Last time I wrote a long discussion of what I think is the best evidence for and against Christianity.
When it comes to the historical testimony to the Resurrection, I analyzed the data in terms of a set of minimal facts which I think even a "liberal" biblical critic ought to accept. By the "liberal" view I mean someone who thinks that the Gospels, Acts, half the letters of Paul, and the general epistles were pretty much all written in the closing decades of the 1st century, by people other than their traditional authors—but that nevertheless the texts are based on identifiable earlier sources which go back to some real, and much earlier, written or oral traditions about Jesus. (Because some parts of the Gospels are obviously rooted in actual historical events.)
This was a concession, not my actual beliefs: in reality I think this "liberal" view, despite its acceptance by the majority of secular biblical critics, is supported mostly by circular reasoning, and is more than a little reminiscent of the claim that Bacon wrote Shakespeare. Nevertheless, if a good case can be made for Christianity even under such hostile circumstances, then obviously things would work even better on a more "conservative" view. For a Bayesian approach based on a somewhat more conservative view of biblical criticism, see this paper by Timothy and Lydia McGrew (they argue for the conservative biblical view, but don't incorporate it into their probability analysis).
I also made no reference to the "inspiration" of the Scriptures by the Holy Spirit; rather I am treating them as ordinary historical documents containing what is claimed to be testimony. Just to be clear, I do believe that the entire Bible is inspired by the Holy Spirit, and that therefore all of it is an important communication from God to us. I disagree with fundamentalists in that I don't think the point of the Bible is to communicate scientific facts, or even matters of historical trivia. For this reason it doesn't bother me that the Gospel accounts contain apparent contradictions: although most of these contradictions would probably be resolvable if we knew the full circumstances, it's not important to me whether if the documents contain mild inaccuracies regarding trivial matters, as all genuine eyewitness testimony does.
So now I have to analyze how I think the probability analysis actually goes. To summarize the previous post, the main factors weighing against Christianity are:
(A1) prior probability due to specificity of Christian doctrine,
(A2) prior probability due to the weirdness of Christian doctrine,
(A3) prior probability due to Jesus being only one of billions of people.
(B) the Argument from Evil
while some of the factors weighing in favor are:
(C) the Fine-Tuning version of the Argument for Design.
(D) circumstantial facts of Jesus' life prior to his death, making him more likely to be the Messiah
(E) multiple testimonies to the Resurrection,
(F) modern-day miracles.
This is by no means a complete list, but I think it contains most of the highlights. I've excluded some other arguments for Theism, and various Christian experiences, as less likely to be persuasive to someone coming from a skeptical point of view, even though I rate some of these things as significant.
Each of these factors leads to a shifting of the relative odds of Christianity : Naturalism. (I'm assuming here that Naturalism is the main rival hypothesis to Christianity for skeptics of a "scientific" sort.) Since these factors are approximately independent, each of them multiplies the relative odds by some number. Since we aren't going to agree on precise ratios, I'm going to just estimate how persuasive these are to me, in terms of orders of magnitude (i.e. powers of 10).
I think that (B) and (C) are both good for about 2-4 orders of magnitude and therefore they approximately cancel each other out. Each of them, despite being grounded in certain observable facts about the universe, is a controversial philosophical argument about metaphysics, and I think it's quite rare for any such argument to establish a conclusion with an error rate less than or . You could say that I'm skeptical about Philosophy, but that's not really fair since shifting the odds by a factor of 1000 should count as a "success" even if this can be easily overwhelmed by more definitive evidence. For similar reasons I don't think (A2) should be much less than .
(A3) and (D) cancel each other out nearly exactly, because if Jesus is in a very small group of people more likely to be the Messiah, then it doesn't matter how many other people there are in the control group. See the discussion of the prosecutor's fallacy.
(E) contains a factor which cancels out
(A1) exactly†. That's because the Christian testimony is to Christianity specifically, and not to any other religion. Whereas a made up religion is going to be selected from some large set of possible religions. See the discussion of the Farmer Jones case in the comments of this thread.
The remaining part of (E) has to do with how plausible the testimony is, leaving aside its specificness. Now the testimony of a single individual is has to be worth at least 2 orders of magnitude, because most of the time people don't lie about things.
