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A Frequentist interprets probability as an estimate of how frequent an event is giving that we can repeat the experiment many times. It is natural for them to try to maximize the expected utility because doing so maximizes the total utility across all the events that occur.

A Bayesian interprets probability as their degree of belief giving all available prior knowledge, so why would they want to maximize expectation?

Matthew Drury
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nalzok
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    There is a lot of confusion in this question. First, a frequentist does not interpret probability as an estimate of anything. Second, a Bayesian doesn't interpret probability as a "degree of belief", which is far too vague a concept to do math with. Third, the question of maximizing expected utility relates to von Neumann and Morgenstern's work : https://en.wikipedia.org/wiki/Von_Neumann–Morgenstern_utility_theorem; later on, a lot of work was done by Savage among others relating utility to subjectivist probability: https://en.wikipedia.org/wiki/Bayesian_probability. – jbowman Mar 14 '20 at 17:50
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    @jbowman Thanks for the pointers! Just for the record, the interpretation part is quoted from [this answer](https://stats.stackexchange.com/a/43418/108877). – nalzok Mar 14 '20 at 17:52
  • Fair enough, I overstated the case. See https://en.wikipedia.org/wiki/Probability_interpretations for some useful information; the "degree of belief" actually has to follow the laws of probability to be coherent, which is a lot more restrictive IMO than "degree of belief" all by itself. An objective Bayesian of course believes in objective probabilities, as does the frequentist. This presentation https://www.as.utexas.edu/jefferys/slides/berger.pdf may be of interest. – jbowman Mar 14 '20 at 18:08
  • As suggested in comments, your question is confusing, since it seems to make incorrect assumptions. Could you please make it more verbose? What exactly do you mean by those terms? – Tim Mar 14 '20 at 18:45
  • A Bayesian is likely to want to base decisions on minimising a loss function, for example in the simple case of being forced to give a point estimate for a parameter and a loss function proportional to the square of the error, a Bayesian will give the mean of the posterior distribution – Henry Mar 14 '20 at 21:17

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