Probability MT22, Bayes' theorem
Flashcards
What does Bayes’ theorem for a partition of $\Omega$ formed by a family of events $\{B _ 1, B _ 2, \ldots, B _ n\}$ say about $\mathbb{P}(B _ k \vert A)$?
\[\mathbb{P}(B _ k \vert A) = \frac{\mathbb{P}(A \vert B _ k) \mathbb{P}(B _ k)}{\sum _ {i\ge1} \mathbb{P}(A \vert B _ i) \mathbb{P}(B _ i)}\]
@Prove Bayes’ theorem:
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 $i \ge 0$. Then for any $A \in \mathcal F$,
\[\mathbb P(B _ k \vert A) = \frac{\mathbb P (A \vert B _ k) \mathbb P(B _ k)}{\sum _ {i \ge 1} \mathbb P(A \vert B _ i)} \mathbb P(B _ i)\]
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 $i \ge 0$. Then for any $A \in \mathcal F$,
@todo (probability, page 11).
@important~