Somewhat counterintuitively, the value of testimony towards implausible claims like miracle reports is actually somewhat greater than this, because people are less likely to make up claims that are implausible. There are many people who claim to have crossed the street, but only a few people claim to be abducted by UFO's. The number of people who claim to have seen people physically resurrected is quite low; certainly much less than of the population. To the extent that this effect exists, (E) also partially cancels (A2). But this cancellation only works up to a certain point, since nothing says that the rate of false claims has to mirror the true a priori probabilities. For the small fraction of the population who is willing to make absurd claims, there is no limit to how unlikely their assertions may be. So some residue of (A2) may remain.
What remains to be considered in (E) is the gross improbability of seeing testimony as strong as this about anything false. The multiplicity of eyewitnesses, the priority of the women, the apparent unanimity in the pre-existing group of Twelve (minus Judas), and the fact that they were willing to die for their testimony all weigh in its favor. All of these facts are strongly supported by the historical data. Their initial skepticism and doubt, mentioned explicitly in almost all of the Gospel accounts, weighs heavily in its favor, so the skeptic is probably better off trying to argue (somewhat less implausibly) that this was made up later to prove a point. The lack of any reasonable ulterior motive also makes the testimony less likely given Naturalism.
St. Paul presents a special case: since we know he was previously hostile, his testimony is more or less independent of the others. That in conjunction with the unusual circumstances surrounding his conversion is a factor of all by itself; the group testimony of the remaining apostles is at least , and that's rather generous, since on that hypothesis we'd expect to find about 100 similarly strong examples of false dramatic claims by large groups, in the most recent generation of people alone.
Taking everything together I think that the implausibility of a set of testimony looking like (E) given Naturalism is at least , and much lower if one takes a more conservative approach towards biblical criticism. If this is right, then (E) pretty much overwhelms every other factor in play, and (F) is just icing on the cake. Therefore, with high probability, the Resurrection seems to have really happened. In other words, (some unspecified version of) Christianity is far more probable than Naturalism is.
[UPDATE 5/30/15. Much later, a reader St. James wrote in to ask for clarification of the probability values. My response to him included the following, which I think is worth appending for any clarification value it may have:
Thanks for your comment. As is clear from the date I'm coming back to this post after a couple years, so I need to reconstruct what I was thinking. I'm pretty sure I meant for (D) to cancel (A3) specifically, and for (E) to include a piece which cancels (A1), but I think I wrote (A3) by mistake so I've fixed it in the post above.
(D) as I've stated it is rather vague, so it actually contains the potential for more evidence than just that which cancels (A3), for example based on the greatness of Jesus' ethical teaching. My point was just that, if we have even vauge circumstantial evidence that the pre-Resurrection Jesus was the type of person who might be the Messiah, then the common atheistic objection (at least people have said it to me multiple times) that Christianity is unlikely because Jesus is just one out of many people, and that resurrections are extremely uncommon in a group of people, is irrelevant. So I shouldn't have said (D) exactly cancels, but I should have said it includes a factor which approximately cancels (A3).
Similarly, testimony (E) has the magical property that it can compensate for almost any amount of improbability due merely to specificity, since P(witness tells the truth) and P(witness is lying) both have to be equally specific. That's why I threw out (A1) and just concentrated on the probability of a group this large having a conspiracy to lie, (or being misled, or being legendary, etc. etc.)
With respect to that probability, I guesstimated 14 orders of magnitude but as you can see from the comments section that was really controversial, and the main point of controversy concerned whether any ancient historical documentation can ever get us probabilities that large. This is an important question and looking back I'm not at all certain I didn't run the numbers much too "hot"; it's possible the whole thing should be scaled back a bit. But then I think the philosophical numbers should probably also be diminished proportionally, since I'm fairly certain that History provides better quality evidence than Metaphysics in general.
But I still think the posterior probability that Julius Caeser wasn't really assassinated must be several orders of magnitude tiny (though not as small as I estimated below), and that the apostolic testimony for the Resurrection has a comparable in strength likelihood ratio. So that without tiny prior probabilities or other substantial arguments against, Christianity winds up being highly probable.
UPDATE 8/30/16: For a more recent take on analysing the probabilities, see my recent post: Just how certain can we be? ]