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# Probability interpretations

Updated: Sep 4, 2018

The word probability is used in a variety of ways since it was first applied as a mathematical concept. Does probability measure the frequency to which something occurs or how strongly one believes it will occur, or both? the simple answer is both, which leads to much debate by professionals. However, riskmaticians apply both definitions in practice

Fuentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in a large number of trials. This interpretation supports the statistical needs of experimental scientists and pollsters; probabilities can be found (in principle) by a repeatable objective process (and are thus ideally devoid of opinion). It does not support all needs; gamblers typically require estimates of the odds without experiments. The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the classical interpretation. In the classical interpretation, probability was defined in terms of the principle of indifference, based on the natural symmetry of a problem, so, e.g. the probabilities of dice games arise from the natural symmetric 6-sidedness of the cube. This classical interpretation stumbled at any statistical problem that has no natural symmetry for reasoning.

Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses, i.e., the propositions whose truth or falsity is uncertain. In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference, a hypothesis is typically tested without being assigned a probability.

Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies some prior probability, which is then updated to a posterior probability in the light of new, relevant data (evidence). The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation. The term Bayesian derives from the 18th century mathematician and theologian Thomas Bayes, who provided the first mathematical treatment of a non-trivial problem of statistical data analysis using what is now known as Bayesian inference.:131 Mathematician Pierre-Simon Laplace pioneered and popularised what is now called Bayesian probability.

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