Frequentist probability


Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials. Probabilities can be found by a repeatable objective process. This interpretation supports the statistical needs of many experimental scientists and pollsters. It does not support all needs, however; 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.

Definition

In the frequentist interpretation, probabilities are discussed 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 to be considered. For any given event, only one of two possibilities may hold: it occurs or it does not. The relative frequency of occurrence of an event, observed in a number of repetitions of the experiment, is a measure of the probability of that event. This is the core conception of probability in the frequentist interpretation.
A claim of the frequentist approach is that, as the number of trials increases, the change in the relative frequency will diminish. Hence, one can view a probability as the limiting value of the corresponding relative frequencies.

Scope

The frequentist interpretation is a philosophical approach to the definition and use of probabilities; it is one of several such approaches. It does not claim to capture all connotations of the concept 'probable' in colloquial speech of natural languages.
As an interpretation, it is not in conflict with the mathematical axiomatization of probability theory; rather, it provides guidance for how to apply mathematical probability theory to real-world situations. It offers distinct guidance in the construction and design of practical experiments, especially when contrasted with the Bayesian interpretation. As to whether this guidance is useful, or is apt to mis-interpretation, has been a source of controversy. Particularly when the frequency interpretation of probability is mistakenly assumed to be the only possible basis for frequentist inference. So, for example, a list of mis-interpretations of the meaning of p-values accompanies the article on p-values; controversies are detailed in the article on statistical hypothesis testing. The Jeffreys–Lindley paradox shows how different interpretations, applied to the same data set, can lead to different conclusions about the 'statistical significance' of a result.
As William Feller noted:
Feller's comment was criticism of Laplace, who published a solution to the sunrise problem using an alternative probability interpretation. Despite Laplace's explicit and immediate disclaimer in the source, based on expertise in astronomy as well as probability, two centuries of criticism have followed.

History

The frequentist view may have been foreshadowed by Aristotle, in Rhetoric, when he wrote:
Poisson clearly distinguished between objective and subjective probabilities in 1837. Soon thereafter a flurry of nearly simultaneous publications by Mill, Ellis, Cournot and Fries introduced the frequentist view. Venn provided a thorough exposition two decades later. These were further supported by the publications of Boole and Bertrand. By the end of the 19th century the frequentist interpretation was well established and perhaps dominant in the sciences. The following generation established the tools of classical inferential statistics all based on frequentist probability.
Alternatively, Jacob Bernoulli understood the concept of frequentist probability and published a critical proof posthumously in 1713. He is also credited with some appreciation for subjective probability. Gauss and Laplace used frequentist probability in derivations of the least squares method a century later, a generation before Poisson. Laplace considered the probabilities of testimonies, tables of mortality, judgments of tribunals, etc. which are unlikely candidates for classical probability. In this view, Poisson's contribution was his sharp criticism of the alternative "inverse" probability interpretation. Any criticism by Gauss and Laplace was muted and implicit.
Major contributors to "classical" statistics in the early 20th century included Fisher, Neyman and Pearson. Fisher contributed to most of statistics and made significance testing the core of experimental science; Neyman formulated confidence intervals and contributed heavily to sampling theory; Neyman and Pearson paired in the creation of hypothesis testing. All valued objectivity, so the best interpretation of probability available to them was frequentist. All were suspicious of "inverse probability" with prior probabilities chosen by the using the principle of indifference. Fisher said, "...the theory of inverse probability is founded upon an error, and must be wholly rejected.". While Neyman was a pure frequentist, Fisher's views of probability were unique; Both had nuanced view of probability. von Mises offered a combination of mathematical and philosophical support for frequentism in the era.

Etymology

According to the Oxford English Dictionary, the term 'frequentist' was first used by M. G. Kendall in 1949, to contrast with Bayesians, whom he called "non-frequentists". He observed
"The Frequency Theory of Probability" was used a generation earlier as a chapter title in Keynes.
The historical sequence: probability concepts were introduced and much of probability mathematics derived, classical statistical inference methods were developed, the mathematical foundations of probability were solidified and current terminology was introduced. The primary historical sources in probability and statistics did not use the current terminology of classical, subjective and frequentist probability.

Alternative views

is a branch of mathematics. While its roots reach centuries into the past, it reached maturity with the axioms of Andrey Kolmogorov in 1933. The theory focuses on the valid operations on probability values rather than on the initial assignment of values; the mathematics is largely independent of any interpretation of probability.
Applications and interpretations of probability are considered by philosophy, the sciences and statistics. All are interested in the extraction of knowledge from observations—inductive reasoning. There are a variety of competing interpretations; All have problems. The frequentist interpretation does resolve difficulties with the classical interpretation, such as any problem where the natural symmetry of outcomes is not known. It does not address other issues, such as the dutch book.