Yes, I mostly agree with everything you said—the limitation with the probabilistic Turing machine approach (it’s usually equivalently described as the a priori probability and described in terms of monotone TM’s) is that you can get samples, but you can’t use those to estimate conditionals. This is connected to the typical problem of computing the normalization factor in Bayesian statistics. It’s possible that these approximations would be good enough in practice though.
Yes, I mostly agree with everything you said—the limitation with the probabilistic Turing machine approach (it’s usually equivalently described as the a priori probability and described in terms of monotone TM’s) is that you can get samples, but you can’t use those to estimate conditionals. This is connected to the typical problem of computing the normalization factor in Bayesian statistics. It’s possible that these approximations would be good enough in practice though.