Saturday, March 29, 2008

Theism and Probability

Or: A Prolegomenon of Prodigious Length to the Further Explanation of my Apostasy

When I'm asked (as I often am) why I left the Faith, the first part of my answer has to do with recognizing that I wasn't being rational in how I confronted the question of God's* existence, how I was seeking a justification to keep believing rather than sincerely investigating, and that my real (social and psychological) reasons for belief were almost irrelevant to the actual existence of God. Thus I resolved to ponder it as rationally and objectively as I could.

But the second part of my answer— that is, why this examination concluded in atheism— is something I don't think I've adequately explained. After all, I continue to reject as proofs all the various arguments pro and con, finding them wanting for pure certitude. But if I haven't found a reason to be certain, one way or another, shouldn't this leave me an agnostic— or make me susceptible to Pascal's Wager?

Well, not all uncertainty is equal; there are plenty of logically possible claims (outlandish conspiracy theories, for example) we can't disprove with certainty, but still feel justified in discarding as highly improbable, and many conjectures that can't be proven with certainty (say, the classical Latin pronunciation) which we take as quite probable. If we're being quite honest, we should acknowledge that there is a possibility of being deceived in simply everything we are told, all we perceive, and even the logical truths we deduce (even our minds can be compromised); we have no guarantee against this, only the estimation that most of these deceptions are vanishingly unlikely.

Thus there is an implicit notion of subjective probability in all the claims we make, even the firmest ones; and the reason that I count myself an atheist today is not because of one knock-down argument against the existence of God, but because the total scope of my human experience leads me to assign the existence of the Christian God a very small probability.

There's nothing original here— this is ground covered both by J.L. Mackie's The Miracle of Theism and Stephen Unwin's recent The Probability of God (which I haven't read), one of which arrives at a probability near zero and the other a 67% likelihood. I hope to discuss my own reasoning along these lines, but first I feel I ought to address an objection to this whole enterprise.

In my comment boxes last year (sadly lost to the ravages of Blogger), I took part in the argument over whether one could even assign a probability to something like the existence of God, since such a thing was either true already and thus deserved 100%, or false already and thus deserved a 0%; I didn't like that objection, but raised my own in that the question was foundational— our very means of interpreting experience depends on whether we take the world as simply material or as the intrinsically meaningful creation of the Christian God (among other possibilities), and all our evidence depends on that to some degree. Since then, I've heard a stronger form of the first objection— that even if we could assign probabilities to things besides future events, the existence of such a God may in fact be logically necessary or logically impossible, and it would be absurd to assign a positive probability to a logical impossibility!

These objections are all essentially based in an intuitive frequentist definition of probability— that probability is an intrinsic property of future events (like the probability that a certain weighted die will roll a 6). However, even in my examples above of historical events, I've been using a quite different application of probability theory, the Bayesian definition†; and this is a method of arriving at subjective probabilities which is applicable to even the question of the existence of God.

Bayesian probability concerns the likelihood you, as a rational agent with certain information, should ascribe to some state of affairs, rather than an intrinsic probability. Two people with different information can arrive at different Bayesian probabilities for the same event, without either being wrong. Of course, in the case where you have full knowledge of all the relevant factors affecting some future event (i.e. the precise weighting of the die), your Bayesian probability should be the frequentist probability.

The difference can be illustrated in this way: Let's say that our friend Xavier comes to us carrying a bag with three blocks in it, and tells us that either two of them are red and one blue (Case R), or two of them are blue and one red (Case B). (To simplify, let's say we're sure Xavier's telling the truth, and that neither of us have any reason to suspect one case over the other.) Well, then the frequentist probability of me drawing out a red block is either 1/3 or 2/3, but we don't yet know which. But the Bayesian probability is 1/2. We can deduce this easily, because I like to gamble on such events. If I offer you better than even odds the first block will be red, you should take the bet; if I offer you worse than even odds, you should refuse it. Thus your probability is 1/2.


But I draw the first block, and it's blue (Event A— sorry to be introducing all these). Now what Bayesian probability should you assign the next block being blue, given that the first was blue? Well, if there were 2 red and 1 blue originally (Case R), the probability would be 0 since the only blue block is gone; and if there were 2 blue and 1 red (Case B), there would be 1 red and 1 blue remaining, so the probability would be 1/2; that is to say, the probability of drawing a blue block given A is 0 times the probability of R given A plus 1/2 times the current probability of B given A. Since we had no reason to favor R over B or vice versa, it's clear that the original probability of B was 1/2. But how do we get the probability of B, given that A took place? Here we can apply Bayes' Theorem (which, if you're wondering, is an actual mathematical theorem about any sort of probabilities, here being applied to Bayesian probabilities in particular): the probability of B given A is the probability of A given B, times the original probability of B, divided by the probability of A. The probability of A given B is 2/3, the probability of drawing out a blue block from a bag of 2 blue and 1 red; the probability of B was our original probability 1/2 that we were in case B; the probability of A— drawing out a blue brick— was figured to be 1/2 in the preceding calculation. So the probability of B given A comes out to 2/3, and our new Bayesian probability of drawing a second blue brick should be half of that, or 1/3. Thus you should accept my bets now if I offer better than 2-to-1 odds for the brick being blue, and not if I offer less. (My example sucks, I know. If you want a better explanation, see the second footnote.)


