My first pillar of Science is that it is based on repeatable observations. In order to see how Christianity measures up, we need to examine whether it is based on observations, and whether it is repeatable.
Observations ultimately boil down to sense-data experienced by individual human beings. You can't go directly from observations to theories without a certain amount of interpretation, but that's true in Science too. In order to be regarded as accurate, theories need to be based on empirical evidence, which has to be specific enough to prove that that theory is accurate, rather than just some similar theory in the class of theories. Not all aspects of the theory need to be directly confirmed to be accepted (otherwise we could never use a theory to make predictions in new circumstances). So long as the untested aspects of the theory are closely bound to aspects of the theory which are testable, the theory has a whole can be considered empirical.
Now Christian theology is empirical in this sense: that most of the core doctrines are the most reasonable explanations of certain ordinary sense-data reported by actual human beings. By "ordinary" sense data, I mean things which appear to be interactions with the normal shared waking world, rather than sense-data seen in dreams or hullucinations, experienced in an altered state of consciousness. (If Christianity is true, God does sometimes communicates using dreams and such, but the most important information was not given primarily in this way.) The experiences might be "visions" in the sense of revealing truths about things in Heaven, but they are not "visions" in the sense of being subjective and intangible features of a single individual's private experiences.
Here's an example. One of the most esoteric doctrines in all Christianity—the one that may at first sight seem to have the least to do with the actual physical world—is the doctrine of the Trinity: that God consists of a loving communion between the Father, Son, and Holy Spirit. But the Gospel of St. Luke tells us that
When all the people were being baptized, Jesus was baptized too. And as he was praying, heaven was opened and the Holy Spirit descended on him in bodily form like a dove. And a voice came from heaven: “You are my Son, whom I love; with you I am well pleased.” (Luke 3:21-22)
There is a specific series of sense-data which communicates something about the relationship between these three entities, two of which are normally invisible. (The Son would also have been invisible, but for his Incarnation as a human being.) Combining this with other similar events, such as the Transfiguration and Pentecost, and the fact that Jesus claimed to be divine but also prayed to his Father, the Church came to the conclusion that the Trinity was the best explanation of the observed facts. If it had been based on philosophical speculation, they would have come up with some more logical way to divine divinity into three aspects (like say, Creator-Preserver-Destroyer).
Note that here I am addressing the question of whether the claimed empirical facts support the theological claim. Although some might argue that the claimed facts support some other claim (e.g. space aliens), most skeptics would probably think that's pretty silly. It's a completely different question whether the purported testimony is actually true, whether it really happened or else was made up by somebody. The accuracy of the accounts is a very important question, but before I go into it, I wanted to make sure to get clear what kind of factual support Christianity claims to have.
Here's a parable to illustrate why this distinction is important. Water normally boils at 100 
C. Suppose I claim that if I drop an orange into water, it boils at 150 C instead. I'm pretty sure that's false, but suppose you claim it is true. And you say that you know it's true because you went into the laboratory and tested it carefully. Now that doesn't mean I have to believe your claim. I could accuse you of lying, or making stupid mistakes in the lab. But if I make fun of you for "believing without any evidence at all" and make derogatory comments about the tooth fairy and Santa Claus, then I haven't understood the nature of your claim.
Admittedly, the degree to which Christian doctrine is founded on sense-data depends on the specific doctrine. For example, the Second Coming will involve a lot of sense-data when it happens, but because it concerns the future the sense-data hasn't been observed yet. It is based primarily on his promise that he will do so. It is related to his Ascension into Heaven, which is itself partially based on sense-data (so far as his leaving Earth is concerned). The fact that he has the power to rise from the dead and ascend into Heaven is certainly relevant to assessing how likely he is to keep this promise to return to Earth.
The appeal to sense-data becomes particularly clear in the case of the Resurrection. Here are all the accounts of Resurrection Accounts contained in the four Gospels, Acts, and 1 Corinthians. (The appendix of the Mark account is a latter addition, which was almost certainly not written by St. Mark. It might well go back to the first century, but for obvious reasons I won't be putting any weight on it. I've included it solely for the sake of completeness.)
Note how the accounts refer specifically to the fact that Jesus had a tangible body which could be viewed by multiple people with multiple senses simultaneously, and that he eats fish in order to reassure the disciples that he is not a ghost. This despite the fact that he was also capable of instantaneously appearing and disappearing. Note also the role that doubt and skepticism plays in several of these stories, as well as the fact that Jesus could eventually be recognized, but was frequently not recognized at first.
This is most obvious, of course, in the "Doubting Thomas" incident I quoted yesterday. St. Thomas had an opportunity to believe in the Resurrection on the testimony of others first, but he rebuffs this by saying that he needs to experience the sense-data himself in order to believe it. He demands that the experiment be repeated. Jesus appears again a week later, and offers the evidence, but at the same time rebukes him gently by pronouncing a blessing on those who believe without seeing.
At this point the story is likely to raise all sorts of questions to a skeptical mind. Why should faith play a role at all? Why not just empiricism?
There's some deep issues here, but let's start with this: Jesus isn't requiring belief without empirical sense-data. He's asking for belief based on the empirical sense data of other people besides yourself. That may involve faith in the sense of trusting others, but not in the sense of "belief without evidence".
Leaving aside for the moment questions of why God would choose to arrange things that way, the situation itself is not that unusual. It is how things always work when you're studying History prior to the 20th century, which is not repeatable. It's how things usually work in Science itself, since only a very few people have replicated the fundamental experiments in physics personally. Most of us believe in scientific experiments on the testimony of other people. As a theorist I certainly haven't done all those experiments myself; in fact, I haven't even interviewed most of the people who did them!
