The possible therefore probable fallacy. This is something that I have often encountered so many times in religious debates that there is no way I could put a finger on when I first ran into it. Not only that, but it was hard to explain why it was fallacious before I started studying probability theory. And there was no official name for this fallacy.
But make no mistake. It's a fallacy. If you study probability theory, you will learn that everything is possible. Everything. Even though everything is possible, not everything is probable. This is the heart of the matter; we should always go by whichever idea or explanation has the highest probability of being correct, since this will minimize the probability of us being incorrect. If we didn't do that, then things like the Base Rate Fallacy wouldn't be, well, a fallacy.
The way it goes, someone posits some hypothesis that has a high probability of being correct. The person on the losing side, not content with being on the low end of the probability stick, spouts out his sore-loser argument: "I have this alternative argument. Even if it's less probable than your argument it's still possible! Therefore it's true/I'm justified in believing it/na-na-nah-boo-boo." While not as obvious as that, this is the basic gist of the thread of conversation.
Since Richard Carrier has recently published his book Proving History (p 26 - 29), there is now an actual peer reviewed publication that addresses this fallacy. Which means it is now an "official" fallacy since he gave it a Latin name lol: Possibiliter ergo probabiliter.
This can be added to the list of fallacies that can be analyzed by Bayes'.