The key points here are that (1) we can use probability theory to discuss things that either are or are not already, in terms of how probable we should deem them given what we know, and that (2) there is a well-defined and uncontroversial process for updating this probability given more information.

Here's the thing: people have been implicitly doing just this all along, just without calling it probability theory, and using subjective feelings of certainty and unconscious heuristics rather than using numbers! In a murder trial, the jury starts with a low prior probability (presumption of innocence, the prior assumption that this defendant is a priori no likelier than any other citizen to have committed the crime), hears the evidence and its link to the accused crime and updates its estimates accordingly, and ultimately decides whether the probability of guilt is above some undefined but near-one value (beyond a reasonable doubt).

Or a historian says that it is unlikely Napoleon was poisoned, based on all the historical sources. This is a perfectly meaningful claim that is another implicit Bayesian probability! Or, better yet, consider the Riemann Hypothesis, which has been shown to hold for the first several trillion of the infinitely many cases, and whose consequences seem quite in keeping with what mathematicians have guessed. Although the Riemann Hypothesis is in fact either logically valid, logically invalid, or logically undecidable, it seems to make sense when mathematicians say amongst themselves that it is "almost certainly true".

There is of course one point where assumptions creep in— we have to have some "prior probability" before we count any information; in our example we had the prior probability of 1/2 that 2 bricks were red and 1 blue, and 1/2 that 2 were blue and 1 red (neglecting the almost-zero probabilities we ascribed to Xavier lying, or the bricks having turned green in the bag, etc). This of course gets very tricky with something like the existence of God; a significant difference between Mackie's estimate and Unwin's above is that Mackie started with a low prior probability, while Unwin started with 50%. Furthermore, the probabilities that Unwin assigns to such things as the probability of a sense of morality in the human mind, given the existence or nonexistence of God, are as up for debate as the terms in the Drake Equation for the number of expected alien civilizations. So this is a complicated matter indeed.

At any rate, the point of this post is to confront the objections to discussing the probability of God's existence. The "foundational" objection is taken into account by Bayes' Rule: even if the same results are thought to be possible with or without such a God, they may be more likely to result from one than from the other, and thus they count as evidence affecting the probability. And on the objection that it might be settled logically one way or another beyond our knowledge— well, this is the same objection that you might make against the mathematicians! If our assertion that given what we know, the Riemann Hypothesis is almost certainly true, isn't absurd, then neither is a best guess about the probability that a certain God might be behind the Universe. It may be a more poorly calibrated guess, of course; but it remains a best guess.

So much for the prolegomenon, I hope.

*Right now, I'll mostly be talking about the Christian idea of God, so forgive my sloppiness in dropping the qualifiers (the Christian God, this or any other God, etc). Later I should discuss the wide panoply of possible religious ideas.

†Bayesian probability is one of the favorite topics of a very interesting and occasionally strange blog, Overcoming Bias; and one of the authors does quite a good job with An Intuitive Explanation of Bayesian Reasoning, and its followup, A Technical Explanation of Bayesian Reasoning.


Adam said...

Oh My! Things have changed. From your time as an undergraduate, my personal odds that you would not only think probabilistically, but as a Bayesian, would have been very very long.

Subjective evaluation brings a troubling lack of certainty, I see. Too bad the blog title "Mathematics After the Fall" is already taken.

some guy on the street said...

I hate to reply so tersely to such an elaborate essay; since you have deftly identified the TWO subjective points in arguing (for or against) God from empirics, I fear I can't say much beyond

1) I don't see how there can be debate between atheists/theists on a "sensible" prior distribution

2) I don't see how atheists/theists can debate "sensible" conditional probabilities, for updating priors to posteriors.

Altogether, this looks from the start like a dead-end for argument. But I wish you every blessing in your hunt for Truth. The fact that it is Truth you're seeking is for the moment my strongest hope that you'll end up somewhere.

Patrick said...


I'd have found it unlikely too. I'm still honored you found "Mathematics After the Fall" such a mot juste.


Very good points. To the first, I'd say that if the drift of evidence one way or the other is strong enough, we could start with prior probabilities different by orders of magnitude and still wind up near the same place.

As for the other, of course everything is going to be a very subjective estimate— but not an arbitrary one. Perhaps this line of argument won't suffice to convince people one way or the other for that reason; perhaps the best one can do is to figure it out as honestly as one can, and ask others to do the same. We'll see!