This year, the Higgs boson was discovered at the LHC. This is Science, so the results have to be "repeatable". That means, if you have billions of dollars, your own team of hundreds of scientists, and a decade of your life to devote to it, you too can discover the Higgs boson. But are you going to wait for that to happen before you believe it? No, you believe it now, on faith! And what I mean by faith here is not a vaporous sentimentality, but trust based on good evidence that the community of particle physicists is reliable in this particular respect.
(I apologize to anyone who was expecting a Bayesian analysis in this post. As you can see, I'm still working up to it.)
A few remarks:
1. Although you say "most of the core doctrines are the most reasonable explanations of certain ordinary sense-data reported by actual human beings", it's highly disputed (a) whether the allegedly reported sense-data really were reported by actual human beings and (b) whether the doctrines really are the most reasonable explanations, or indeed are reasonable explanations at all, of the alleged observations.
2. You give as an example Luke 3:21-22. But at least some of what's said there very decidedly isn't a matter of sense-data. Assuming arguendo that the thing really happened, whoever was watching and listening didn't see that it was the Holy Spirit descending on Jesus. Nor did anyone observe who, if anyone, spoke. I'm sure you understand this perfectly well, of course.
3. (Related to 1.) You say "He's asking for belief based on the empirical sense data of other people besides yourself", but again it's highly disputable whether that's so. For instance, it's not out of the question that the story in Luke 3 might just have been made up, in which case no sense-data at all need be involved. So there's more to it than just trusting, say, "doubting" Thomas. You have to trust the author of the biblical story, and his sources, and perhaps some long chain of oral tradition back to (if we're lucky) Thomas. And then you need to trust whatever chain of reasoning gets you from Thomas's alleged experiences to some conclusion about Jesus.
g,
I think that almost everything you say here was already admitted to in my post itself. I admitted that the sense-data requires interpretation, and that things like the Baptism need to be combined with other events in order to formulate doctrines. I also explicitly conceded that "It's a completely different question whether the purported testimony is actually true, whether it really happened or else was made up by somebody," which is much the same as your (a).
Regarding (b), however, I'm not sure it's all that controversial. Are you saying that if tomorrow you learned that the Gospels and Acts were 100% accurate in all matters relating to empirical sense-data (but not necessarily interpretation), that it would be reasonable to conclude from that anything other than some version of Christianity? At least, the large majority of non-Christians choose strategy (a) rather than strategy (b).
By the way, no hard feelings regarding the other thread. I wasn't really upset, just momentarily plaintive, and too much "on my dignity". Don't worry about it.
You did, at least kinda, already admit those things -- but it seems to me that the point of what you wrote depends on not taking them too seriously. So I wanted to make a bit of a fuss about them. :-)
If I discovered that everything empirical-looking in the Gospels and Acts is exactly true then I would agree that something at least somewhat like Christianity is probably right. But even most Christians agree that nothing so extreme is the case -- at best it might be that most of the empirical-looking stuff, including lots of the parts that seem to call for miraculous expansions, is true; and from your example I thought, perhaps wrongly, that you were suggesting not only that Christianity is the best explanation of all the alleged observations in the NT but that individual chunks of Christian doctrine are the best explanations of individual batches of observations, which is quite a different matter; and even if all the empirical bits in the NT were right (or, let's say to avoid an irrelevant side-issue, as near to that as possible without that accuracy itself calling for miraculous explanation) I'm not sure that *exactly* Christianity would be the best explanation. (Maybe it would; I haven't thought it through in detail.)
As for the other thread, I'm far enough from having hard feelings about it that it took a considerable time for me to work out what you were referring to.
g,
Not sure what "most Christians" think about this---in evangelical circles many of them would probably endorse the claim that everything in the Bible has to be 100% accurate historically when taken literally, whenever it isn't explicitly or obviously metaphorical. But that's based on some inerrantist theory of inspiration, which is a bit different from my version of inspiration (which I haven't been using in these posts!) I believe that the Gospels and Acts are high-quality historical sources whose details are accurate in roughly the same sense that good historical sources usually are, which gives a lot of material to work with. But obviously what one concludes depends on what one thinks about that.
But I think maybe the real disconnect between us here is that (doubtless due to my poor explanations) you haven't realized the kind of thing this post is supposed to be. It's a statement of "theological method" which is supposed to be parallel to my posts on "scientific method", explaining the sense in which theology is an empirical subject. It is directed towards people who are inclinded to dismiss theology as pure speculation, and is explaining how in principle one can progress towards the goal of theological doctrines based on empirical data.
The post is not intended as a complete argument for specific Christian doctrines, any more than my posts on "scientific method" were arguing for specific scientific theories. I mentioned specific doctrines and theories for illustrative purposes only, in order to make specific points about how the method works. You could endorse the methods and still disagree with many of the conclusions that the scientific or theological community has come to.
In my Bayesian calculation found in the other posts, I don't go into the details of any doctrines besides Christ's specialness and the Resurrection, so obviously the calculation can't prove anything more than that something kinda-sorta-like Christianity is true (including versions of Christianity which I would regard as extremely heretical). But that's obviously the main sticking point for most skeptics---one can quibble about the details later.
I meant that I had no hard feelings. I was trying to make a gracious reply to your (unnecessary) apology in the Pillars of Science: Summary and Questions thread, but I guess my reply wasn't necessary either.
Aron, I think I've understood pretty well what you're aiming at here, and in particular I didn't think this single post was meant to be an argument for any particular conclusion. I'm not sure there *is* a disconnect (as opposed to a disagreement), but if you think there is then perhaps I've missed some way in which connection is failing...