Probability MT22, Bayes' theorem
Flashcards
@State Bayes’ theorem for a partition of $\Omega$ formed by a family of events $\{B _ 1, B _ 2, \ldots, B _ n\}$, i.e. the formula for $\mathbb{P}(B _ k \vert A)$.
\[\mathbb{P}(B _ k \vert A) = \frac{\mathbb{P}(A \vert B _ k) \mathbb{P}(B _ k)}{\sum _ {i\ge1} \mathbb{P}(A \vert B _ i) \mathbb{P}(B _ i)}\]
@Prove Bayes’ theorem (∆bayes-theorem): suppose $B _ 1, B _ 2, \ldots$ is a partition of $\Omega$ by sets from $\mathcal F$, such that $\mathbb P(B _ i) > 0$ for all $i \ge 1$. Then for any $A \in \mathcal F$,
\[\mathbb P(B _ k \vert A) = \frac{\mathbb P (A \vert B _ k) \mathbb P(B _ k)}{\sum _ {i \ge 1} \mathbb P(A \vert B _ i) \mathbb P(B _ i)}\]
@todo (probability, page 11).