# What is Bayes's theorem, and how can it be used to assign probabilities to questions such as the existence of God? What scientific value does it have?

Note that although this is receiving much attention now, quantifying one's judgments for use in Bayesian calculations of the existence of God is not new. Richard Swinburne, for example, a philosopher of science turned philosopher of religion (and Dawkins's colleague at Oxford), estimated the probability of God's existence to be more than 50 percent in 1979 and, in 2003, calculated the probability of the resurrection [presumably of both Jesus and his followers] to be "something like 97 percent." (Swinburne assigns God a prior probability of 50 percent since there are only two choices: God exists or does not. Dawkins, on the other hand, believes "there's an infinite number of things that some people at one time or another have believed in ... God, Flying Spaghetti Monster, fairies, or whatever," which would correspondingly lower each outcome's prior probability.) In reviewing the history of Bayes's theorem and theology, one might wonder what Reverend Bayes had to say about this, and whether Bayes introduced his theorem as part of a similar argument for the existence of God. But the good reverend said nothing on the subject, and his theorem was introduced posthumously as part of his solution to predicting the probability of an event given specific conditions. In fact, while there is plenty of material on lotteries and hyperbolic logarithms, there is no mention of God in Bayes's "Essay towards Solving a Problem in the Doctrine of Chances," presented after his death to the Royal Society of London in 1763 (and available online at www.stat.ucla.edu/history/essay.pdf ).

One primary scientific value of Bayes's theorem today is in comparing models to data and selecting the best model given those data. For example, imagine two mathematical models, A and B, from which one can calculate the likelihood of any data given the model (p(D|A) and p(D|B)). For example, model A might be one in which spacetime is 11-dimensional, and model B one in which spacetime is 26-dimensional. Once I have performed a quantitative measurement and obtained some data D, one needs to calculate the relative probability of the two models: p(A|D)/p(B|D). Note that just as in relating p(+|s) to p(s|+), I can equate this relative probability to p(D|A)p(A)/p(D|B)p(B). To some, this relationship is the source of deep joy; to others, maddening frustration.

The source of this frustration is the unknown priors, p(A) and p(B). What does it mean to have prior belief about the probability of a mathematical model? Answering this question opens up a bitter internecine can of worms between "the Bayesians" and "the frequentists," a mathematical gang war which is better not entered into here. To oversimplify, "Bayesian probability" is an interpretation of probability as the degree of belief in a hypothesis; "frequentist probability is an interpretation of probability as the frequency of a particular outcome in a large number of experimental trials. In the case of our original doctor, estimating the prior can mean the difference between more-than-likely and less-than-likely prognosis. In the case of model selection, particularly when two disputants have strong prior beliefs that are diametrically opposed (belief versus nonbelief), Bayes's theorem can lead to more conflict than clarity.

More generally, Bayes's theorem is used in any calculation in which a "marginal" probability is calculated (e.g., p(+), the probability of testing positive in the example) from likelihoods (e.g., p(+|s) and p(+|h), the probability of testing positive given being sick or healthy) and prior probabilities (p(s) and p(h)): p(+)=p(+|s)p(s)+p(+|h)p(h). Such a calculation is so general that almost every application of probability or statistics must invoke Bayes's theorem at some point. In that sense Bayes's theorem is at the heart of everything from genetics to Google, from health insurance to hedge funds. It is a central relationship for thinking concretely about uncertainty, and--given quantitative data, which is sadly not always a given--for using mathematics as a tool for thinking clearly about the world.

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1. 1. johnsorflaten 01:54 AM 3/3/08

Let's apply Bayes's theorem to politicians. For example, what is the marginal probability settting public policy, given an attitude of conservative or liberal (OK, let's be more accurate: Republican or Democrate). So, the debate hinges on "why" is someone Republican or Democrate. Is it personal experience? If so...then why should public policy, which affects the public, be contingent on a person's extrapolation of their ego-bound view of the world? Or, more to the point, why should the unexpert Republican or Democrate be "voting" on acceptability of data such as "the oceans will rise 2 meters in the next x years"? Seems to me that driving a taxi in Manhatten is not meant to be a group activity by the passangers. Instead, they set the goal, and let an intelligent driver get them there. We'll allow for the passengers to judge whether the driver is ethical, and taking the most economical route, however.

2. 2. eco-steve 07:50 PM 10/4/08

The question we should be asking is not the probability of an intelligent God existing, but the probability that Darwinism is true. The former cannot be demonstrated, as its proponents declare that it is solely a question of belief. The latter can be proved without a shadow of doubt by the study of genes.

3. 3. Frank@existenceofgod.org 11:37 AM 3/20/12

Interesting piece. First time I've come across an article about a mathematical equation regarding to the existence of God. :) I can see why some people would need something like this, others gravitate towards arguments such as the one found on Stanfords.edu http://plato.stanford.edu/entries/teleological-arguments/

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