Editing Occam's razor (section)

==== Mathematical arguments against Occam's razor ====
{{Technical|date=February 2024|section}}
The [[No free lunch theorem|no free lunch]] (NFL) theorems for inductive inference prove that Occam's razor must rely on ultimately arbitrary assumptions concerning the prior probability distribution found in our world.<ref name="Adam2019">Adam, S., and Pardalos, P. (2019), [https://www.researchgate.net/profile/Stamatios-Aggelos-Alexandropoulos-2/publication/333007007_No_Free_Lunch_Theorem_A_Review/links/5e84f65792851c2f52742c85/No-Free-Lunch-Theorem-A-Review.pdf No-free lunch Theorem: A review], in "Approximation and Optimization", Springer, 57-82</ref> Specifically, suppose one is given two inductive inference algorithms, A and B, where A is a [[Bayesian inference|Bayesian]] procedure based on the choice of some prior distribution motivated by Occam's razor (e.g., the prior might favor hypotheses with smaller [[Kolmogorov complexity]]). Suppose that B is the anti-Bayes procedure, which calculates what the Bayesian algorithm A based on Occam's razor will predict  – and then predicts the exact opposite. Then there are just as many actual priors (including those different from the Occam's razor prior assumed by A) in which algorithm B outperforms A as priors in which the procedure A based on Occam's razor comes out on top. In particular, the NFL theorems show that the "Occam factors" Bayesian argument for Occam's razor must make ultimately arbitrary modeling assumptions.<ref name="WOLP95">Wolpert, D.H (1995), On the Bayesian "Occam Factors" Argument for Occam's Razor, in "Computational Learning Theory and Natural Learning Systems: Selecting Good Models", MIT Press</ref>