条件付き独立
, Z n = z n ) F Y ∣ Z = z ( y ) = Pr ( Y 1 ≤ y 1 , ... , Y m ≤ y m ∣ Z 1 = z 1 , ... , Z n = z n ) {\displaystyle {\begin{aligned}F_{\mathbf {X} ,\mathbf {Y} \mid \mathbf {Z} =\mathbf {z} }(\mathbf {x} ,\mathbf {y} )&=\Pr(X_{1}\leq x_{1},\dots ,X_{l}\leq x_{l},\;Y_{1}\leq y_{1},\dots ,Y_{m}\leq y_{m}\mid Z_{1}=z_{1},\dots ,Z_{n}=z_{n})\\F_{\mathbf {X} \,\mid \,\mathbf {Z} =\mathbf {z} }(\mathbf {x} )&=\Pr(X_{1}\leq x_{1},\dots ,X_{l}\leq x_{l}\mid Z_{1}=z_{1},\dots ,Z_{n}=z_{n})\\F_{\mathbf {Y} \,\mid \,\mathbf {Z} \,=\,\mathbf {z} }(\mathbf {y} )&=\Pr(Y_{1}\leq y_{1},\dots ,Y_{m}\leq y_{m}\mid Z_{1}=z_{1},\dots ,Z_{n}=z_{n})\end{aligned}}}
P ( X ∣ Z ) = ∑ w ∈ W P ( X ∣ Z , W = w ) P ( W = w ∣ Z ) = ∑ w ∈ W P ( X ∣ Z , Y ) P ( W = w ∣ Z ) = P ( X ∣ Z , Y ) ∑ w ∈ W P ( W = w ∣ Z ) = P ( X ∣ Z , Y ) {\displaystyle {\begin{aligned}P(X\mid Z)&=\sum _{w\in W}P(X\mid Z,W=w)\;P(W=w\mid Z)\\&=\sum _{w\in W}P(X\mid Z,\,Y)\;P(W=w\mid Z)\\&=P(X\mid Z,\,Y)\;\sum _{w\in W}P(W=w\mid Z)\\&=P(X\mid Z,\,Y)\end{aligned}}}
ここで、
X ⊥ ⊥ Y ∣ Z , W ⟹ P ( X ∣ Z , W , Y ) = P ( X ∣ Z , Y ) {\displaystyle X\perp \!\!\!\perp Y\mid Z,\,W\quad \implies \quad P(X\mid Z,\,W,\,Y)=P(X\mid Z,\,Y)}
に注意して、次式を得る。
P ( X ∣ Z , W , Y ) = P ( X ∣ Z ) ( ⟺ X ⊥ ⊥ Y , W ∣ Z ) {\displaystyle P(X\mid Z,\,W,\,Y)=P(X\mid Z)\quad (\iff X\perp \!\!\!\perp Y,\,W\mid Z)}
X ⊥ ⊥ A ∣ B ∧ X ⊥ ⊥ B ⟹ X ⊥ ⊥ A , B {\displaystyle X\perp \!\!\!\perp A\mid B\quad \land \quad X\perp \!\!\!\perp B\implies \quad X\perp \!\!\!\perp A,\,B}
X ⊥ ⊥ A , B ⟹ X ⊥ ⊥ A ∣ B ∧ X ⊥ ⊥ B ∣ A {\displaystyle X\perp \!\!\!\perp A,\,B\quad \implies \quad X\perp \!\!\!\perp A\mid B\quad \land \quad X\perp \!\!\!\perp B\mid A}
X ⊥ ⊥ Y ∣ Z , W ∧ X ⊥ ⊥ W ∣ Z , Y ⟺ P ( X ∣ Z , W , Y ) = P ( X ∣ Z , W ) ∧ P ( X ∣ Z , W , Y ) = P ( X ∣ Z , Y ) ⟹ P ( X ∣ Z , Y ) = P ( X ∣ Z , W ) {\displaystyle {\begin{aligned}&X\perp \!\!\!\perp Y\mid Z,\,W\quad \land \quad X\perp \!\!\!\perp W\mid Z,\,Y\\&\iff P(X\mid Z,\,W,\,Y)=P(X\mid Z,\,W)\quad \land \quad P(X\mid Z,\,W,\,Y)=P(X\mid Z,\,Y)\\&\implies P(X\mid Z,\,Y)=P(X\mid Z,W)\end{aligned}}}
このとき、 P ( X ∣ Z ) {\displaystyle P(X\mid Z)} に全確率の法則を適用することで、