Bayes' theorem

Bayes' Theorem

• Let $A_1,A_2,\dots ,A_k$ be a collection of
  • $k$ mutually exclusive and exhaustive events
  • with prior probabilities $P\left(A_i\right)\left(i=1,\dots ,k\right)$ .
• Then for any other event $B$ for which $P\left(B\right)>0$ ,
  • the posterior probability of $A_j$ given that $B$ has occurred is

$$\begin{array}{rl}P\left(A_j\mid B\right)& =\frac{P\left(A_j\cap B\right)}{P\left(B\right)}\\ & =\frac{P\left(B\mid A_j\right)P\left(A_j\right)}{\sum _{i=1}^{k}P\left(B\mid A_i\right)\cdot P\left(A_i\right)}\forall j=1,\dots ,k\end{array}\text{\hspace{1em}(2.6)}$$

• The transition from the second to the third expression in (2.6) rests on using
  • the multiplication rule in the numerator
  • and the law of total probability in the denominator.
• The proliferation of events and subscripts in (2.6) can be a bit intimidating to probability newcomers.