Manfred comments on Stupid Questions Open Thread Round 4 - Less Wrong Discussion
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I finally decided it's worth some of my time to try to gain a deeper understanding of decision theory...
Question: Can Bayesians transform decisions under ignorance into decisions under risk by assuming the decision maker can at least assign probabilities to outcomes using some kind of ignorance prior(s)?
Details: "Decision under uncertainty" is used to mean various things, so for clarity's sake I'll use "decision under ignorance" to refer to a decision for which the decision maker does not (perhaps "cannot") assign probabilities to some of the possible outcomes, and I'll use "decision under risk" to refer to a decision for which the decision maker does assign probabilities to all of the possible outcomes.
There is much debate over which decision procedure to use when facing a decision under ignorance when there is no act that dominates the others. Some proposals include: the leximin rule, the optimism-pessimism rule, the minimax regret rule, the info-gap rule, and the maxipok rule.
However, there is broad agreement that when facing a decision under risk, rational agents maximize expected utility. Because we have a clearer procedure for dealing with decisions under risk than we do for dealing with decisions under ignorance, many decision theorists are tempted to transform decisions under ignorance into decisions under risk by appealing to the principle of insufficient reason: "if you have literally no reason to think that one state is more probable than another, then one should assign equal probability to both states."
And if you're a Bayesian decision-maker, you presumably have some method for generating ignorance priors, whether or not that method always conforms to the principle of insufficient reason, and even if you doubt you've found the final, best method for assigning ignorance priors.
So if you're a Bayesian decision-maker, doesn't that mean that you only ever face decisions under risk, because at they very least you're assigning ignorance priors to the outcomes for which you're not sure how to assign probabilities? Or have I misunderstood something?
This reminds me of a recent tangent on Kelly betting. Apparently it's claimed that the unusalness of this optimum betting strategy shows that you should treat risk and ignorance differently - but of course the difference between the two situations is entirely accounted for by two different conditional probability distributions. So you can sort of think of situations (that is, the probability distribution describing possible outcomes) as "risk-like" or "ignorance-like."
If you're talking about what I think you're talking about, then by "risk", you mean "frequentist probability distribution over outcomes", and by "ignorance", you mean "Bayesian probability distribution over what the correct frequentist probability distribution over outcomes is", which is not the way Luke was defining the terms.