Probability, Evidence, and the Confirmation of Unlikely Hypotheses.

The year is 1919. You are waiting for the results of Arthur Eddington's solar eclipse observations. Einstein's theory of gravitation -- recently proposed -- maintains that light bends around massive objects. As a result, we expect that stars located within the region of the sky where the solar eclipse occurs will appear to be slightly shifted away from their original locations.

Let G represent Einstein's theory of gravitation. Let E represent the phenomenon that the stars near the solar eclipse appear slightly shifted away from their original locations. Let's assign the following value to the prior probability of G: P(G) =  .01. This shows that the prior probability of Einstein's theory is rather low. Let P(E|G) = .99. In other words, if G is true, then it is nearly certain that we will see the apparent shift in the stars. If G is not the case, then P(E|~G) = .001 (i.e., it is very, very improbable that we will see the shift in the stars if G is false). E is observed to be the case by Arthur Eddington.

What is the probability that Einstein's theory of gravitation is true given the observed shift in the location of the stars near the eclipse? We can use Baye's theorem to assess the weight of the evidence, E, on Einstein's theory, G.

P(G|E) = P(G&E) / ((P(G)P(E|G) +  P(~G)P(E|~G))

P(G|E) = (.01 x .99)/((.01 x .99) + (.99 x .001))

P(G|E) = .9 (approximately)

Although Einstein's theory of gravitation was given a low prior probability, the specific evidence significantly affected the posterior probability, making a highly probable hypothesis out of what was otherwise an improbable hypothesis. 

APPLICATION IN OTHER CONTEXTS

The principle to glean is this: even a theory with a low prior probability can become probable given good enough evidence working in its favor. This is especially relevant with respect to miracles. However improbable some miracle may be, if there is good and sufficient evidence for it, then, according to Baye's theorem, the probability of the miracle's occurrence can ascend to probable heights. As Tim and Lydia McGrew note, "...any real, nonzero prior improbability can be overcome by sufficient evidence" [1]. While a low prior probability for some theory is significant, we must consider the weight that specific evidence has on its posterior probability, which, in some cases, can make very improbable events become probable.

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Footnotes:

[1] McGrew and McGrew, "The Argument From Miracles." The Blackwell Companion to Natural Theology, ed. William Lane Craig and J.P. Moreland.