In the previous section, we have discussed just one of many sorts of deities that might exist. This one happens to be very powerful and rather inscrutable (and is intended to be a model of a generic Judeo-Christian-Islamic sort of deity, though believers are welcome to disagree and propose--and justify--their own interpretations of their favorite deity). However, there are many other sorts of deities that might be postulated as being responsible for the existence of the universe. There are somewhat more limited deities, such as Zeus/Jupiter, there are deities that share their existence with antagonistic deities such as the Zoroastrian Ahura-Mazda/Ahriman pair of deities, there are various Native American deities such as the trickster deity Coyote, there are Australian, Chinese, African, Japanese and East Indian deities, and even many other possible deities that no one on Earth has ever thought of. There could be deities of lifeforms indigenous to planets around the star Arcturus that we should consider, for example.
Now when considering a multiplicity of deities, say D1,D2,...,Di,..., we would have to specify a value of the likelihood function for each individual deity, specifying what the implications would be if that deity were the actual deity that created the universe. In particular, with the "fine-tuning" argument in mind, we would have to specify P(F|Di&L) for every i (probably an infinite set of deities). Assuming that we have a mutually exclusive and exhaustive list of deities, we see the hypothesis ~N revealed to be composite, that is, it is a combination or union of the individual hypotheses Di (i=1,2,...). Our character set doesn't have the usual "wedge" character for "or" (logical disjunction), so we will use 'v' to represent this operation. We then have
~N = D1 v D2 v...v Di v...
Now, the total prior probability of ~N, P(~N|L), has to be divvied up amongst all of the individual subhypotheses Di:
P(~N|L) = P(D1|L) + P(D2|L) + ... + P(Di||L) + ...
where 0<P(Di)<P(~N|L)<1 (assuming that we only consider deities that might exist, and that there are at least two of them). In general, each of the individual prior probabilities P(Di|L) would be very small, since there are so many possible deities. Only if some deities are a priori much more likely than others would any individual deity have an appreciable amount of prior probability.
This means that in general, P(Di|L)<<1 for all i.
Now when we originally considered just N and ~N, we calculated the posterior probability of N given L&F from the prior probabilities of N and ~N given L, and the likelihood functions. Here it would be simpler to look at prior and posterior odds. These are derived straightforwardly from probabilities by the relation
Odds = Probability/(1 - Probability).
This yields a relationship between the prior and posterior odds of N against ~N [using P(N|F&L)+P(~N|F&L)=1]:
P( N|F&L) P(F| N&L) P( N|L)
Posterior Odds = --------- = ---------- x -------
P(~N|F&L) P(F|~N&L) P(~N|L)
= (Bayes Factor) x (Prior Odds)
The Bayes Factor and Prior Odds are given straightforwardly by the two ratios in this formula.
Since P(F|N&L)=1 and P(F|~N&L)<=1, it follows that the posterior odds are greater than or equal to the prior odds (this is a restatement of our first theorem, in terms of odds). This means that observing that F is true cannot decrease our confidence that N is true.
But by using odds instead of probabilities, we can now consider the individual sub-hypotheses that make up ~N. For example, we can calculate prior and posterior odds of N against any individual D_i. We find that
P( N|F&L) P(F| N&L) P( N|L)
Posterior Odds = --------- = --------- x -------
P(Di|F&L) P(F|Di&L) P(Di|L)
This follows because (by footnote 2)
P(N |F&L) = P(F| N&L)P( N|L)/P(F|L),
P(Di|F&L) = P(F|Di&L)P(Di|L)/P(F|L),
and the P(F|L)'s cancel out when you take the ratio.
Now, even if P(F|Di&L)=1, which is the maximum possible, the posterior odds against Di may still be quite large. The reason for this is that the prior probability of ~N has to be shared out amongst a large number of hypotheses Dj, each one greedily demanding its own share of the limited amount of prior probability available. On the other hand, the hypothesis N has no others to share with. In contrast to ~N, which is a compound hypothesis, N is a simple hypothesis. As a consequence, and again assuming that no particular deity is a priori much more likely than any other (it would be incumbent upon the proposer of such a deity to explain why his favorite deity is so much more likely than the others), it follows that the hypothesis of naturalism will end up being much more probable than the hypothesis of any particular deity Di.
This phenomenon is a second manifestation of the Bayesian Ockham's Razor discussed in the Jefferys/Berger article (cited above).
In theory it is now straightforward to calculate the posterior odds of N against ~N if we don't particularly care which deity is the right one. Since the Di form a mutually exclusive and exhaustive set of hypotheses whose union is ~N, ordinary probability theory gives us
P(~N|F&L) = P(D1|F&L) + P(D2|F&L) + ...
= [P(F|D1&L)P(D1|L) + P(F|D2&L)P(D2|L) + ...]/P(F|L)
Assuming we know these numbers, we can now calculate the posterior odds of N against ~N by dividing the above expression into the one we found previously for P(N|F&L). Of course, in practice this may be difficult! However, as can be seen from this formula, the deities Di that contribute most to the denominator (that is, to the supernaturalistic hypothesis) will be the ones that have the largest values of the likelihood function P(F|Di&L) or the largest prior probability P(Di|L) or both. In the first case, it will be because the particular deity is closer to predicting what naturalism predicts (as regards F), and is therefore closer to being a "God-of-the-gaps" deity; in the second, it will be because we already favored that particular deity over others a priori.