Yeah, you’re right. I was assuming that Bayes(M, P) is continuous in both arguments in some sense. But we haven’t proved that, and now I think I found a counterexample.
Consider the language that consists of countably many sentences, each of which contradicts all the others. So the only valid models are M_k = “the kth sentence is true, all the rest are false” for each k, and M_inf = “all sentences are false”. Let’s say that the kth sentence has weight 2^-k. Now let’s define a probability assignment P that assigns to the kth sentence the probability P_k = (2^(-2^k))/(1+2^(-2^k)). You can check that Bayes(M_k, P) has the same value for all k, but Bayes(M_inf, P) has a different and higher value.
That seems to falsify most naive statements like “Bayes(M, P) is continuous in M for a given P, where the distance between two M’s is the total weight of sentences where they disagree”. There’s probably some simple argument that falsifies continuity in P as well, once we define a metric on P’s (which I don’t know how to do).
Yeah, you’re right. I was assuming that Bayes(M, P) is continuous in both arguments in some sense. But we haven’t proved that, and now I think I found a counterexample.
Consider the language that consists of countably many sentences, each of which contradicts all the others. So the only valid models are M_k = “the kth sentence is true, all the rest are false” for each k, and M_inf = “all sentences are false”. Let’s say that the kth sentence has weight 2^-k. Now let’s define a probability assignment P that assigns to the kth sentence the probability P_k = (2^(-2^k))/(1+2^(-2^k)). You can check that Bayes(M_k, P) has the same value for all k, but Bayes(M_inf, P) has a different and higher value.
That seems to falsify most naive statements like “Bayes(M, P) is continuous in M for a given P, where the distance between two M’s is the total weight of sentences where they disagree”. There’s probably some simple argument that falsifies continuity in P as well, once we define a metric on P’s (which I don’t know how to do).