The probability of God

The Notices of the American Mathematical Society features a twin book review this month: Stephen Unwin's notorious 2004 work, The Probability of God: A simple calculation that proves the ultimate truth, and Steven Brams's more recent work, Superior Beings: If they exist, how would we know?

Unwin's book uses Bayes's theorem to calculate a 67% probability for the existence of God! One popular form of Bayes's theorem is the following:

P(H|E) = P(E|H)P(H)/(P(E|H)P(H) + P(E|~H)P(~H))

H denotes a hypothesis, ~H denotes the negation of the hypothesis, E denotes a piece of evidence, P(H) denotes the prior probability of the hypothesis, P(~H) denotes the prior probability of the falsity of the hypothesis, P(E|H) denotes the conditional probability of the piece of evidence E given the truth of the hypothesis H, P(E|~H) denotes the conditional probability of the piece of evidence given the falsity of the hypothesis, and P(H|E) denotes the conditional probability of the hypothesis given the piece of evidence E.

The numerator on the right-hand side here gives the probability that the piece of evidence could be explained by the hypothesis H, while the denominator gives the probability that the piece of evidence could be explained by anything, (in fact, the denominator is the same as P(E), the unconditional probability of E). The ratio of these two values gives the posterior probability of H, given the piece of evidence E.

The basic idea is that the prior probability of the hypothesis P(H) is updated to P(H|E) in light of new evidence E. The process is iterated for each new piece of evidence. In the case of interest here, H is the hypothesis that God exists. Unwin begins by assigning a 50% prior probability P(H) that God exists, and a 50% prior probability P(~H) that the hypothesis is false. It seems, then, that any questions over whether the hypothesis is even meaningful, are not to be considered.

Unwin then considers six pieces of evidence, and plucks, out of thin air, conditional probabilities P(E|H)/P(E) for such evidence given the existence of God, relative to the unconditional probability P(E) of each piece of evidence. Unwin specifies that for E = the existence of goodness, P(E|H)/P(E) = 10; for E = existence of moral evil, P(E|H)/P(E) = 0.5; for E = existence of natural evil, P(E|H)/P(E) = 0.1; for E = intranatural miracles (prayers), P(E|H)/P(E) = 2; for E = extranatural miracles (resurrection), P(E|H)/P(E)= 1; and for E = religious experiences P(E|H)/P(E) = 2. Given that the denominator from the expression of Bayes's theorem above is the same as P(E), Bayes's theorem simplifies to:

P(H|E) = P(H) × P(E|H)/P(E)

By iterating this calculation six times, for each piece of evidence, Unwin updates the probability for the existence of God to 67%. Michael Shermer assigns alternative values to the conditional probabilities, and uses Bayes's theorem to calculate that the probability of God is 2%.

As Shermer comments, "all such scientistic theologies are compelling only to those who already believe. Religious faith depends on a host of social, psychological and emotional factors that have little or nothing to do with probabilities, evidence and logic. This is faith's inescapable weakness. It is also, undeniably, its greatest power."

Bayes God