conditional odds
定式化
def. $ O(A \mid B) ≔ \frac{P(A \mid B)}{1-P(A \mid B)} = \frac{P(A \cap B)}{P(B)-P(A \cap B)}
$ = \frac{P(A∩B)}{P(A^\complement ∩ B)} = \frac{P(A \mid B)}{P(\neg{A} \mid B)}
where $ P(B) = P(B∩(A∪A^\complement)) = P(B∩A)+P(B∩A^\complement)
最後に確率論理式を使った。
both: $ O(A:B \mid E) ≔ O(A:B)\cdot\Lambda(A:B \mid E)
one: $ O(A \mid B) ≔ O(A)\cdot\Lambda(A \mid B)
posterior odds = prior odds × likelihood ratio
ref.