Probabilities have to sum to 1, but hypotheses are infinite, and you can consider only a handful. In AI, this “truncation” of possible factors was often called “a closed-world assumption.” In philosophy, it’s referred to as “ceteris paribus conditions” or “the background.”
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John Searle and Hubert Dreyfus repeatedly invoked the unbounded “background” of possible practical considerations as a reason to dismiss the possibility of AI. I don’t think it works as an in-principle argument, but it is a powerful reason to doubt rationalist approaches.
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Misuse of statistics in science often also begins with failing to recognize that hypotheses are infinite. Any application of probability theory begins by excluding almost all. If you don’t understand you are doing that, you will have unjustified confidence in your results.
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“Just tell me which statistical method I’m supposed to use” implicitly assumes that probability theory guarantees that there is a correct method. But probability theory itself does not, cannot, apply to the real world—only to finite (or real-valued) abstractions of it.
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If you find some bit of reality in which the finite-hypothesis abstraction is roughly accurate, probability theory can work extraordinarily well! You just need to be aware that you can never be sure it will continue to work, and that in lots of situations it plain doesn’t.
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“Expanding the hypothesis space” is typically a meta-rational operation. There is no systematic method. You have to rummage around in the unbounded, undefined background to see what you can find. This was Dreyfus’s central, best objection to rationalism, I think.
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@_Srijit Probability density only works when you have a measure space on the possible outcomes, which mostly means R^n. The relevant cases aren’t like that. How many different ways could your job interview go badly? You can’t enumerate them, and they aren’t like R^n. - 1 more reply
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even infinite minds struggle to assign priors over countably infinite hypotheses. i.e., an infinite mind has meta-rational work to do in the selection of a prior that gives nonzero mass to infinitely many hypotheses
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Do you think this provides insight into why people are so polarized lately? The amount of information (priors, posteriors) is so great, that people just oversimplify (poorly classify) the large number of probability spaces into 1/0, Rep/Dem, Good/Bad, Us/Them ?
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