xi_2^c{}) {}\big) {}_{\overline{\mathcal A} } = {}_{{\mathcal A}^*} \big(\rho, {}F_{12}^{(2)} (\Xi_2^c {}) {}\big) {}_{\overline{\mathcal A} } \end{align*} に注意せよ．

\begin{align} & \; \; {\rm{Rep}}_\rho^{\Xi_1\times \Xi_2}[{\mathsf O}_{12}] = \left[\begin{array} _{{\mathcal A}^*} \big(\rho, F_{12} (\Xi_1 \times \Xi_2) \big) _{\overline{\mathcal A} } & _{{\mathcal A}^*} \big(\rho, F_{12} (\Xi_1 \times \Xi_2^c) \big) _{\overline{\mathcal A} } \\ _{{\mathcal A}^*} \big(\rho, F_{12} (\Xi_1^c \times \Xi_2) \big) _{\overline{\mathcal A} } & _{{\mathcal A}^*} \big( \rho, F_{12} (\Xi_1^c \times \Xi_2^c) \big) _{\overline{\mathcal A} } \end{array}\right] \nonumber \\ = & \left[\begin{array}{ll} \alpha^{^{\Xi_1 \times \Xi_2}}_\rho & \rho( F_1 (\Xi_1)) - \alpha^{^{\Xi_1 \times \Xi_2}}_\rho \\ \rho( F_2 (\Xi_2))- \alpha^{^{\Xi_1 \times \Xi_2}}_\rho & 1+ \alpha^{^{\Xi_1 \times \Xi_2}}_\rho - \rho( F_1 (\Xi_1)) - \rho( F_2 (\Xi_2)) \end{array}\right] \tag{8.2} \end{align} かつ， 次が成立する： \begin{align} & \max \{ 0, \rho( {}F_1 (\Xi_1))+ \rho( {}F_2 (\Xi_2)) -1 {} \} {{\; \leqq \;}} \alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho {{\; \leqq \;}} \nonumber \\ & \min \{ \rho( {}F_1 (\Xi_1)) , \; \rho( {}F_2 (\Xi_2)) \} \nonumber \\ & (\forall \Xi_1 \in {\cal F}_1, \forall \Xi_2 \in {\cal F}_2, \forall \rho \in {\frak S}^p({\mathcal A}^*) ) \tag{8.3} \end{align} また, 次が成り立つ：

\begin{align} \rho( {}F {}( \Xi_{1} \times \Xi^c_{2}{}) {}{} {}) =0 \; & \Longleftrightarrow \; \alpha^{{}^{\Xi_1 \times \Xi_2}}_\rho = \rho( {}F_1 ( \Xi_{1} {}{}) {}) \nonumber \\ &\Longrightarrow \; \rho( {}F_1 ( \Xi_{1} {}{}) {}) {{\; \leqq \;}} \rho( {}F_2 ( \Xi_{2} {}{}) {}) \tag{8.4} \end{align}

$\square \quad$

さて， ${\mathsf O}_{1}$ ${{=}}$ $( X_1, {\cal F}_1 , F_{1}{})$ と ${\mathsf O}_{2}$ ${{=}}$ $( X_2, {\cal F}_2 , F_{2}{} )$ を $\overline{\mathcal A}$ 内の 観測量として， ${\mathsf O}_{12}$ ${{=}}$ $( X_1 \times X_2, {\mathcal F}_1 \boxtimes {\mathcal F}_2, F_{12}{}{{=}} F_1 \mathop{\times}^{qp} F_2)$ を ${\mathsf O}_{1}$ と ${\mathsf O}_{2}$ の擬積観測量とする． ここで， 測定 ${\mathsf M}_{\overline{\mathcal A}} ({\mathsf O}_{12}$ ${{=}}(X_1 \times X_2, {\cal F}_1 \boxtimes {\cal F}_2, F_{12}{}{{=}} F_1 {\mathop{\times}^{qp}} F_2),$ $S_{[\rho]})$) により， 測定値$(x_1, x_2)$ $(\in X_1 \times X_2 )$ が得られたとする． このとき， $x_1 \in \Xi_1$ であることを 知ったとき， $x_2 \in \Xi_2$ である確率$P$ (すなわち， 条件付き確率 )は

