@swamidass, @Faizal_Ali, @Rumraket, @John_Harshman,
I’d like to make a few quick points. First of all, Drs. Tim and Lydia McGrew, in their carefully argued article, The Argument from Miracles: A Cumulative Case for the Resurrection of Jesus of Nazareth, use a slightly different pair of hypotheses from those used by @swamidass (namely, H1 and H2). Their hypotheses are:
R: Jesus of Nazareth rose miraculously from the dead
and
~R: It is not true that Jesus of Nazareth rose miraculously from the dead.
They then invite readers to consider the ratio P(F|R) / P(F|~R), for any fact F which is relevant to the Resurrection. (Once again, this is similar to @swamidass’s ratio P(E | H1) / P(E | H2. He argues that the numerator is relatively high, while the denominator is very low.)
The McGrews then assemble what they consider to be very powerful evidence for the Resurrection - namely, the women’s discovery of the empty tomb, the combined appearances to the apostles and the appearance to Paul - and proceed to argue that this evidence is 10^44 times more likely under hypothesis R than under hypothesis ~R, before concluding:
Sheer multiplication through gives a Bayes factor of 10^44, a weight of evidence that would be sufficient to overcome a prior probability (or rather improbability) of 10^-40 for R and leave us with a posterior probability in excess of .9999…
We have argued above that, given the Bayes factors of the various pieces of evidence, their cumulative impact would overcome a prior probability of R of 10^-40 while leaving us with a posterior probability of approximately .9999. Even an exceptionally low prior may be overcome by extremely strong evidence. That argument deserves to be answered on its own terms, and it illustrates quite handily the fact that there is no such thing as a finite prior probability that is so low as to be “slippery” and hence impossible to overcome by evidence.
Now, supposing purely for argument’s sake that the McGrews’ calculations were correct, @swamidass’s contention that one can argue for the existence of God on the strength of the evidence for the Resurrection would be correct. All that the skeptic needs to grant is that the prior probability of hypothesis R is greater than 10^-40, and we end up with a very high posterior probability for a theistic hypothesis (Jesus of Nazareth rose miraculously from the dead), after considering the evidence.
Crucially, the McGrews treat each apostle who saw Jesus as an independent witness, which then allows us to multiply the individual probabilities of each of them having the same hallucination. For instance, if the probability of one apostle (say, John) seeing, hearing and feeling the same thing as Simon Peter did when they had an apparition of Jesus is (say) only 1 in 1,000, and if there were ten other apostles (barring Judas Iscariot) who experienced the same thing as Peter did when he had an apparition of Jesus, then we can calculate the probability of them all having the same apparition by chance as (1 in 1,000) raised to the power of 10, or 1 in 10^30. That’s the chief booster used by the McGrews, to beef up their ratio P(F|R) / P(F|~R) to 10^44.
The McGrews argue that the apostles must have all independently experienced Jesus, and that each of them must have had the same experience of their risen Master – otherwise, they wouldn’t have all been ready to suffer and die for him. However, this argument relies heavily on a psychological counterfactual about the conditions under which the apostles would have been ready to die for their belief in the Resurrection, coupled with the psychological assumption that the disciples would have all carefully compared notes about the details of their experience after Jesus appeared to them (maybe, but who knows?), plus two more factual assumptions: the historical assumption (which has been called into question by Professor Candida Moss in recent years) that the apostles were continually under threat of being tortured or martyred, and another historical assumption : namely, that the specific reason why they were martyred was that they believed in and preached the message that Jesus had risen from the dead . The McGrews’ case for the Resurrection is a solid one only if all four assumptions are true.
Now, let’s suppose instead that the evidence for the Resurrection is “only” 10^10 times more likely under hypothesis R than under hypothesis ~R. (10^10 is still a pretty high likelihood ratio.) And let’s suppose that one rates the prior probability of R as 1 in 10^11, which is the maximum value that Rumraket is prepared to generously consider. Now the probability of the Resurrection, in the light of the evidence, falls to 1 in 10, roughly.
Given the unreliability of human memory over a period of decades, or even weeks (if it is contaminated by listening to other people’s recollections of the same event), as well as the massive uncertainties regarding (i) who wrote the Gospels and when, (ii) whether the authors of the Gospels interviewed any eyewitnesses, (iii) what the apostles saw when Jesus appeared to them, and where and when they saw him, (iv) whether they were previously expecting a post-mortem divine vindication of Jesus, and (v) the manner of Jesus’ burial, I honestly do not see how one can argue for a likelihood ratio of greater than 10^10.
@swamidass acknowledges that there is no systematic way of defining the ratio of P(H1) to P(H2) - or as the McGrews would say, P( R) to P(~R). But to make his case for the Resurrection, he needs to explain why he thinks it is more than 1 in 100 billion.