One way of defining prior and utility is just by first taking a preference over the events of sample space, and then choosing any pair prior+utility such that expected utility calculated from them induces the same order on events.
Um, isn’t that obviously wrong? It sounds like your are suggesting that we say “I like playing blackjack better than playing the lottery, so I should choose a prior probability of winning each and a utility associated with winning each so that that preference will remain consistent when I switch from ‘preference mode’ to ‘utilitarian mode’.” Wouldn’t it be better to choose the utilities of winning based on the prizes they give? And choose the priors for each based on studying the history of each game carefully?
Any observations and updating consist in choosing what events you work with. Once prior is fixed, it never changes.
Events are sets of outcomes, right? It sounds like you are suggesting that people update their probabilities by reshuffling which outcomes go with which events. Aren’t events just a layer of formality over outcomes? Isn’t real learning what happens when you change your estimations of the probabilities of outcomes, not when you reclassify them?
It almost seems to me as if we are talking past each other… I think I need a better background on this stuff. Can you recommend any books that explain probability for the layman? I already read a large section of one, but apparently it wasn’t very good...
Although I do think there is a chance you are wrong. I see you mixing up outcome-desirability estimates with chance-of-outcome estimates, which seems obviously bad.
If you don’t want the choice of preference to turn out bad for you, choose good preference ;-) There is no freedom in choosing your preference, as the “choice” is itself a decision-concept, defined in terms of preference, and can’t be a party to the definition of preference. When you are speaking of a particular choice of preference being bad or foolish, you are judging this choice from the reference frame of some other preference, while with preference as foundation of decision-making, you can’t go through this step. It really is that arbitrary. See also: Priors as Mathematical Objects, Probability is Subjectively Objective.
You are confusing probability space and its prior (the fundamental structure that bind the rest together) with random variables and their probability distributions (things that are based on probability space and that “interact” with each other through the definition in terms of the common probability space, restricted to common events). Informally, when you update a random variable given evidence (event) X, it means that you recalculate the probability distribution of that variable only based on the remaining elements of the probability space within event X. Since this can often be done using other probability distributions of various variables lying around, you don’t always see the probability space explicitly.
Um, isn’t that obviously wrong? It sounds like your are suggesting that we say “I like playing blackjack better than playing the lottery, so I should choose a prior probability of winning each and a utility associated with winning each so that that preference will remain consistent when I switch from ‘preference mode’ to ‘utilitarian mode’.” Wouldn’t it be better to choose the utilities of winning based on the prizes they give? And choose the priors for each based on studying the history of each game carefully?
Events are sets of outcomes, right? It sounds like you are suggesting that people update their probabilities by reshuffling which outcomes go with which events. Aren’t events just a layer of formality over outcomes? Isn’t real learning what happens when you change your estimations of the probabilities of outcomes, not when you reclassify them?
It almost seems to me as if we are talking past each other… I think I need a better background on this stuff. Can you recommend any books that explain probability for the layman? I already read a large section of one, but apparently it wasn’t very good...
Although I do think there is a chance you are wrong. I see you mixing up outcome-desirability estimates with chance-of-outcome estimates, which seems obviously bad.
If you don’t want the choice of preference to turn out bad for you, choose good preference ;-) There is no freedom in choosing your preference, as the “choice” is itself a decision-concept, defined in terms of preference, and can’t be a party to the definition of preference. When you are speaking of a particular choice of preference being bad or foolish, you are judging this choice from the reference frame of some other preference, while with preference as foundation of decision-making, you can’t go through this step. It really is that arbitrary. See also: Priors as Mathematical Objects, Probability is Subjectively Objective.
You are confusing probability space and its prior (the fundamental structure that bind the rest together) with random variables and their probability distributions (things that are based on probability space and that “interact” with each other through the definition in terms of the common probability space, restricted to common events). Informally, when you update a random variable given evidence (event) X, it means that you recalculate the probability distribution of that variable only based on the remaining elements of the probability space within event X. Since this can often be done using other probability distributions of various variables lying around, you don’t always see the probability space explicitly.