Sunday, July 04, 2010

God and Bayes, Part 2

In my last post I mentioned an argument I once saw for the existence of God based on Bayes’ theorem. In fact I’m talking as far back as 20 years when in 1990 I read a book called “Reason and Faith” published in 1989 by Roger Forster and Paul Marston. Their argument can be found on the penultimate page of the book, page 427.

Expressed in terms of the Venn diagram notation I introduced in my last post Forster and Marston set P(A) = P(God exists) and P(B) = P(the universe is inhabitable). Let’s represent that as P(G) and P(H) respectively (“G” for “God” and “H” for “habitable”). Rendering this in Venn diagram form we have:

…where the domain labeled G represents the cases favourable to God’s existence and the domain labeled H represents the cases favourable to an inhabitable cosmos.

As they say, a picture is worth a thousand words (or a thousand calculations!): In configuring this diagram in the way that I have I’ve actually given away Forster's and Marston’s game: Notice that the area for an inhabitable universe is rather small and that very little of it overlaps onto the “No God” area. We can immediately see, then, that in this application of Bayes’ theorem most cases of H cluster with cases for G. It is therefore almost trivially obvious from the above diagram that given H, G is very probable! That is, given we live in an inhabitable universe we deduce that the existence of God is very likely! However, I must concede that giving H and G such a close relationship is a moot point and I'll come back to it below.

The first query that should be raised here is the philosophical one: Viz: Is this really a valid application of probability? In particular, in the context of an interpretation of probability based on the simple enumeration of cases is it really coherent to talk of “T items” some of which may have the property of God existing? Just what are these T items? Are they possible cosmoses? Are we saying that it is possible to imagine cosmoses where there is no God? Now, if we were drawing items from a jumbled bag of objects it is fairly obvious that this Venn interpretation works: Here talk of “T items” is intelligible and these items may have equally as intelligible properties such as colour and shape; for example, A could stand for “orange” and B could stand for “square”. However, it is far from obvious that such a banal model scales up to whole cosmoses and beyond. The upshot is that when people use probability to argue for God they tend to employ a much more philosophically diffuse concept of probability than the somewhat prosaic (and coherent) model involving shuffled objects and the enumeration of cases; usually the rather enigmatic Bayesian “degree of belief” view of probability is adopted, which I think is the view adopted by Forster and Marston. My own guess is that the human mind is built to deal with those prosaic junctures that can be treated with case enumerations; if so then the mind’s probability intuitions, when applied to the outermost cosmic frame, may not be an intelligible application.

Anyway, in spite of the foregoing philosophical disquiet, let me proceed with the actual argument used by Forster and Marsden. Firstly they actually employ a modified form of Bayes’ equation. They don’t present a proof of this modified form so I have provided it below: (I use notation like !G to mean the negation of G; that is “G false”)

Forster and Marsden present the latter equation (equation 5) with their guesstimated values already plugged in. The values they use I explain below:

P(G) = 0.000001. Here F&M are trying to take the point of view of the atheist who, although he might claim that he doesn’t believe there is a God, nevertheless may admit in his heart of hearts that his “degree of disbelief” is not absolute. He may express this level of uncertainty as, say, a “1 in a million of chance of God existing”.

P(H|G) = 0.00001 Again, F&M plug in a very conservative value for the chance of God creating an inhabitable universe if He exists.

P(!G) = 1 – P(G) = 0.99999

P(H|!G) = 0.000….0001 where “…” represents billions of zeros! It is the value of this quantity that is crucial (and contentious) to F&M’s argument. It is this value which determines, as I have noted above, that the Venn circle of inhabitable cosmoses largely resides in the circle for God.

When the above values are plugged into the equation it is fairly obvious why we arrive at a probability very close to 1 even for very small values of P(G): It is consequence of the following inequality:

P(H|G) P(G) >> P(H|!G) P(!G)

The huge difference between these two terms, which are summed in the dominator of equation 5, ensures that the denominator is all but equal to the numerator thus returning P(G|H) ~ 1. The strategy of F&M is fairly clear here: Because P(H|!G) is so small the required conclusion, P(G|H) ~ 1, even works for atheists! For one might expect even the most hardened atheist to have just a tiny, tiny little bit of doubt about his position – conceivably he might acknowledge that he feels there is a one in a billion, billion chance that God might exist. Thus according to F&M even with these small odds we still get P(G|H) ~1 because P(H|!G) is far, far smaller than even a billion billionth, having as it does billions of zeroes after the decimal point. Therefore, since it is clear that the universe is inhabitable it follows that G must exist!

