Good point, so I added a second answer.
You could come up with a probability (I think the use of 'prior' in the OP is incorrect), but if anybody disagrees, they're much more likely to be questioning your definition of 'just war' (or how to interpret various facts in light of that definition) than your use of Bayes' Theorem.
Could you use Bayes Theorem to figure out whether or not a given war is just?
If so, I was wondering how one would go about estimating the prior probability that a war is just.
Thanks for any help you can offer.