jsteinhardt comments on Non-trivial probability distributions for priors and Occam's razor - Less Wrong
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You don't need finite, only countable. Assuming that our conception of the universe is approximately correct, we are only capable of generating a countable set of hypotheses.
Huh? There are only countably many statements in any string-based language, so this includes decidedly non-Occamian scenarios like every statement being true, or every statement being true with the same probability.
I'm confused. What is the countable set of hypotheses you are considering? My claim is merely that if you have hypotheses H1, H2, ..., then p(Hi) > 1/n for at most n-1 values of i. This can be thought of as a weak form of Occam's razor.
In what sense is "every statement being true" a choice of a countable set of hypotheses?
I think maybe the issue is that we are using hypothesis in a different sense. In my case a hypothesis is a complete model of the world, so it is not possible for multiple hypotheses to be true. You can marginalize out / observe a bunch of variables to talk about a subset of the world, but your hypotheses should still be mutually exclusive.