On the Prior Probability of the Resurrection
Recently a friend on Facebook argued that Christians have no business declaring the Resurrection of Jesus to be the most probable (a posteriori) explanation for the relevant facts, since they are unable to first pin down the prior probability of the Resurrection independent of those facts. I think that's a reasonable enough objection and deserves a reply. After all, posterior probability by definition is a function of both likelihood on the evidence and prior probability. Clearly, then, one cannot determine posterior probability without some idea of the prior.
My friend went on to say that the prior probability of a hypothesis is typically established as a ratio of previous instances of the event and total opportunities for the event to have occurred: "Normally we determine the probability of X by how many occasions of X we have seen out of how many opportunities for X there have been. Is the resurrection of Jesus some kind of exception?" This amounts to an appeal to frequentism for finding the prior. Right here is where I begin to take issue with the typical skeptical-Bayesian approach to miracle questions like the Resurrection. What appears missing from so many of these calculations is any consideration of relevant background knowledge. I do agree with Swinburne when he says that "any division of evidence between e [observational evidence] and k [background knowledge] is a somewhat arbitrary one." That said, numerous facts indirectly relevant to the question of the Resurrection are too often overlooked, perhaps lost between very specific, directly relevant evidence (like the empty tomb or post-resurrection appearances of Jesus to the disciples) and very general knowledge about the world.
First, there are the evidence and arguments from natural theology that suggest the existence of God. Evidence in the way of fine-tuning in the physical universe, specifiable complexity of biological systems, and the universal moral intuition of human beings, among other things, suggests that the probability that God exists is quite high. Moreover the prophetic history of Israel, in which the Jewish people were scattered throughout the world and then re-gathered to her ancestral home in the "last days," suggests the existence of the God invoked by Jesus in particular. Though it is certainly right to bear in mind the number of previous recorded and confirmed resurrections in history (arguably zero), the evidence for the God of Israel is important information to bring to the question of prior probability.
Next, consider the particular personality and historical circumstances of the central figure involved. The question before us is not, "What is the probability that some random guy rose from the dead?" but "What is the probability that Jesus of Nazareth rose from the dead?" I do not dispute that the probability of a random guy rising from the dead is negligible. But from all indications Jesus of Nazareth was not some random guy. Jesus claimed for himself, both explicitly and implicitly, to be the Son of God and the King of the Jews, the Messiah; and elements of his life indeed fulfilled various messianic prophecies from the Old Testament. Jesus was widely reported by followers and detractors alike to have performed healing miracles and miracles of provision. (The Pharisees attributed these to the work of Beelzebub, but did not deny their occurrence; and the later rabbis of the Tannaitic period likewise attributed the miracles of Jesus to "sorcery.") In addition, Jesus frequently foretold his own crucifixion and resurrection – most often to disciples who refused to believe it. These considerations together would seem to make the Resurrection of Jesus much more antecedently probable than the resurrection of some random guy.
Finally, I would suggest there is precedent for a miracle, even a "raising of the dead" of sorts, in the origin of life. The fact is that at one point in our prehistory a dead collection of elements became a living organism – whether by God breathing life into the "dust of the earth" as recorded in Genesis, or by some sort of chemical evolution. And of course no origin of life event has ever been witnessed by anyone (not even in principle). The prior probability of the origin of life just before life actually originated therefore must have been at or very near zero. Yet here we are reflecting on the fact that life originated. Thus our continually being alive constitutes evidence strong enough to overcome the seemingly overwhelming prior probability against life originating. In a very real sense the origin of life is evidence of a miracle.
Taken together, these background factors arguably make the prior probability of the Resurrection much higher than any prior probability that would be reached by a frequentist interpretation of probability alone. When that higher prior probability is conjoined with a similarly high likelihood ratio (a measure of explanatory power), the posterior probability that Jesus Christ actually rose from the dead increases accordingly. I would argue, in fact, that while intuitively implausible, the Resurrection is evidentially probable. This is just the sort of thing we should expect of a God who intends to reveal himself through the "sign" of a miracle in history.
 This is essentially an informal statement of Bayes' Theorem,
P(E│H & K) x P(H│K)
P(H│E & K) = -----------------------------
where P(H│E & K) is the posterior probability of hypothesis H, given new evidence E and background knowledge K; P(E│H & K) is the probability of the evidence given the truth of the hypothesis and background knowledge; P(E│K) is the probability of the evidence given background knowledge alone; and P(H│K) is the prior probability of the hypothesis, again conditional on background knowledge.
 Richard Swinburne, The Existence of God (New York: Oxford, 2004), p. 67.
 See "Understated Evidence and the Resurrection of Jesus" for reasons to think that the explanatory power of the Resurrection hypothesis is very high relative to competing hypotheses.