And yet if you add those two posterior distributions, weighted by your current probability of ending up with each, you get your prior back. Magic!

(Witch burners don't get their prior back when they do this because they expect to update in the direction of "she's a witch" in either case, so when they sum over probable posteriors, they get back their real prior which says "I already know that she's a witch", the implication being "the trial has low value of information, let's just burn her now".)