条件付き独立
条件付き確率の定義から、
P ( A , B ∣ C ) = P ( A ∣ C ) P ( B ∣ C ) ⟺ P ( A , B , C ) P ( C ) = P ( A , C ) P ( C ) P ( B , C ) P ( C ) ⟺ P ( A , B , C ) P ( B , C ) = P ( A , C ) P ( C ) ⟺ P ( A ∣ B , C ) = P ( A ∣ C ) {\displaystyle {\begin{aligned}&P(A,\,B\mid C)=P(A\mid C)\,P(B\mid C)\\&\iff {\frac {P(A,\,B,\,C)}{P(C)}}=\ {\frac {P(A,\,C)}{P(C)}}\,{\frac {P(B,\,C)}{P(C)}}\\&\iff {\frac {P(A,\,B,\,C)}{P(B,\,C)}}={\frac {P(A,\,C)}{P(C)}}\\&\iff P(A\mid B,\,C)=P(A\mid C)\end{aligned}}}