## Bayes’ Rule

Let A and B be sets in &. Because the sets A and A form a partition of the sample space ^, we have B = (B П A) U (B П A); hence,

P(B) = P(B П A) + P(B П A) = P(B|A)P(A) + P(B |A)P(A).

Moreover,

P(AB)= P(A П B) P(B|A)P(A)

( 1 ) P (B) P (B) ■

Combining these two results now yields Bayes’ rule:

P (B | A) P (A)

P(B | A)P(A) + P(B| A)P(A) ■

Thus, Bayes’ rule enables us to compute the conditional probability P(A |B) if P(A) and the conditional probabilities P(B | A) and P(B | A) are given.

More generally, if Aj, j = 1, 2,…,n (< to) is a partition of the sample space ^ (i. e., the Aj’s are disjoint sets in Ж such that ^ = Un=j Aj), then

P (B | Ai) P (Ai)

TTj=1 P (B| Aj) P (Aj) •

Bayes’ rule plays an important role in a special branch of statistics (and econometrics) called Bayesian ...

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