Pope Francis, Thomas Bayes, and the Existence of God

Should liberal atheists find their Bayesian level of rational credence in the existence of God to go up because of the election of the pope?  Should conservative theists find it to go down?

I do not anticipate that what follows is going to have any significant effect on anyone's level of belief or disbelief. It is only an exercise in treating the existence of God as an empirical question. There is a wide ranging, if seldom enunciated, agreement that this question is empirical in one way or another. The dissenters from this proposition are only the supporters of Anselm's ontological argument or of the unmovable stone argument and its siblings. Admittedly many theists think that God's existence is an empirical matter of a special sort, decisively settled by uncontroversial universally available evidence, e.g. the existence of something rather than nothing, or just as decisively, if not universally, by their own direct experience of God. For these people, further evidence is irrelevant to the God question. For all the rest of us, and for any of the foregoing willing to put their commitments aside for the sake of argument, evidence from events in the world, for example from miracles, might well have a bearing on God's existence. This is an exercise in applying generally accepted evidence handling methodology to one world event well covered by the media.

Methodology: If you want to apportion rationally the confidence you place in the truth of a given proposition in the light of new evidence, you pretty much have to use a formula of conditional probability associated with the name “Bayes.” Thomas Bayes was a Presbyterian minister in England at a time when Presbyterians were more popularly known as “nonconformists.” He died in 1761, and would have to be placed high on the list of those who could have had no idea how celebrated their work would be by posterity. "An Essay towards solving a Problem in the Doctrine of Chances” was unpublished and unknown at Bayes' death. My Google of “Bayseian” just now produced 11.6 million results.

That you must use the Bayes formula if you want your degrees of confidence to be coherent and rational in a pretty plausible sense has been elaborated by Ramsey, de Finneti, David Lewis, and Paul Teller. The details are a little technical, but worth working through when you have the time.

Applying the Bayes formula, a large number of people of skeptical and liberal frame of mind should have their degree of belief (subjective probability, degree of confidence, degree of credence) in the existence of God increased by the election of Pope Francis.

A quick summary of pertinent facts: those enfranchised for the papal election, were, by and large, a conservative group. It would not be stretching things far to say that Pope Benedict had packed the College with conservatives. Jorge Mario Bergoglio, was a relative liberal, a reformer, and regarded as a very unlikely choice for Pope. He was not even on the list of possible candidates drawn up by expert Vatican watchers. Yet a supermajority of two thirds of the cardinals voted on an early ballot for Bergoglio, who became Francis. This wants an explanation.

One possible explanation is given by Cardinal Schönbrom, the Archbishop of Vienna, and himself possessor of the name most mentioned as the likely successor of Benedict. “We were driven by the Holy Spirit to this man – he was sitting in the last corner of the Sistine Chapel: This man he is the chosen one.” Schönbrom said that he had “at least two strong signs” that Bergoglio was God's choice, and expressed confidence that the other voting cardinals had experienced similar signs, and that this explains the quick election of Francis.

It is a traditional understanding in the church that when the conclave functions properly it is the Holy Spirit that guides the outcome. That is why the conclave is designed to exclude as much outside influence as possible. If there is supernatural influence, that should maximize its leverage. (Why an omnipotent God would need special measures to increase his leverage will have to be answered by careful deployment of the free will doctrine.)

It is important for one step in the application of Bayes's theorem that Bergoglio was not just any unlikely candidate. It has to be plausible that the Holy Spirit would guide the vote in his direction. There is a natural reading of the Gospels, especially of the sayings of Jesus, on which Francis seems exactly the sort of pope whom Jesus might select, at least from among the candidates available. He is a humble man, given to moderating the splendor of the papal office and to seeing a common humanity across doctrinal differences. What he is most known for is his compassion for and ministry to the poor and oppressed.

One might try to make out of all this an argument to the best explanation. The election of this very unlikely, but arguably divinely favored, candidate cannot be plausibly explained except by the intervention of God. Hence God exists. That doesn't seem too bad as such arguments go. It is fair game, however, in assessing an argument to the best explanation, to take into account other circumstances in which the same explanans ought to be appropriate if it is so in this case. This runs into the difficulty that there have been so many popes in history that were so very different from Francis. Some of these popes were uncontroversially wicked, Alexander VI, for example, whether or not the charge can be made good that he endorsed slavery. (This counterexample, again, would be defended against by relying upon the free will doctrine.)

In any event it is cleaner to retreat from a demonstration by inference to the best explanation, and see if the fact of Francis's election supports some increase in rational credence for the existence of God via Bayes.

Let me take as an example an only modestly confident atheist, one who would grant that there is one chance in a thousand that she is wrong on the God question. Perhaps her subjective probability is not lower because she is aware of the existence of many intelligent and sincere believers and the large volume of testimony of God experiences. So .001 is her Bayesian “prior.” Let us also suppose that she thought the election of a humanitarian liberal to the Papacy if God does not exist would about the same probability. That is, election of Francis without divine intervention was, for her, also .001. Finally, as she reads the Bible, the probability of intervention by God in favor of an otherwise little regarded candidate who would emphasize ministry to the poor and eschew the trappings of papal splendor was middling, say .5.

Let us run her subjective probabilities through the Bayes formula.

P(G/F) = P(G) P(F/G)

                  P(F)

Her credence level for the proposition that God exists given the election of Francis should be her prior credence for God's existence times her estimate that Francis would be elected if God exists divided by the total probability of Francis being elected. Her estimate of the total probability of the Francis election, in turn, should be its probability if God exists plus its probability if God does not, each adjusted by her prior probability of God's existence. So

P(G/F) =             P(G) P(F/G)     

                 P(F/G)P(G) + P(F/~G)P(~G)

Numerically

                    .001 * .5  

            .001*.5 +.001*.999

                  .0005                  = .36

              .0005 + .0009

So her assessment of the probability that God exists should jump, on the basis of the election of Francis, from one in a thousand to better than one in three.

