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David_Chapman comments on What is the best paper explaining the superiority of Bayesianism over frequentism? - Less Wrong Discussion
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Can you recommend an explanation of the complete class theorem(s)? Preferably online. I've been googling pretty hard and I've turned up almost nothing. I'd like to understand what conditions they start from (suspecting that maybe the result is not quite as strong as "Bayes Rules!"). I've found only one paper, which basically said "what Wald proved is extremely difficult to understand, and probably not what you wanted."
Thank you very much!
Maybe try this one? Let me know if that helps or if you're looking for something different.
The complete class theorem states, informally: any Pareto optimal decision rule is a Bayesian decision rule (i.e. it can be obtained by choosing some prior, observing data, and then maximizing expected utility relative to the posterior).
Roughly, the argument is that if I have a collection W of possible worlds that I could be in, and a value U(w) to taking a particular action in world w, then any Pareto optimal strategy implicitly assigns an "importance" p(w) to each world, and takes the action that maximizes the sum of p(w)*U(w). We can then show that this is equivalent to using the Bayesian decision rule with p(w) as the prior over W. The main thing needed to formalize this argument is the separating hyperplane theorem, which is what the linked paper does.
Does the complete class theorem thus provide what Peterson (2004) and Easwaran (unpublished) think is missing in classical axiomatic decision theory: namely, a justification for choosing a prior, observing data, and then maximizing expected utility relative to the posterior?