Bayes theorem

Definition:

• Used to infer B|A when knowing A|B, A, B
• Can only be used on CDF not PDF
• Let $E$ and $F$ be events, $E=EF\cup E{F}^{\prime }$
  • Then $P(E)=P(EF)+P(E{F}^{\prime })$ $=P(E|F).P(F)+P(E|F^{\prime }).P(F^{\prime })$ $⟹P(E)=P(E|F).P(F)+P(E|F^{\prime }).P(F^{\prime })$
• $P(F|E)=\frac{P(F).P(E|F)}{P(F).P(E|F)+P(F^{\prime }).P(E|F^{\prime })}=\frac{P(F\cap E)}{P(E)}$
• $P(F|E)=\frac{P(E|F).P(F)}{P(E)}$

For more than 2 variables:

• $P(F_i|E)=\frac{P(F_i).P(E|F_i)}{P(F_1).P(E|F_1)+...+P(F_n).P(E|F_n)}$
• $F_1,F_2,...,F_n$ where all events F are mutually exclusive events (that is disjoint events) whose union is the sample space where $E$ is an event that has occurred