I recently read this beautiful explanation of Bayes’ theorem. I’d always thought it was a statement of philosophy, but it isn’t: it comes from plain old probabilities.

The formula for the conditional probability of

*A*being true given that*B*is true is*P(A | B) = P(A & B) / P(B)*

That is, the proportion of things that are

*A*that are in*B*is equal to the fraction of the proportion of things that are*A and**B*over the proportion of things that are*B*(I like to think of these things in terms of Venn diagrams).We can rearrange the above to get

*P(A & B) = P(A | B).P(B)*

Now for Bayes’ theorem: let’s write

*H*for our hypothesis and*E*for our evidence. We want to know how seeing the evidence*E*affects the probability of our hypothesis*H*being true. From the first rule above we have*P(H | E) = P(H & E) / P(E)*

Now we can apply the second rule to

*P(H & E)*to get*P(H | E) = P(E | H).P(H) / P(E)*

Et voila! No magic at all.

As an aside, the raven paradox has convinced me that Bayesianism is philosophically superior to frequentism.