Swinburne makes use of Bayes’ Theorem in presenting most of the a posteriori arguments for and against God in The Existence of God (EOG), and he makes significant use of it in summing up his case for God.


Bayes’ Theorem:

P (h I e & k) = P(e I h & k) x P(h I k) / P(e I k)

By the symmetry of equality we can restate Bayes’ Theorem with the “answer” on the right hand side of the equation:

P(e I h & k) x P(h I k) / P(e I k) = P (h I e & k)

We have previously discussed the conditional probability that constitutes the answer that we seek:

P (h I e & k)

This is the posterior probability of the hypothesis h, or the probability of the hypothesis, given the specific evidence e and our background knowledge k.

Bayes’ Theorm (with the answer on the right-hand side) has the following simple form:

A x B / C = X

If one of the factors in the numerator (either A or B) increases, then X also increases.
If the denominator (C) increases, then X will decrease. The same relationships hold in
Bayes’ Theorem. If either P(e I h & k) increases or P(h I k) increases, then P(h I e & k) will also increase. If P(e I k) increases, then P(h I e & k) will decrease.

This makes sense if you think about what each of these conditional probability expressions means.

P(e I h & k)

This conditional probability in the numerator is the probability that the evidence would occur given that the hypothesis was true and given our background knowledge. If the evidence is likely to occur given the truth of the hypothesis, then the occurence of the evidence is favorable towards the truth of the hypothesis. So an increase in this probability should mean an increase in the probability of the hypothesis, i.e. P(h I e & k).

P(h I k)

This factor in the numerator is the probability of the hypothesis given only our background knowledge. If the hypothesis is already very probable, prior to considering evidence in support of the hypothesis, then the probability of the hypothesis after consideration of the evidence will also tend to be high, at least higher than it would be if the hypothesis was improbable to begin with. So, it makes sense that the higher this prior probability of the hypothesis h is, the higher the posterior probability of the hypothesis will be.

P(e I k)

This conditional probability is in the denominator. It means the probability of the evidence occurring given only our background knowledge. If the evidence was likely to occur whether or not the hypothesis is true, then the evidence is a poor indicator of the probability of the hypothesis. Thus it makes sense that an increase in this prior probability of the evidence e would decrease the posterior probability of the hypothesis, i.e P(h I e & k).