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Decius 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?
Bayesian decisions cannot be made under an inability to assign a probability distribution to the outcomes.
As mentioned, you can consider a Bayesian probability distribution of what the correct distributions will be; if you have no reason to say which state, if any, is more probable, then they have the same meta-distribution as each other: If you know that a coin is unfair, but have no information about which way it is biased, then you should divide the first bet evenly between heads and tails, (assuming logarithmic payoffs).
It might make sense to consider the Probability distribution of the fairness of the coin as a graph: the X axis, from 0-1 being the chance of each flip coming up heads, and the Y axis being the odds that the coin has that particular property; because of our prior information, there is a removable discontinuity at x=1/2. Initially, the graph is flat, but after the first flip it changes: if it came up tails, the odds of a two-headed coin are now 0, the odds of a .9999% heads coin are infinitesimal, and the odds of a tail-weighted coin are significantly greater: Having no prior information on how weighted the coin is, you could assume that all weightings (except fair) are equally likely. After the second flip, however, you have information about what the bias of the coin was- but no information about whether the bias of the coin is time-variable, such that it is always heads on prime flips, and always tails on composite flips.
If you consider that the coin could be rigged to a sequence equally likely as that the result of the flip could be randomly determined each time, then you have a problem. No information can update some specific lacks of a prior probability.