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CBHacking comments on Making Beliefs Pay Rent (in Anticipated Experiences) - Less Wrong

110 Post author: Eliezer_Yudkowsky 28 July 2007 10:59PM

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Comment author: CBHacking 18 January 2016 11:31:39PM 0 points [-]

I don't think "converse" is the word you're looking for here - possibly "complement" or "negation" in the sense that (A || ~A) is true for all A - but I get what you're saying. Converse might even be the right word for that; vocabulary is not my forte.

If you take the statement "most beliefs are false" as given, then "the negation of most beliefs is true" is trivially true but adds no new information. You're treating positive and negative beliefs as though they're the same, and that's absolutely not true. In the words of this post, a positive belief provides enough information to anticipate an experience. A negative belief does not (assuming there are more than two possible beliefs). If you define "anything except that one specific experience" as "an experience", then you can define a negative belief as a belief, but at that point I think you're actually falling into exactly the trap expressed here.

If you replace "belief" with "statement that is mutually incompatible with all other possible statements that provide the same amount of information about its category" (which is a possibly-too-narrow alternative; unpacking words is hard sometimes) then "true statements that are mutually incompatible with all other possible statements that provide the same amount of information about their category are vastly outnumbered by false statements that are mutually incompatible with all other possible statements that provide the same amount of information about their category" is something the I anticipate you would find true. You and Eliezer do not anticipate a different percentage of possible "statements that are mutually incompatible with all other possible statements that provide the same amount of information about their category" being true.

As for universal priors, the existence of many incompatible possible (positive) beliefs in one space (such that only one can be true) gives a strong prior that any given such belief is false. If I have only two possible beliefs and no other information about them, then it only takes one bit of evidence - enough to rule out half the options - to decide which belief is likely true. If I have 1024 possible beliefs and no other evidence, it takes 10 bits of evidence to decide which is true. If I conduct an experiment that finds that belief 216 +/- 16 is true, I've narrowed my range of options from 1024 to 33, a gain of just less than 5 bits of evidence. Ruling out one more option gives the last of that 5th bit. You might think that eliminating ~96.8% of the possible options sounds good, but it's only half of the necessary evidence. I'd need to perform another experiment that can eliminate just as large a percentage of the remaining values to determine the correct belief.