Bayesian inference applies to quantum mechanics in the "obvious" way. You have some hypothesis space (of possible wavefunctions). You plan to perform an observa

Bayesian inference applies to quantum mechanics in the "obvious" way. You have some hypothesis space (of possible wavefunctions). You plan to perform an observation (a measurement). Different hypotheses assign different probabilities to the various possible outcomes. You perform the observation, get a result, and perform a standard Bayesian update based on the outcome. There are two caveats somewhat unique to quantum mechanics. First, measurement has a known kickback effect on the state. So you always have to be careful to distinguish inferences of what the value of the state was before measurement vs inferences of the state after measurement. In general, after the observation has been made and the back-action accounted for, you know more about the post-measurement state than the pre-measurement state. Second, many apparently different probability distributions of wavefunctions are observationally indistinguishable. For example, a qubit with a 50% chance of being definitely ON and a 50% chance of being definitely OFF is observationally indistinguishable from a qubit known to be in a state of the form 1/sqrt(2)|ON> + e^(i theta)/sqrt(2) |OFF> where theta is a uniform random variable ranging from 0 to 2pi. Those two hypotheses give identical predictions for all possible measurements. So people usually work with equivalence classes of probability distributions, represented by mathematical objects called density matrices, instead of individual probability distributions. Density matrices massively simplify everything, to the point where you might not even recognize the Bayes bits happening underneath.