1.3 Bayes' theorem
From the ECE600 Pre-requisites notes of Sangchun Han, ECE PhD student.
• = Bayes' rule
• $ P\left(A|B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}=\frac{P\left(B|A\right)P\left(A\right)}{P\left(B\right)} $
• If $ P\left(B\right)=\sum_{i}P\left(B\cap A_{i}\right)=\sum_{i}P\left(B|A_{i}\right)P\left(A_{i}\right) $ , then $ P\left(A_{i}|B\right)=\frac{P\left(A_{i}\cap B\right)}{P\left(B\right)}=\frac{P\left(B|A_{i}\right)A_{i}}{\sum_{j}P\left(B|A_{j}\right)A_{j}} $
