Bayes Theorem

Definition

Let

• $A_1,\dots ,A_k$ : Events
• $P(A_i)>0,i=1,\dots ,k$
• $B$ : Event

Suppose $A_1,\dots ,A_k$ form a partition of $\mathcal{C}$

Then

$$\begin{array}{rl}P(A_j|B)& =\frac{P(A_j)P(B|A_j)}{{\sum }_{i=1}^{k}P(A_i)P(B|A_i)}\\ & \\ P(A|B)& =\frac{P(B|A)P(A)}{P(B)}\end{array}$$