# Bayesian Inference
the probability we Usually Want is **Pr( Hypothesis | given the Evidence)**

$$P(H|E)=\frac{P(H)P(E|H)}{P(E)}$$

Bayes' Theorem for Probability distribution is often stated as:
$$Posterior \propto Likelihood \times Prior $$

In the problem of whether the true side has been chosen in the last decision. the hypothesis are A, B and the evidence is C :

event A : chose the **false** side in the last decision,
with probability **0.3** of detecting "**small distance error**" in the current decision.

event B : chose the **true** side in the last decision,
with probability **0.9** of detecting "**small distance error**" in the current decision.

event C : detecting "**small distance error**" in the current turning decision.

**Prior** : $P(A)=P(B)=0.5$

**Likelihood** : $P(C|A)=0.3 , \enspace P(C|B)=0.9$

the **Posteriors** we want are :

$$P(A|C)=\frac{P(A)P(C|A)}{P(C)}=0.25 ,\enspace P(B|C)=\frac{P(B)P(C|B)}{P(C)}=0.75 $$

where $P(C)= P(A)P(C|A)+P(B)P(C|B)$ by the law of total probability