The binary thing isn't important, what's important is that there are real situations where likelihood based methods (including Bayes) don't work well (because by assumption there is only strong info on the part of the likelihood we aren't using in our functional, and the part of the likelihood we are using in our functional is very complicated).
I think my point wasn't so much the technical specifics of that example, but rather that these are the types of B vs F arguments that actually have something to say, rather than going around and around in circles. I had a rephrase of this example using causal language somewhere on LW (if that will help, not sure if it will).
Robins and Ritov have something of paper length, rather than blog post length if you are interested.
I think I'm beginning to see the problem for the Bayesian, although I not yet sure what the correct response to it is. I have some more or less rambling thoughts about it.
It appears that the Bayesian is being supposed to start from a flat prior over the space of all possible thetas. This is a very large space (all possible strings of 2^100000 probabilities), almost all of which consists of thetas which are independent of pi. (ETA: Here I mistakenly took X to be a product of two-point sets {0,1}, when in fact it is a product of unit intervals [0,1]. I don't...
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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