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The problems and paradoxes of the classical interpretation of probability motivated the development of the relative frequency concept of probability.
Most of the mathematics commonly used to make statistical estimates or tests are developed by statisticians who use this concept. They are usually called frequentists, and their position is called frequentism. A statistician who uses these methods of inference is therefore referred to as a frequentist statistician. According to the Oxford English Dictionary, the term 'frequentist' (one who believes that the probability of an event should be defined as the limit of its relative frequency in a large number of trials) was first used by M. G. Kendall * in 1949, who observed
"It might be thought that the differences between the frequentists and the non-frequentists (if I may call them such) are largely due to the differences of the domains which they purport to cover. "*.
and went on to say
I assert that this is not so ... The essential distinction between the frequentists and the non-frequentists is, I think, that the former, in an effort to avoid anything savouring or matters of opinion, seek to define probability in terms of the objective properties of a population, real or hypothetical, whereas the latter do not. emphasis in original
Frequentists talk about probabilities only when dealing with well-defined random experiments. The set of all possible outcomes of a random experiment is called the sample space of the experiment. An event is defined as a particular subset of the sample space that you want to consider. For any event only two things can happen; it occurs or it occurs not. The relative frequency of occurrence of an event, in a number of repetitions of the experiment, is a measure of the probability of that event.
This school is often associated with the names of Jerzy Neyman and Egon Pearson who described the logic of statistical hypothesis testing. Other influential figures of the frequentist school include John Venn, R.A. Fisher, and Richard von Mises.
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Bayesianism
The main alternative view, Bayesianism is more popular among decision theorists. Frequentists can't assign probabilities to things outside the scope of their definition. In particular, frequentists attribute probabilities only to events while Bayesians apply probabilities to arbitrary statements. For example, if one were to attribute a probability of 1/2 to the proposition that "there was life on Mars a billion years ago with probability 1/2" one would violate frequentist canons, because neither an experiment nor a sample space is defined here. However, such degree-of-belief assignments of probability to statements are the basis of Bayesian probability theory.
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See also
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Bibliography
P W Bridgman, The Logic of Modern Physics, 1927
Alonzo Church, The Concept of a Random Sequence, 1940
Harald Cramér, Mathematical Methods of Statistics, 1946
M. G. Kendall, On The Reconciliation Of Theories Of Probability, Biometrika 1949 36: 101-116; doi:10.1093/biomet/36.1-2.101
P Martin-Löf, On the Concept of a Random Sequence, 1966
Richard von Mises, Probability, Statistics, and Truth, 1939 (German original 1928)
Jerzy Neyman, First Course in Probability and Statistics, 1950
Hans Reichenbach, The Theory of Probability, 1949 (German original 1935)
Bertrand Russell, Human Knowledge, 1948
John Venn, The Logic of Chance, 1866
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