1 What is Bayes’ Theorem?
Bayes’ theorem states that for events \(A\) and \(B\) with \(\mathbb{P}(B)\neq 0\) then the conditional probability of \(A\) given \(B\) can be computed with \[\mathbb{P}(A|B)=\frac{\mathbb{P}(B|A)\mathbb{P}(A)}{\mathbb{P}(B)}.\]
Bayes’ Theorem is a fundamental result in Bayesian inference where is is used to express the posterior distribution in terms of the prior distribution and the sampling distribution \[ p(\theta|y)=\frac{p(\theta,y)}{p(y)}=\frac{p(\theta)p(y|\theta)}{p(y)}\propto p(\theta)p(y|\theta). \]