Of course the degree of this affect will vary with the prior probabilities. It will vanish altogether, indeed go the other way, if P(F/G) is very low instead of very high.

The conservatives who believed that if God exists there is very little probability that he would permit a man like Bergoglio onto St. Peter's throne should have found the credence of their belief in the existence of God to decrease. Suppose our religious conservative is as modest as our atheist, granting that there is one chance in a thousand that he is wrong about the existence of God. (Perhaps he finds it sobering that there seem to be sincere, intelligent atheists, and that it is quite possible that he might be one himself, had he been born into a different family, just as he might well have been a Hindu, or Buddhist, and might well have been quite confident in his religious beliefs in those possible worlds.)

The conservative before the election should take the probability of Francis's being elected if God exists to be very low because God would not approve of Francis's liberalism. Suppose it is also .001. Suppose he regards the election of Francis were there no God to be .01. Politics happen. Again:

P(G/F) =              P(G) P(F/G)

                  P(F/G)P(G) + P(F/~G)P(~G)

In this case:

                .999 * .001

          .999*.001 + .01*.001

               .000999                    = .98

          .000999+.00001

So this conservative's rational credence in the possibility that God not exist should go from one in a thousand to 20 in a thousand.

More generally, all those who acknowledge that Bergoglio was an unlikely selection and that he might well be the candidate God would chose are rationally required to increase the credence they put in the proposition that God exists. Those who insist that Bergoglio was not at all the sort of person that God would want for pope should have their rational credence in God decreased by the election.

There are exceptions for those who thought that the probability of God's existence was a perfect 1 or a flat 0. It is a matter of definition, of course, that someone with a credence of 1 in a proposition cannot have it increased by additional supporting evidence, and a credence of 0 cannot be reduced by new disconfirming evidence. More interesting is that a 1 prior does not get reduced by negative evidence or a zero prior increased by positive evidence. Look at the formula to confirm this.

There is a faction of theorists who believe that this treatment of limit priors is a defect in the Bayes formula in its advertised role as mediator of rational degrees of confidence. After all, sufficiently good evidence should affect even the greatest certainty. Another faction, led by Lindley, would say that credence of 0 or of 1 is never rational.There is always a probability that one is mistaken – even if it requires a vast number of zeroes or nines after the decimal point to represent it. When doing Bayesian degree of confidence calculations, the zero and one priors should be assigned only to the unmovable dogmatists, and these have left rationality behind.

Lindley would grant an exception for logically or mathematically true propositions, but in this, I think, he is mistaken. Assuming for now that mathematical truths depend in no way upon the state of the physical universe, empirical evidence will have no bearing upon their truth. This is a reason to call them “necessary.” Still, what we are dealing with here is subjective probabilities, degrees of credence or confidence. Empirical reality is very much involved when it comes to our affirming of propositions mathematical. It is a matter of brain functioning, and brain glitches are always possible. We readily acknowledge that mistakes can creep in when we multiply 4 digit numbers in our heads. As the calculation gets simpler, our rational confidence in our results properly gets higher. When we get to Kant's “5 + 7 = 12,” there is a temptation to say that the probability of a mistake is zero. It is a temptation, however, that should be overcome. There is always a probability, be it very, very small, of very big malfunctions.

So, on my view, there should always be an end to the 9s or the 0s following the decimal point for any degree of confidence matter. The problem is not the Bayesian treatment of the flat zero or perfect one, it is the perfect one and flat zero themselves.

You may have been waiting patiently through all these mildly technical details to object that this is all silly. Perhaps the the election of Francis had a small effect on a few fence sitters, but none on any theist or atheist you know. I am wrong, you will say, not only about the dogmatic atheists or theists. Even those that are open to evidence, and who feel that Francis was a remarkably good or a remarkably bad choice, will, you anticipate, sincerely report,after searching their intuitions, that they can detect no difference in their pre-election and post-election degree of belief.

What you imagine is just what I would expect. Let me here confess that I tuned my examples to make the Bayesian updating dramatic. Truth to tell, it is unrealistic for an atheist to go from a God credence of one in a thousand to 360 in a thousand. Even the theist who went from 999 to 980 is a little hard to believe. The reason for this, however, is that my examples are outliers. Almost all atheists have much lower priors, and almost all theists much higher.

For the atheist with every zero you add after the decimal point to the prior, you get an additional zero in the updated credence. So if the atheist of our example had been of the 10 zero variety, she would still have ended up with 8 zeroes. We have so little ability to navigate our intuitions around very small numbers. We really don't know what the difference between a probability of .000001 and .00000001 would feel like. So we can anticipate the atheist who says that he finds no intuitive difference in his subjective probability of God's existence, post Francis. All I ask the atheist to assent to is that there should be a positive increase in her degree of credence because of the election of Francis, on all our assumptions, even if she cannot detect that increase in her intuition of her before and after credence levels because of their imponderability.

Similarly, the conservative, with the contrary belief about divine favor, should concede that his rational credence is somewhat diminished, even if that credence remains imponderably high. The imponderability of the low probability of God for (most) atheists I argue is a significant, if little commented upon, feature of the atheist's epistemology. Symmetrically, imponderability is a feature of the very high degree of credence of (most) believers. See post of Sep 10, 2014.

In addition, empirical studies show that people tend to update their subjective probabilities less aggressively than Bayes (and best practice) would counsel. We tend, in this respect, to be "conservative," which is also, no doubt, involved in the intuition of both atheists and theists that such evidence as the election of Francis or the 2004 Indian Ocean tsunami do not affect their belief.