The Nonlinear Library LW - What is an "anti-Occamian prior"? by Zane
Oct 23, 2023
02:21
Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: What is an "anti-Occamian prior"?, published by Zane on October 23, 2023 on LessWrong.
I've seen references to "anti-Occamian priors"
in the Sequences
, where Eliezer was talking about how not all possible minds would agree with Occam's Razor. I'm not sure how such a prior could consistently exist.
It seems like Occam's Razor just logically follows from the basic premises of probability theory. Assume the "complexity" of a hypothesis is how many bits it takes to specify under a particular method of specifying hypotheses, and that hypotheses can be of any length. Then for any prior that assigns nonzero probability to any finite hypothesis H, there must exist some level of complexity L such that any hypothesis more complex than L is less likely than H.
(That is to say, if a particular 13-bit hypothesis is 0.01% likely, then there are at most 9,999 other hypotheses with >= 0.01% probability mass. If the most complicated of these <10,000 hypotheses is 27 bits, then every hypothesis that takes 28 bits or more to specify is less likely than the 13-bit hypothesis. You can change around the numbers 13 and 0.01% and 27 as much as you want, but as long as there's any hypothesis whatsoever with non-infinitesimal probability, then there's some level where everything more complex than that level is less likely than that hypothesis.)
This seems to prove that an "anti-Occamian prior" - that is to say, a hypothesis that always assigns more probability to more complex hypotheses and less to the less complex - is impossible. Or at least, that it assigns zero probability to every finite hypothesis. (You could, I suppose, construct a prior such that {sum of probability mass from all 1-bit hypotheses} is 1/3 of {sum of probability mass from all 2-bit hypotheses}, which is then itself 1/3 of {sum of probability mass from all 3-bit hypotheses}, and on and on forever, and that would indeed be anti-Occamian - but it would also assign zero probability to every finite hypothesis, which would make it essentially meaningless.)
Am I missing something about what "anti-Occamian prior" is really supposed to mean here, or how it could really be consistent?
Thanks for listening. To help us out with The Nonlinear Library or to learn more, please visit nonlinear.org
I've seen references to "anti-Occamian priors"
in the Sequences
, where Eliezer was talking about how not all possible minds would agree with Occam's Razor. I'm not sure how such a prior could consistently exist.
It seems like Occam's Razor just logically follows from the basic premises of probability theory. Assume the "complexity" of a hypothesis is how many bits it takes to specify under a particular method of specifying hypotheses, and that hypotheses can be of any length. Then for any prior that assigns nonzero probability to any finite hypothesis H, there must exist some level of complexity L such that any hypothesis more complex than L is less likely than H.
(That is to say, if a particular 13-bit hypothesis is 0.01% likely, then there are at most 9,999 other hypotheses with >= 0.01% probability mass. If the most complicated of these <10,000 hypotheses is 27 bits, then every hypothesis that takes 28 bits or more to specify is less likely than the 13-bit hypothesis. You can change around the numbers 13 and 0.01% and 27 as much as you want, but as long as there's any hypothesis whatsoever with non-infinitesimal probability, then there's some level where everything more complex than that level is less likely than that hypothesis.)
This seems to prove that an "anti-Occamian prior" - that is to say, a hypothesis that always assigns more probability to more complex hypotheses and less to the less complex - is impossible. Or at least, that it assigns zero probability to every finite hypothesis. (You could, I suppose, construct a prior such that {sum of probability mass from all 1-bit hypotheses} is 1/3 of {sum of probability mass from all 2-bit hypotheses}, which is then itself 1/3 of {sum of probability mass from all 3-bit hypotheses}, and on and on forever, and that would indeed be anti-Occamian - but it would also assign zero probability to every finite hypothesis, which would make it essentially meaningless.)
Am I missing something about what "anti-Occamian prior" is really supposed to mean here, or how it could really be consistent?
Thanks for listening. To help us out with The Nonlinear Library or to learn more, please visit nonlinear.org