#analysis/probability-statistics # derivation from [[conditional probability]] Let $A,B$ be [[event]]s within a [[sample space]] $\Omega$. We have $\begin{align} P(A \vert B) =&\frac{P(A \cap B)}{P(B)} \\ =& \frac{P(B \cap A)}{P(B)} \\ = &\frac{P(B \vert A)P(A)}{P(B)}. \end{align}$ - $P(A | B)$ is the [[posterior probability distribution]] - $P(A)$ is the [[prior probability distribution]]. - Using the above terms we can write the theorem in 'Bayesian English' as $\text{posterior}=\frac{\text{prior} \times \text{likelihood}}{\text{evidence}}.$