ベイズの定理

$ P(A \cap B) = P(B)P(A|B)

$ P(B_i|A)=\frac{P(A \cap B)}{P(A)}=\frac{P(A|B_i)P(B_i)}{P(A)}

$ =\frac{P(A|B_i)P(B_i)}{P(A \cap B_1) + P(A \cap B_2) + ... + P(A \cap B_N)}

$ =\frac{P(A|B_i)P(B_i)}{\sum_j{P(B_j)P(A|B_j)}}