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In a Bayesian framework, can a subjective probability be considered as a proportion of certainty, or does a proportion only make sense if we are, say, counting the number of "successes" out of the number of "trials", as in a frequentist framework? Thank you.

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To clarify: Bayesianism is the philosophical tenet that all uncertainties about the truth or falsity of propositions can be quantified as probabilities and that these follow the usual mathematical rules of probabilities. Frequentism is the belief that mathematical probability should be applied to the world only when probabilities can be interpreted as proportions of a population (construing that last word very broadly; e.g. the set of all neutrinos in the universe is a population, as is the set of all fish living in bodies of water on Earth). – Michael Hardy Aug 15 '13 at 1:14
Thus, for example, the frequentist may say that the probability of getting an ace when a die is thrown is $1/6$ because it happens $1/6$ of the time, and the set of all such trials is of course a population. But the frequentist will refuse to say that the probability that there was life on Mars a billion years ago is $1/2$, not because that number is too big or too small, but because it makes no sense to say that it happens is half of all cases. There is no such population of cases. – Michael Hardy Aug 15 '13 at 1:16
up vote 1 down vote accepted

It can. As defined and used, "subjective probability" quantifies in a relative but cardinal way "how much" certain we are, and this number is comparable to other such numbers and the magnitude of their difference is meaningful: thus if I say "I believe outcome A has 0.4 probability and outcome B has 0.2 probability", it is meaningful to also say "outcome A is twice as probable than outcome B". Perhaps this is the strongest (philosophical and anthropological) doubt related to subjective probability: do people really calculate and "experience" such cardinal probabilities, or do they just use "ordinal probability" (which is fully developed theoretically, but it only orders events in terms of their "subjective certainty", without quantifying their distance)... but I am getting carried away, this final part was not in the question.

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Very good point about human cognition and ordinal certainty. For my purposes, we want to use a true probability. – Brash Equilibrium Aug 15 '13 at 6:30

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