I think you're saying that the Turing machine that reasons by Bayesian inference (i.e. repeated application of the product rule of probability theory) would turn out to be the best one.

What I meant to say was that if, per your hypothetical, we designed the best utility optimizer, we could see if it was "Bayesian" by checking whether it acts in accordance with the Bayesian probabilities (a la the Turing test). I guess?

The only real point of my post was that your stated objection to demonstrate that Turing machines clearly couldn't perform Bayesian updates wasn't really meaningful, in that the same objection could be leveled at any system more granular than the universe itself. On the other hand, I do expect that Turing-equivalent systems will have trouble with calculating priors.

I suppose you could have one Turing machine for each Planck duration in your lifetime (really this isn't so much to ask, as each machine is infinite in size anyway), each returning its calculation one after the other?