Wednesday, August 1, 2007

Bogus Probabilistic Judgments and God

There are frequentist and Bayesian approaches to probabilistic judgments about the likelihood that some claim is true.

In frequentist probability judgments we must have prior observations of the relevant data to produce a statistical claim such as “In the United States, 51% of babies born are female.” With that data about the previous rate of female babies, we can pick a random baby being born and predict that there is a 51% chance that it will be female. This claim is represented as P(f) = .51.

So if the property in question is certain, then the P(x) = 1, and if there is no chance that the property will occur then P(x) = 0.

In many cases, however, where we have not been able to observe prior relevant instances of the property or phenomena in question the Bayesian probability assessments allow us to form a probability judgment. If we were trying to predict the likelihood that a new technology would break down such as a missile that had never been launched before, or the likelihood that a basketball team in one division could beat a team in another division where they had never played before, we would have no previous data upon which to base the estimation.

We would need to use our other knowledge of factors that would be likely to affect the outcome. Suppose that I am trying to judge the probability of c = Hilary Clinton wins the United States presidency in 2008. And I have a body of relevant background knowledge K. We are asking what is the probability that c is true given background information K? or P(c|K)? If we had frequency data, like in the case of the percentage of female babies born in previous years, we could use that in our background knowledge.

The Bayesian formulations of probability are particularly useful because they allow us to distinguish between different probabilities that different individuals would assign to the same event given different subjective kinds and amounts of background information. If Susan has a great deal of information about Clinton’s bid for the presidency, and that information, as she sees it, strongly indicates that Clinton will win, then P(c|K) for her will be higher than Mary who has a lot of information that appears to indicate that Clinton will lose. So naturally the kinds of background information, the amount, the quality and reliability of it will affect the resulting estimations that each woman gives to the event. If Susan primarily reads left wing, liberal, pro-Clinton news sources, yet she thinks that those sources give her an accurate picture, then the likelihood of c|K will approach 1.

Another important component of point about probabilistic reasoning is that the total probability of all the possible outcomes must add up to 1. So if there are only two teams, A and B, competing to win a game, then since one of them must win, P(A|K) + P(B|K) = 1. And if there are currently 12 candidates for the Republican, Democrat, and all third party nominations, then the probability of all of those added together must be 1. So as the options increase, the likelihood that any one outcome will result is diminished. This aspect of probabilistic reasoning is relevant in the God cases we will consider.

What about God? Bayesian reasoning has been used lately to calculate the probability that God does or does not exist. In a famous modern version of the design argument, Richard Swinburne has argued that our background information renders it exceedingly unlikely that all of the matter in the universe would behave according to such a simple, elegant, consistent, and orderly set of natural laws.

Swinburne says,

“That there should be material bodies is strange enough; but that they should all have such similar powers which they inevitably exercise, seems passing strange. It is strange enough that physical objects should have powers at all—why should they not just be, without being able to make a difference in the world? But that they should all, throughout infinite time and space, have some general powers identical to those of all other objects (and they all be made of components of very few fundamental kinds, each component of a given kind being identical in all characteristics with each other such component) and yet there be no cause of this at all seems incredible. . . . Yet this orderliness, if there is no explanation in terms of the action of God, is the orderliness of coincidence.” (70 Pojman—The Existence of God, 1979)

As Swinburne sees it, there are only two possibilities. Either God is responsible for the existence of an orderly universe, or it is the product of random chance. So we are comparing the probability that the orderly universe came about by God’s hand to the probability that it just happened. Most of us would agree that it does seem exceedingly unlikely that chance could have produced all natural law for all matter over all time. And Swinburne maintains that we have background information about God’s preferences for beauty, order, simplicity, and structure, and we have reason to believe that God would desire an environment that would be conducive to our existence and the growth of our knowledge and moral virtue. So the resulting probability that we would find a universe like this one as a result of God’s act of creation is vastly greater than the probability that we would find one that is the product of random chance. Thus, the probability that God exists is very high.

The problem with this sort of reasoning, even though it may appear to be compelling at first, is that it reflects a profound lack of imagination and myopia about the range of possibilities.

Suppose I am visiting a foreign country and I wander into a stadium where a very popular professional sports game is being played. I don’t recognize the sport, I know nothing about the rules, and I have very little information about the context of the match. But it does appear to me after watching the plays for a bit that there are two teams, and each is trying to outscore the other. I also see a big scoreboard at the end of the field and an enormous trophy that appears to be waiting for the winners. I look at the teams, see the red jersey team score a few times, and figure out that they are ahead by many points on the scoreboard. So I conclude that given that one team’s losing means the other one loses, and that red is substantially ahead, the probability that the red team will win the trophy is very high, and the probability that the green team will win the trophy is very low. The two probabilities must add up to 1, so I make an educated guess that the red team has a .8 chance of winning the trophy and the green team has the other .2 chance.

