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.