\begin{align*} P= \frac{ \rho ({}F_{12} (\Xi_1 \times \Xi_2{})) } { \rho ({}F_{1} (\Xi_1 {})) } = \frac{ \rho ({}F_{12} (\Xi_1 \times \Xi_2{})) } { \rho ({}F_{12} (\Xi_1 \times \Xi_2 {})) + \rho ({}F_{12} (\Xi_1 \times \Xi_2^c)) } \end{align*} で与えられて， (8.3)により 次のように評価される: \begin{align*} & \frac{\max \{ 0, \rho ({}F_1 ( \Xi_{1} {}{}) ) + \rho ({}F_2 ( \Xi_{2} {}{}) ) -1 {} \} }{{ \rho ({}F_{12} (\Xi_1 \times \Xi_2)) + \rho ({}F_{12} (\Xi_1 \times \Xi^c_2 {})) }} {{\; \leqq \;}} P {{\; \leqq \;}} \\ & \frac{ \min \{ \rho ({}F_1 ( \Xi_{1} {}{}) ) , \; \rho ({}F_2 ( \Xi_{2} {}{}) ) \} }{{ \rho ({}F_{12} (\Xi_1 \times \Xi_2{})) + \rho ({}F_{12} (\Xi_1 \times \Xi_2^c)) }} \end{align*}

\begin{align} {\mathsf O}_{12} = (X_{_{RD}} \times X_{_{SW}} , 2^{ X_{_{RD}} \times X_{_{SW}} }, F {{=}} F_{_{RD}} \times F_{_{SW}}) \end{align} すなわち, \begin{align} & \; \; \mbox{Rep}^{\{( y_{{_{RD}}} , y_{{_{SW}}}) \}}_{\omega_k} [{\mathsf O}_{12}] = \left[\begin{array}{ll} [F (\{( y_{{_{RD}}} , y_{{_{SW}}}) \}) ] ({\omega_k}) & [F (\{( y_{{_{RD}}} , n_{{_{SW}}}) \}) ] ({\omega_k}) \\ [F (\{( n_{{_{RD}}} , y_{{_{SW}}}) \}) ] ({\omega_k}) & [F (\{( n_{{_{RD}}} , n_{{_{SW}}}) \}) ] ({\omega_k}) \\ \end{array}\right] \\ = & \left[\begin{array}{ll} \alpha_{_{\{( y_{_{RD}} , y_{{_{SW}}}) \} }} & [F_{_{RD}} (\{ y_{{_{RD}}} \}) ] - \alpha_{_{\{( y_{{_{RD}}} , y_{{_{SW}}}) \} }} \\ [F_{_{SW}} (\{ y_{{_{SW}}} \}) ] - \alpha_{_{\{( y_{{_{RD}}} , y_{{_{SW}}}) \} }} & 1+ \alpha_{_{\{( y_{{_{RD}}} , y_{{_{SW}}}) \} }} - [F_{_{RD}} (\{ y_{{_{RD}}} \}) ] - [F_{_{SW}} (\{ y_{{_{SW}}} \}) ] \\ \end{array}\right] \end{align}

ここに， $\alpha_{\omega_k}^{{}^{\{( y_{{{}_{RD}}} , y_{{{}_{SW}}}{}) \} }} ({\omega_k})$ は(8.3)を満たす． したがって， トマト ${\omega_k}$ が「赤い」 とわかったとき， そのトマト ${\omega_k}$ が 「甘い」ことがわかる確率$P$は

\begin{align} P= \frac{ [F (\{(y_{_{RD}} , y_{_{SW}}) \})]({\omega_k}) } { [F (\{(y_{_{RD}} , y_{_{SW}}) \})]({\omega_k}) + [F (\{(y_{_{RD}} , n_{_{SW}}) \})]({\omega_k}) } = \frac{ [F (\{(y_{_{RD}} , y_{_{SW}}) \})]({\omega_k}) } { [F_{_{RD}} (\{ y_{{_{RD}}} \}) ] ({\omega_k}) } \end{align}

であり， $[{}F (\{(y_{{}_{RD}} , y_{{}_{SW}}{}) \}{}){}]({\omega_k}{})= \alpha^{{}^{\{(y_{{}_{RD}} , y_{{}_{SW}}{}) \}} } (\omega_k)$であるから， 条件付確率$P$の評価：

\begin{align} & \frac{\max \{ 0, [F_1 ( \{ y_{{_{RD}}} \} ) ] ({\omega_k}) + [F_2 ( \{ y_{{_{SW}}} \} ) ] ({\omega_k}) -1 \} }{{ [F_{_{RD}} (\{ y_{{_{RD}}} \}) ] ({\omega_k}) }} {{\; \leqq \;}} P {{\; \leqq \;}} \frac{ \min [F_1 ( \{ y_{{_{SW}}} \} ) ] ({\omega_k}) , \; [F_2 ( \{ y_{{_{SW}}} \} ) ] ({\omega_k}) \} }{{ [F_{_{RD}} (\{ y_{{_{RD}}} \}) ] ({\omega_k}) }} \end{align}