Great question. I'm glad this is an area you're exploring. All of us can use more systematic tools for thinking in our toolkits. In the modern world, we all need such tools. In politics, for instance, we may encounter partisan scholars whose work always comes out supporting a given position, showing that they are in fact advocates as opposed to seekers. And tools for analysis like this can help us recognize that and make better judgments.
I personally love Bayesian analysis. There are many questions on which it can be very useful, and these, to my mind, include some historical questions and even some religious questions. I used Bayesian thinking years ago in assessing certain apologetic-critical questions, and it still forms an important part of my own thinking.
One problem with applying it formally (numerically) to history is that in history, as compared with the sciences, it is much more difficult to quantify probabilities on the questions we want answers to. So, interpreters of history using Bayesian reasoning end up assigning probabilities based on personal, subjective factors, but the use of actual numbers to represent these can grant them the illusion of objectivity.
So, most questions in history are not very amenable to formal Bayesian analysis purporting to generate concrete numerical probabilities. Yet, while formal Bayesian equations usually don't render sound historical probabilities, where broadly Bayesian thinking is actually useful in history is generally just in helping us see the direction that a given evidence points. We can employ Bayesian thinking to help us look at our evidence systematically, recognizing that a given piece of evidence should be viewed as making a given hypothesis more likely or less likely.
It's also important to use Bayesian thinking by taking into account the "prior probability" of an outcome. For instance, a Latter-day Saint linguist, Brian Stubbs (whose work I think is generally rigorous and whom I like) once noted that the Hebrew language shares more roots with the Uto-Aztecan Native American language family than certain neighboring African tribes' languages share with each other and pointed out that linguists have concluded that these tribes' languages share a common origin. The implication he's putting across is that the relationship between Hebrew and Uto-Aztecan exceeds the standard needed to conclude that two languages share a common origin, and this is therefore strong evidence for the historicity of the Book of Mormon.
While I actually think this is one of the most interesting lines of evidence adduced for Book of Mormon historicity, I see Stubbs making a major oversight here: he has failed to account for prior probabilities. Linguists have concluded that these neighboring tribes' languages are related because the probability that neighboring groups' languages will have a common origin is high to start with--i.e., in Bayesian terms, this has a high probability. Meanwhile, when we have two groups that are separated by oceans and don't share evident genetic origin or clearly share other (non-linguistic) culture in common, the prior probability that their languages share a common origin is low. It's not that it can't be so--maybe it is! It's just a hypothesis that requires s a higher standard of evidence to overcome the low probability prior to the assessment of the new evidence (the sharing of linguistic roots).
Linguistics is far, far more amenable to formal (numerical) Bayesian analysis than are the great majority of questions in history, but this example can perhaps help show the kind of insight that Bayesian thinking can provide within history.
As for applying Bayesian analysis to ancient history, in my own area of history, Mormonism, where we have probably literally about 100,000 times as much documentation on the faith's origins than we do for those of early Christianity, I think it would still be fiendishly difficult to design a proper formal Bayesian analysis--one that could generate numerical probabilities that can be remotely relied on. Unlike measurements made in the natural sciences, historical narratives overwhelmingly consist of things that are not properly quantifiable, much less quantified. So the analyst would essentially have to just make up numbers to plug into the equations. Given how difficult this would be for Mormon history, I have trouble seeing how it would really be possible for ancient Christianity.
One philosopher who has considerable expertise with Bayesian analysis is Richard Swinburne. Swinburne is also a Christian theologian. He has applied Bayesian analysis to the resurrection of Jesus. So, paradoxically, Richard Carrier has used Bayesian analysis to conclude that Jesus never existed, while Richard Swinburne has used it to argue that Jesus literally rose from the dead! So it's use regarding early Christianity thus far is not only problematic but thus far also contradictory. It may be that historians will learn enough about the past and refine techniques for establishing prior probabilities enough in, say, the next fifty years that they will be able to more systematically illuminate questions of ancient history, but I think we are a long way from doing that with any objectivity now.
While early Christianity is not my area of history, there's a scholar I can tell you is legit from both his work in early Mormon history and in early Christian history--Daniel Gullotta. As a historian of Mormonism, I'm in an excellent position to judge the quality of his thinking, and he's first-rate. He primarily does early Christianity, and given what I know of him from his work in Mormon history, I trust him as a scholar with rigorous thinking. It turns out he has written a review of Carrier's work. You can find his review on his Academia.edu site. (Oh shoot, I just checked and the link takes you to a pay wall. I believe parts of the review are on Carrie's Wikipedia entry.) I know he credits Carrier's approach as being highly sophisticated, but he also has a number of strong criticisms.
Richard Carrier seems like a highly intelligent and good natured guy. It bears noting that his projects seem to all be controversial and all point in the same direction. He has written arguing against broad-consensus views among historians in each case that Jesus didn't exist, that science thrived under the Romans and stopped progressing under Christianity, and that Hitler was a Christian. In my experience, in exploring controversial topics regarding a given ideology's worldview, like a religious or antireligious worldview or political worldview, the results sometimes come out more favorable to the worldview and sometimes less favorable. Where we have someone solely and consistently finding evidence that supports a given worldview, we should question their objectivity. Carrier's consistent conclusions that are both unfavorable to Christianity and cut against broad-consensus conclusions about Christianity even among religiously nonbelieving scholars (e.g., Bart Ehrman) raise real questions here. If we're looking at probabilities, we can assess some probabilities here. What is the probability that a particular maverick scholar who consistently argues to negative conclusions about a worldview even against the judgment of most other scholars (whether those scholars embrace that worldview or not) is objective and right and they are wrong?
There are a great many scholars of early Christianity who don't appear to be driven by ideology and often make findings that problematize their own position, showing their independence of ideology. I believe John Kloppenborg to be one of these. If you want to know about the historical Jesus, I'd recommend one of those guys.