1. The General Case
One of the most important (and equally most often forgotten) lessons that Bayes’s Theorem can teach us about evidence is that the strength of evidence is a ratio. To be precise, let H1 and H2 be rival explanatory hypotheses, B be the relevant background information, and E be the evidence to be explained. Now consider the following ratio:
Pr(E | B & H1)
Pr(E | B & H2)
If Pr(E | B & H1) > Pr(E | B & H2), then this ratio is greater than one and the evidence favors H1 over H2. If Pr(E | B & H2) < Pr(E | B & H1), then this ratio is less than one and the evidence favors H2 over H1. And if Pr(E | B & H1) = Pr(E | B & H2), then this ratio is equal to one and the evidence favors neither H1 nor H2.
Paul Draper has taught me that this ratio has some interesting implications for topics that come up in debates between theists and naturalists. Suppose that Pr(E | B & H1) is really high and Pr(E | B & H2) is middling. In this case, there will be an evidential asymmetry: if E is true, E is not strong evidence for H1 over H2, but if E is false, then ~E is strong evidence for H2 over H1. This can be shown with a couple of examples.
First, suppose that E is true. If E is true, then E is not strong evidence for H1 over H2. This follows because the ratio of Pr(E | B & H1) to Pr(E | B & H2) is not high.
Second, suppose that E is false. This is where things get interesting. Because Pr(E | B & H1) is really high, the negation of E would strongly favor H2. Again, this follows from the ratio of the likelihoods: the ratio of Pr(~E | B & H1) to Pr(~E | B & H2) is high.
It is mathematically necessary that this evidential asymmetry will always be present when the evidence has a middling probability on one hypothesis and a very high (or very low) probability on the negation of that hypothesis. (The basic idea is that is the range of probabilities is zero to one, a high probability divided by a middle one must be relatively small while a middling probability divided by a low one must be relatively large.)
2. The Efficacy of Prayer and Scientific Confirmation
Let’s assume, as appears to be the case, that recent scientific studies have failed to confirm the efficacy of prayer. If facts about evil and divine hiddenness are included in our background knowledge, then those study results do not strongly favor naturalism over theism because the ratio of the likelihood of the evidence on naturalism to the likelihood of the evidence on theism is not really high. (This follows because the probability of such studies given theism and that background information is middling.) But now imagine the results had turned out differently and the studies had confirmed the efficacy of prayer. In that case, such results would strongly favor theism over naturalism.
This result shows that atheists are making a mistake when they accuse theists of using an unjustified double standard by dismissing the (negative) study results as evidentially insignifcant. Rather, theists are correct that if the study results had confirmed prayer, then such positive results would have been strong evidence favoring theism over naturalism, but the absence of such results is only weak evidence favoring naturalism over theism.
3. Horrific Evils
The topic of horrific evils provides another example of evidential asymmetry. Horrific evils have a middling probability given naturalism but a very low probability given theism. So if horrific evils were absent, that fact wouldn’t strongly favor theism over naturalism, whereas the presence of horrific evils strongly favors naturalism over theism.
I owe the main point of this post, as well as both examples, to Paul Draper. Any errors in this post are, of course, mine.