Dr. Richard Carrier - a historian with a expertise in ancient science as well as the historical Jesus - has applied Bayes' theorem to his work with the historical method. Off the top of my head, I know he's used Bayesian reasoning in the articles he contributed to the books The Christian Delusion and The End of Christianity. (In the end notes, he explicitly puts his arguments in terms of Bayesian probability theory.) If memory serves, he's also authoring a book which explains Bayes' theorem and its application to the historical method.
Coincidentally, it's through his work that Iearned about Bayes' theorem and I first learned about Dr. Carrier through lukeprog's podcast "Conversations from the Pale Blue Dot."
Edit: The podcast is located here http://commonsenseatheism.com/?p=10150 . I highly recommend it, as he discusses his application of Bayes' theorem to the historical method and its superiority to abductive reasoning.
Connecting grouchymusicologist's comment, Carrier showed his work on Bayes' Theorem for historians to C. Behan McCullagh and McCullagh responded well to it. (McCullagh has thus far not used a Bayesian approach to historical method.)
I had lunch with Carrier recently and his first of two books on Bayes' Theorem for historians is in progress for publication with Prometheus Press.
There has been some work done in this area, such as by the philosopher of history C. Behan McCullagh. You can get the flavor of his approach from the Wikipedia article on historical method, as well as some other potentially formalizable views that have been expressed by others. The book-length version of McCullagh's method is Justifying Historical Descriptions (Cambridge, 1984), and there is a more recent book of his called Truth of History that I haven't read. You could also consult the section "Causation in History" in the SEP article Philosophy of History and get a lot more suggestions for further reading there.
In general, my impression is that explicitly Bayesian approaches are currently rare (not nonexistent, see ParagonProtege's mention above of Richard Carrier), but most philosophers of history who've thought seriously about these issues have arrived at "casual formalisms" that are probably capable of being translated into Bayesian terms.
no two people who are completely rational should ever disagree?
No two people who are completely rational and have the same information should ever disagree.
I am an aspiring historian and I'm very interested in ways to apply Bayesian reasoning to history. When I say "history" I mean the study of history -- as a historian, allowing my map of what has happened in the past to match the territory, and being able to represent more accurately the relative strength of historical evidence for and against various historical models.
I know that historical evidence works quite a bit differently from scientific evidence. But I think that historical evidence is also useful. Historians, in recording and assembling secondary sources, assess the relative strength of evidence (mostly primary sources) with regards to a topic already. But there must be a way to do it more formally. Shouldn't there be a right answer, just as no two people who are completely rational (and have the same information) should ever disagree?
This is a post (or series of posts) I might write in the future, and I have put a bit of thought into it so far, but I need to do quite a bit more research. Is there anyone interested in reading something on this topic? Has it been done before? Is there anyone who is knowledgeable about how historians treat evidence who might be able to offer some insights?