All of awenonian's Comments + Replies

You're completely right that hypotheses with unconstrained Murphy get ignored because you're doomed no matter what you do, so you might as well optimize for just the other hypotheses where what you do matters. Your "-1,000,000 vs -999,999 is the same sort of problem as 0 vs 1" reasoning is good. Again, you are making the serious mistake of trying to think about Murphy verbally, rather than thinking of Murphy as the personification of the "inf" part of the EΨ[f]:=inf(m,b)∈Ψm(f)+b definition of expected value, and writing actual equations. Ψ is the available set of possibilities for a hypothesis. If you really want to, you can think of this as constraints on Murphy, and Murphy picking from available options, but it's highly encouraged to just work with the math. For mixing hypotheses (several different Ψi sets of possibilities) according to a prior distribution ζ∈ΔN, you can write it as an expectation functional via ψζ(f):=Ei∼ζ[ψi(f)] (mix the expectation functionals of the component hypotheses according to your prior on hypotheses), or as a set via Ψζ:={(m,b)|∃(mi,bi)∈Ψi:Ei∼ζ(mi,bi)=(m,b)} (the available possibilities for the mix of hypotheses are all of the form "pick a possibility from each hypothesis, mix them together according to your prior on hypotheses") This is what I meant by "a constraint on Murphy is picked according to this probability distribution/prior, then Murphy chooses from the available options of the hypothesis they picked", that Ψζ set (your mixture of hypotheses according to a prior) corresponds to selecting one of the Ψi sets according to your prior ζ, and then Murphy picking freely from the set Ψi. Using ψζ(f):=Ei∼ζ[ψi(f)] (and considering our choice of what to do affecting the choice of f, we're trying to pick the best function f) we can see that if the prior is composed of a bunch of "do this sequence of actions or bad things happen" hypotheses, the details of what you do sensitively depend on the probability distribution over hypothese

Maximin, actually. You're maximizing your worst-case result. It's probably worth mentioning that "Murphy" isn't an actual foe where it makes sense to talk about destroying resources lest Murphy use them, it's just a personification of the fact that we have a set of options, any of which could be picked, and we want to get the highest lower bound on utility we can for that set of options, so we assume we're playing against an adversary with perfectly opposite utility function for intuition. For that last paragraph, translating it back out from the "Murphy" talk, it's "wouldn't it be good to use resources in order to guard against worst-case outcomes within the available set of possibilities?" and this is just ordinary risk aversion. For that equation argmaxπinfe∈BEπ⋅e[U], B can be any old set of probabilistic environments you want. You're not spending any resources or effort, a hypothesis just is a set of constraints/possibilities for what reality will do, a guess of the form "Murphy's operating under these constraints/must pick an option from this set." You're completely right that for constraints like "environment must be a valid chess board", that's too loose of a constraint to produce interesting behavior, because Murphy is always capable of screwing you there. This isn't too big of an issue in practice, because it's possible to mix together several infradistributions with a prior, which is like "a constraint on Murphy is picked according to this probability distribution/prior, then Murphy chooses from the available options of the hypothesis they picked". And as it turns out, you'll end up completely ignoring hypotheses where Murphy can screw you over no matter what you do. You'll choose your policy to do well in the hypotheses/scenarios where Murphy is more tightly constrained, and write the "you automatically lose" hypotheses off because it doesn't matter what you pick, you'll lose in those. But there is a big unstudied problem of "what sorts of hypotheses