Followup to: Making Beliefs Pay Rent in Anticipated Experiences
In the comments section of Making Beliefs Pay Rent, Eliezer wrote:
I follow a correspondence theory of truth. I am also a Bayesian and a believer in Occam's Razor. If a belief has no empirical consequences then it could receive no Bayesian confirmation and could not rise to my subjective attention. In principle there are many true beliefs for which I have no evidence, but in practice I can never know what these true beliefs are, or even focus on them enough to think them explicitly, because they are so vastly outnumbered by false beliefs for which I can find no evidence.
If I am interpreting this correctly, Eliezer is saying that there is a nearly infinite space of unfalsifiable hypotheses, and so our priors for each individual hypothesis should be very close to zero. I agree with this statement, but I think it raises a philosophical problem: doesn't this same reasoning apply to any factual question? Given a set of data D, there must be an nearly infinite space of hypotheses that (a) explain D and (b) make predictions (fulfilling the criteria discussed in Making Beliefs Pay Rent). Though Occam's Razor can help us to weed out a large number of these possible hypotheses, a mind-bogglingly large number would still remain, forcing us to have a low prior for each individual hypothesis. (In philosophy of science, this is known as "underdetermination.") Or is there a flaw in my reasoning somewhere?
Surely, this is dealt with by considering the amount of information in the hypothesis? If we consider each hypothesis that can be represented with 1,000 bits of information, there will only be a maximum of 2^1,000 such hypotheses, and if we consider each hypothesis that can be represented with n bits of information, there will only be a maximum of 2^n - and that is before we even start eliminating hypotheses that are inconsistent with what we already know. If we favor hypotheses with less information content, then we end up with a small number of hypothese...
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