Probability interpretations
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The word probability has been used in a variety of ways since it was first coined in relation to games of chance.
There are two broad categories of probability interpretations: Frequentists talk about probabilities only when dealing with well defined random experiments. The relative frequency of occurrence of an experiment's outcome, when repeating the experiment, is a measure of the probability of that random event. Bayesians, on the other hand, assign probabilities to any statement whatsoever, even when no random process is involved, as a way to represent its subjective plausibility.
Epistemological controversy
The use of Bayesian probability raises the philosophical debate as to whether it can contribute valid justifications of belief.
Bayesians point to the work of Ramsey and de Finetti as proving that subjective beliefs must follow the laws of probability if they are to be coherent.
The use of Bayesian probability involves specifying a prior probability. This may be obtained from consideration of whether the required prior probability is greater or lesser than a reference probability associated with an urn model, or a thought experiment. The issue is that for a given problem, multiple thought experiments could apply, and choosing one is a matter of judgement. Thus different people may assign different prior probabilities. The "sunrise problem" illustrates the issue. A particular version of this issue is the reference class problem.
The frequentist view has its own problems. For example, the apparently reasonable question "what is the probability that Jupiter has a solid core?" (in the sense of "is it likely that Jupiter has a solid core?") is strictly meaningless from the frequentist point of view, since there is only one planet Jupiter and repeated experiments where multiple Jupiters are inspected cannot be performed, even in theory. (On the other hand, the question "what is the probability that a Jupiter-like planet has a solid core?" could be meaningful under the frequentist interpretation, since more than one such Jupiter-like planet could exist.)
Practical controversy
This difference in point of view has also many implications both for the methods by which statistics is practiced, and for the way in which conclusions are expressed. When comparing two hypotheses and using some information, frequency methods would typically result in the rejection or non-rejection of the original hypothesis at a particular significance level, and frequentists would all agree that the hypothesis should be rejected or not at that level of significance. Bayesian methods would suggest that one hypothesis was more probable than the other, but individual Bayesians might differ about which was the more probable and by how much, by virtue of having used different priors. Bayesians would argue that this is right and proper - if the issue is such that reasonable people can put forward different, but plausible, priors and the data are such that the likelihood does not swamp the prior, then the issue is not resolved unambiguously at the present stage of knowledge and Bayesian statistics highlights this fact. They would argue that any approach that purports to produce a single, definitive answer to the question at hand in these circumstances is obscuring the truth.
An alternative solution, is the eclectic view, which accepts both interpretations: depending on the situation, one selects one of the 2 interpretations for pragmatic, or principled, reasons.
Axiomatic probability
The mathematics of probability can be developed on an entirely axiomatic basis that is independent of any interpretation: see the articles on probability theory and probability axioms for a detailed treatment.
See also
External links
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