As a PhD student in condensed matter physics, I am very familiar with Dirac's bra-ket notation, but not so much with Bayesian inference. One of the first things that struck me when I started studying the Bayesian framework is the similarity that some formula bear with bra-ket formulas. The equivalences generally take the form :
$$p(A|B) = \int p(A|x) \cdot p(x|B) dx $$ which is formally equivalent to $$\langle A| B\rangle = \langle A | 1 | B\rangle = \langle A | \left(\int |x\rangle\langle x|dx \right)|B\rangle = \int \langle A|x\rangle\langle x|B\rangle dx$$
My question is how deep does this go? Has there been any systematic investigation of the matter? and under what conditions is it safe to use formulas from quantum mechanics to derive results on probabilities?