Frequency probability

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John Venn
John Venn

Frequency probability is the interpretation of probability that defines an event's probability as the limit of its relative frequency in a large number of trials. The problems and paradoxes of the classical interpretation 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 subscribe to this view of probability. They are usually called frequentists, and their position is called frequentism.

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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[edit] Definition

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 one of two possibilities can happen; it occurs or it does not occur. 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.

Thus, if nt is the total number of trials and nx is the number of trials where the event x occurred, the probability P(x) of the event occurring will be approximated by the relative frequency as follows:

P(x) \cong \frac{n_x}{n_t}

A further and more controversial claim is that in the "long run," as the number of trials approaches infinity, the relative frequency will converge exactly to the probability:[1]

P(x) = \lim_{n\rightarrow \infty}\frac{n_x}{n_t}

One objection to this is that we can only ever observe a finite sequence, and thus the extrapolation to the infinite involves unwarranted metaphysical assumptions. This conflicts with the standard claim that the frequency interpretation is somehow more "objective" than other theories of probability.

[edit] Scope

This is a highly technical and scientific definition and doesn't claim to capture all connotations of the concept 'probable' in colloquial speech of natural languages. Compare how the concept of force is used by physicists in a precise manner despite the fact that force is also a concept in many natural languages, used in religious texts for example. However, this seldom causes problems or confusion, as the context usually reveals if it's the scientific concept that is intended or not.

As William Feller noted:

There is no place in our system for speculations concerning the probability that the sun will rise tomorrow. Before speaking of it we should have to agree on an (idealized) model which would presumably run along the lines "out of infinitely many worlds one is selected at random..." Little imagination is required to construct such a model, but it appears both uninteresting and meaningless.

[edit] Etymology

According to the Oxford English Dictionary, the term 'frequentist' was first used by M. G. Kendall [1] 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. [2].

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 of 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]

[edit] Alternative views

Main article: Probability interpretations

[edit] 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" 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.

[edit] See also

[edit] External links

[edit] References

  1. ^ Richard von Mises, Probability, Statistics, and Truth, 1939. p.14

[edit] 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
  • William Feller, An introduction to Probability Theory and its Applications, 1957
  • 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