But even leaving aside the philosophical contention over whether or not it is right to interpret probability in a vague Bayesian “degree of belief” sort of way, F&M’s analysis still begs questions: Viz: Where do they get the extremely low value of P(H|!G)? This seems to be based on a quote by Paul Davies where he remarks that random choices are extremely unlikely to arrive at the selection of factors required by a universe capable of evolving life. In fact the odds against such a configuration of factors coming together, if selected at random, is, according to Davies, “…one followed by a thousand billion, billion, billion zeros at least”. I believe the basic perception of Davies here is correct: The configurations we see around us are taken from an extremely rare class of possibility; therefore left to random selection alone we are looking at some absolutely minute probabilities of formation. Evolution makes no dent on this improbability: it merely trades configurational improbabilities for the improbability of selecting the right physical regime that favours evolution. Thus whilst P(Life|Evolution) and P(Evolution|Laws of Physics) may return realistic probabilities, the space of possible physical regimes seems to be so large that on the basis of random selection alone P(Laws of Physics) is vanishingly small!

Given this irreducible improbability it is no surprise that the atheist is tempted to make recourse to (very speculative) infinite universes and/or multiverses in an attempt to balance these miniscule values with huge numbers of trails. Although I am myself not impressed with such resorts (on the basis of them being effectively a kind of “turtles all the way down” type explanation), we see that Forster’s and Marston’s assumption that P(H|!G) is extremely small is not unassailable if one starts to postulate huge numbers of trials.

Whatever the weaknesses in Forster’s and Marston’s argument it is significant, however, that it was submitted in 1989, just before William Dembski and friends made their “ID” splash. At the top of the section of F&M’s book where they use Bayes’ Theorem we find the title “The Improbable Us”. By way of introduction they ask: “How apart from design, can we explain the mind boggling improbability of the inhabitable universe?” In short F&M are using a design argument here and yet they are at least sympathetic toward evolution if not evolutionists themselves. Clearly these were the days before anti-evolutionism had become so strongly identified with ID. But F&M’s basic idea is essentially the same as William Dembski’s argument; the apparent presence of highly improbable classes of outcome are assumed to point to intelligent contrivance. But, and this is something you don’t hear enough from William Dembski and friends, this improbability exists whether or not evolution has occurred.

ID arguments based on the probabilities of likely causes flow out of, I submit, a subliminal deistic theology containing a built in weakness that actually helps to undermine ID. This class of argument taps into an ulterior understanding that there is a distinction between patterns that are intelligently designed and those that are not. This can be seen in my “intelligent design” Venn diagram where it is assumed that there are possible cases which are not sourced in the intelligent contrivance of God. William Dembski’s so called “explanatory filter” also brings out this dichotomy: The filter presupposes a distinction between patterns sourced in the “natural forces” of law and chance and those that have been intelligently designed. By promoting this apparently polarized view of the artificial vs. the natural this form of ID actually suggests the way in which it can be challenged; for if it can be shown that a pattern (such as life) is a product of law and chance, then to all appearances it would seem the ID theorists have lost the argument. These theorists have inadvertently adopted a kind of dualism that gives natural forces a quasi autonomous status and in so doing they have made a rod for their own backs. As I have remarked before, this dualist philosophy very naturally leads to a premium being placed on very explicit divine interventions in the natural order: The implication is that these natural processes, in the absence of Divine intervention, are able to at least caretaker the cosmic show themselves, thus implicitly raising their status to a quasi autonomous level; thus the proof of God's presence is only manifest when He explicitly intervenes in the day to day running of the cosmos. In short this form of ID philosophy has a very close subliminal relationship with deism and atheism; for if a case can be made that natural forces are sufficient to explain, say, the configurations of living things, then it is a very short step to conclude that either God is an absentee landlord or does not exist. It is no surprise, then, that contemporary dualist ID theorists have effectively equated ID theory with anti-evilutionism, and therefore by default, are very vociferous in their attacks on evolution. Moreover, on the flip side of these attacks we find that for atheists evolution is the proof of a “blind watchmaker”. Both parties conceive natural forces in all but deistical terms, terms that ultimately subvert the belief in a part-time interventionist God.

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