I have I already presented this to Abram Demski, and he and I have been working together on trying to prove my conjecture. (He and I are both in Los Angeles, and coincidentally are interested in the same question, so it is likely to be the direction that the MIRIxLosAngeles workshop continues to focus on.)
Your proposal is equivalent to Abram's proposal. We believe the two distributions are not the same. I think we checked this for some small finite analogue.
Your "general" setting does not seem that much more general to me, It seems like it is pretty much identical, only reworded in terms of set theory instead of logic. There is one way in which it is more general. In my system, the set of subsets must be closed under union, intersection, and complement, and this is not true for a general collection of subsets. However, my construction does not work without this assumption. I actually use the fact that \mu is nowhere zero, and not being closed under union intersection and complement is kind of liking having some sets have measure 0.
I think the language of subsets instead of logic makes things a little easier to think about for some people. I think I prefer the logic language. (However when I wanted to think about small finite examples I would sometimes think about it in the subset language instead)
Firstly, I'd love to see the counterexample for the distributions being the same.
Secondly, are you sure \mu nowhere zero is essential? Intuitively, your uniqueness result must work whenever for every two models M1, M2 there is a sentence \phi separating them with \mu(\phi) non-zero. But I haven't checked it formally.
In this post, I propose an answer to the following question:
Given a consistent but incomplete theory, how should one choose a random model of that theory?
My proposal is rather simple. Just assign probabilities to sentences in such that if an adversary were to choose a model, your Worst Case Bayes Score is maximized. This assignment of probabilities represents a probability distribution on models, and choose randomly from this distribution. However, it will take some work to show that what I just described even makes sense. We need to show that Worst Case Bayes Score can be maximized, that such a maximum is unique, and that this assignment of probabilities to sentences represents an actual probability distribution. This post gives the necessary definitions, and proves these three facts.
Finally, I will show that any given probability assignment is coherent if and only if it is impossible to change the probability assignment in a way that simultaneously improves the Bayes Score by an amount bounded away from 0 in all models. This is nice because it gives us a measure of how far a probability assignment is from being coherent. Namely, we can define the "incoherence" of a probability assignment to be the supremum amount by which you can simultaneously improve the Bayes Score in all models. This could be a useful notion since we usually cannot compute a coherent probability assignment so in practice we need to work with incoherent probability assignments which approach a coherent one.
I wrote up all the definitions and proofs on my blog, and I do not want to go through the work of translating all of the latex code over here, so you will have to read the rest of the post there. Sorry. In case you do not care enough about this to read the formal definitions, let me just say that my definition of the "Bayes Score" of a probability assignment P with respect to a model M is the sum over all true sentences s of m(s)log(P(s)) plus the sum over all false sentences s of m(s)log(1-P(s)), where m is some fixed nowhere zero probability measure on all sentences. (e.g. m(s) is 1/2 to the number of bits needed to encode s)
I would be very grateful if anyone can come up with a proof that this probability distribution which maximizes Worst Case Bayes Score has the property that its Bayes Score is independent of the choice of what model we use to judge it. I believe it is true, but have not yet found a proof.