I was fascinated to discover that the Bayesian reasoner's probability of Bayes' formula being true calculates to zero.
I should start by mentioning that my probability estimate for Bayes' formula being useful is 100%. At least I hope so, since I often use it.
To calculate the probability using Bayesian analysis requires assuming the Bayes formula - in classical terms, begging the question. But, I thought, perhaps the series can converge? Every run-through changes the probability assigned to the Bayes' formula step of the calculation, so it needs to be run until it settles down. However, any sub-unity number multiplied by itself enough times converges to zero. It's an article of Bayesian reasoning that no priors can be unity. Oops.
Bayes' formula was discovered and proved within the context of classical logic, and indeed even the Bayesian reasoner must use one prior step of classical logic before they go on their statistical voyage. They must assume that Bayes' formula is unconditionally true.
There's a similar problem estimating the odds of yourself being mistaken. If you run the calculation once, perhaps you get a reasonable number, like 2%. But this calculation is reflexive - the odds that you're mistaken about being that mistaken is 2%. Works out to a 3.96% chance of being mistaken. But it's reflexive, so...a Bayesian calculation shows I have a 100% chance of being mistaken on every subject. Therefore, I'm wrong that you can understand the individual words I'm now typing.
Though clearly nonsense, I enjoy this result because I've often suspected that self-doubt in the usual sense can't be logically upheld. Especially if I'm not evaluating evidence, but taking action, the correct course is insensitive to doubt. If I'm faced with a black box with several buttons, one of which will serve my goals if pushed, whether I assume I have a 30% chance of being wrong or a 0% chance, I push the same button. It only changes if I have a higher estimate of being right for some other button - but it changes instantly and entirely, so again my action indistinguishable from one with utter certainty.