as it turns out, we have results like the complete class theorem, which imply that EU maximization with respect to an appropriate prior is the only “Pareto efficient” decision procedure (any other decision can be changed so as to achieve a higher reward in every possible world).
I found several books which give technical coverage of statistical decision theory, complete classes, and admissibility rules (Berger 1985; Robert 2001; Jaynes 2003; Liese & Miescke 2010), but I didn’t find any clear explanation of exactly how the complete class theorem implies that “EU maximization with respect to an appropriate prior is the only ‘Pareto efficient’ decision procedure (any other decision can be changed so as to achieve a higher reward in every possible world).”
Do you know any source which does so, or are you able to explain it? This seems like a potentially significant argument for EUM that runs independently of the standard axiomatic approaches, which have suffered many persuasive attacks.
The formalism of the complete class theorem applies to arbitrary decisions, the Bayes decision procedures correspond to EU maximization with respect to an appropriate choice of prior. An inadmissable decision procedure is not Pareto efficient, in the sense that a different decision procedure does better in all possible worlds (which feels analogous to making all possible people happier). Does that make sense?
There is a bit of weasel room, in that the complete class theorem assumes that the data is generated by a probabilistic process in each possible world. This doesn’t seem like an issue, because you just absorb the observation into the choice of possible world, but this points to a bigger problem:
If you define “possible worlds” finely enough, such that e.g. each (world, observation) pair is a possible world, then the space of priors is very large (e.g., you could put all of your mass on one (world, observation) pair for each observation) and can be used to justify any decision. For example, if we are in the setting of AIXI, any decision procedure can trivially be described as EU maximization under an appropriate prior: if the decision procedure outputs f(X) on input X, it corresponds to EU maximization against a prior which has the universe end after N steps with probability 2^(-N), and when the universe ends after you seeing X, you receive an extra reward if your last output was f(X).
So the conclusion of the theorem isn’t so interesting, unless there are few possible worlds. When you argue for EUM, you normally want some stronger statement than saying that any decision procedure corresponds to some prior.
I found several books which give technical coverage of statistical decision theory, complete classes, and admissibility rules (Berger 1985; Robert 2001; Jaynes 2003; Liese & Miescke 2010), but I didn’t find any clear explanation of exactly how the complete class theorem implies that “EU maximization with respect to an appropriate prior is the only ‘Pareto efficient’ decision procedure (any other decision can be changed so as to achieve a higher reward in every possible world).”
Do you know any source which does so, or are you able to explain it? This seems like a potentially significant argument for EUM that runs independently of the standard axiomatic approaches, which have suffered many persuasive attacks.
The formalism of the complete class theorem applies to arbitrary decisions, the Bayes decision procedures correspond to EU maximization with respect to an appropriate choice of prior. An inadmissable decision procedure is not Pareto efficient, in the sense that a different decision procedure does better in all possible worlds (which feels analogous to making all possible people happier). Does that make sense?
There is a bit of weasel room, in that the complete class theorem assumes that the data is generated by a probabilistic process in each possible world. This doesn’t seem like an issue, because you just absorb the observation into the choice of possible world, but this points to a bigger problem:
If you define “possible worlds” finely enough, such that e.g. each (world, observation) pair is a possible world, then the space of priors is very large (e.g., you could put all of your mass on one (world, observation) pair for each observation) and can be used to justify any decision. For example, if we are in the setting of AIXI, any decision procedure can trivially be described as EU maximization under an appropriate prior: if the decision procedure outputs f(X) on input X, it corresponds to EU maximization against a prior which has the universe end after N steps with probability 2^(-N), and when the universe ends after you seeing X, you receive an extra reward if your last output was f(X).
So the conclusion of the theorem isn’t so interesting, unless there are few possible worlds. When you argue for EUM, you normally want some stronger statement than saying that any decision procedure corresponds to some prior.
That was clear. Thanks!