Bayes' Theorem

mathsstats

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

Proof

By definition of conditional probability, $P(A|B)=\frac{P(A\cap B)}{P(B)},P(B)\ne 0$ Similarly, $P(B|A)=\frac{P(A\cap B)}{P(A)},P(A)\ne 0$ Therefore $P(A|B)=\left(\frac{1}{P(B)}\right)\left(P(B|A)P(A)\right),P(B)\ne 0$

Bayesian inference

Posterior = (likelihood x prior) / evidence Where $A$ is our hypothesis, P(A) is the prior probability
How do we update when we get new information. \begin{align*}\text{updated probability} &= \frac{\text{probability of new information for a given event}}{\text{unconditional probability of new information}} \\ &\times \text{prior probability of event} \end{align*}

https://en.wikipedia.org/wiki/Bayesian_inference