Baye’s Theorem

$$P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$$

Partitioned Baye’s Theorem

If $A_{\mathrm{1..}n}$ for a Partition of Universe $S$ and $B$ is any event, then

$$\forall j\in [1,n]P(A_j\mid B)=\frac{P(A_j\cap B)}{P(B)}=\frac{P(A_j)\cdot P(B\mid A_j)}{{\sum }_{i=1}^{n}P(A_i)P(B\mid A_i)}$$

• where $P(A_j)$ is the prior probabilities, “before $B$”
• where $P(A_j\mid B)$ are the posterior probabilities, “after B”

See Conditional Probability