I leave before the match ends, and later I ask one of my local friends about the game. He tells me that in fact there is a very large tournament that is being played. The match between the red and green teams is just one of over a 1,000 matches that will be played between 2,000 teams. The winners of each round move on to the play other winners until a winner from among all of the 2,000 original teams is awarded the big trophy that I saw. Obviously, this new information and all these new possibilities that I did not previous know about affect my estimation of the likelihood that the red team would win the trophy. They may have been likely to win that match, but the likelihood of their beating every other team in the tournament for the trophy is an entirely different matter. And with my gross lack of information about the game and the other teams, I realize I am not in any sort of position to assign a likelihood to red’s winning the trophy now. With games, hazarding a guess in these circumstances is harmless. But if the outcomes in question were more important, say about whether the navigation system in a new passenger plane will work, it be would be irresponsible and foolish of me to offer my ridiculously under informed opinion about the outcome.

The mistake I made in estimating their chances of red’s winning as so high is comparable to the mistake that people, including Swinburne, frequently make with regard to God’s existence. If in fact there were only two possible outcomes—either the red or the green team wins, or either God created the universe or it happened by chance—then odds that one of those two outcomes will result will be much higher than if there are 1,000 or 2,000 total possible outcomes. If there are an infinite number of possible outcomes or hypotheses, then it would not appear that we could reliably or successfully apply Bayesian probabilities to the scenario at all.

That is the problem with leaping to the God option as the only other possibility besides random chance. What we are trying to explain, according to Swinburne, is the cause of all the orderly matter in nature. How many different supernatural forces might be responsible? It might have been some supernatural but sub-omni being—one that lacks all knowledge, or it might have lacked all power, or it certainly might have lacked all goodness; there might have been 2 of them, or 10, or 50; it might not have been a blind, unconscious force, not a personal being; it might have been aliens; it might have been 2 aliens; or a stupid alien; or a negligent alien, and on and on and on. What we have seen with the teleological and cosmological arguments is that the evidence before us underdetermines the God inference. God might have done—it seems possible that an omni-being could have done it. But many other forces that are lesser than God are consistent with the results we find too.

Since this list is infinitely long, we can’t really get any help from Bayesian reasoning because we just don’t have adequate background information to assign probabilities to all of these hypotheses, and there appears to be an infinitely long list of them. Over and over again in God arguments that invoke Bayes theorem to motivate their conclusion we see this failure of imagination and artificially narrow assessment of the possibilities. Stephen Unwin is guilty of the same mistake in The Probability of God. It happens so often that we have to suspect that they are being disingenuous in introducing probabilistic reasoning of any kind in the first place. People are easily overwhelmed by all the technical talk and formulas. But when the results at the end of the page show that the probability that God exists = .85 or some such high number, we’re all suitably impressed with how “scientific” it all seems. But if we are serious about considering all the possible explanatory hypotheses and trying to sort between them, then we’ll have to treat the God story as just one competing among many. And if the list of possibilities is infinitely long, then we should reject this bogus application of Bayes theorem to the matter in principle.

3 comments:

Jon said...

It is interesting to note that concerning the amount of possibilities of laws of nature that the main body of physicists at Stanford - notably Susskind (who independently intvented string theory) and Linde (inventor of inflationary theory) have put together a strong theory that suggests that our universe may 'probably' be apart of a megaverse that has at least 10 to the 500 power of various 'universes' in it, almost all with varying laws of physics.

S D Owen said...

Great post, one of your best.

I want to raise something new here; at least I think it is.

How can any theist invoke probability in the first place?

Probability presupposes a probabilistic universe, and that would seem contradictory, at least to me, with the idea of absolute truth/god.

A PU, lol, a Probablistic Universe, implies that certainty is a mistaken concept, one which should be theoretically eliminated.

In a PU, which we apparently exist in, "certainty" can only mean "high probability."

Now, how does absolute truth/god govern or affect a PU at all?

It would seem to me logically incoherent that absolutism in any form can function causally in an ontologically probabilistic environment.

Any thoughts on this, thinkers?

>:D

paulve said...

I have never liked proofs for the existance of non-existance of god(s). It just seems like such a waste of time.

Could omnipotent beings design a universe in which they could not be detected? If not, they are not really that omnipotent. If yes, then how can you be so sure that he/they didn't.
(Its your interpretation of the evidence vs. an omnipotent being)

When you find evidence of a cockroach in your appartment, the probability that there is only one is quite small.

Not only is the God story one possibility amoung many, but the one god story is only one possibility amoung many.

I am not so sure that certainty is eliminated in a probalistic universe. From what I remember of quantum mechanics, it sometimes gives solutions that certain results cannot occur, because the sum of the squares of the probability amplitude is zero. In fact it is a powerfull argument, for the standard model that it correctly predicts that certain transitions cannot occur, thus explaining (or perhaps just predicting) the spectral lines of certain gases.