Back of the Envelop Mathematical Model Number 2: Bayes' Theorem.

Bayes; a Man of the Cloth.

Some years ago I came across an argument for the existence of God that used Bayes' Theorem. In fact the theorem seems to have been associated with arguments for the existence of God from day one; for example see the end of the wiki page on Bayes' Theorem. Before I can consider this argument we first need to get a handle on the theorem.

A straight forward derivation of Bayes' Theorem can be provided using a frequency understanding of profanity probability. To this end we imagine we have a total of T items. Let these T items have properties A and B distributed over them. Let the number of items with property A be F(A) and the number of items with property B be F(B).

Now, there may be some items that possess both properties A and B. If we take the set of items that possess property B, then let the number of items in this set possessing property A be represented by F(A|B). If we now take the set of items that possess properties A, then let the number of items in this set possessing property B be represented by F(B|A).

These quantities can be pictured using the following Venn diagram:

One fairly obvious deduction is that:

F(A|B) = F(B|A)

Thus, given these quantities we now imagine that we throw the T items into a bag, agitate them, and then select an item. If we assume a priori equal probabilities, then using the Laplacian definition of probability we then have probabilities P(A), P(B), P(A|B) and P(B|A) and Bayes Theorem follows as below:

The latter expression is Bayes' Theorem. Notice the symmetry of the proof and moreover the symmetry of Bayes equation; as one would expect given that A and B are interchangeable. I have arranged the derivation in order to bring out this symmetry.