Bayes' theorem follows directly from the product rule $$ $$ p(A|B,I) = \frac{p(B|A,I) p(A|I)}{p(B|I)}. $$ $$ The importance of this property to data analysis becomes apparent if we replace \( A \) and \( B \) by hypothesis(\( H \)) and data(\( D \)): $$ \begin{align} p(H|D,I) &= \frac{p(D|H,I) p(H|I)}{p(D|I)}. \tag{1} \end{align} $$ The power of Bayes’ theorem lies in the fact that it relates the quantity of interest, the probability that the hypothesis is true given the data, to the term we have a better chance of being able to assign, the probability that we would have observed the measured data if the hypothesis was true.