Equiprobable
Encyclopedia
Equiprobability is a philosophical
concept in probability theory
that allows one to assign equal probabilities to outcomes when they are judged to be equipossible
or to be "equally likely" in some sense. The best-known formulation of the rule is Laplace's principle of indifference
(or principle of insufficient reason), which states that, when "we have no other information than" that exactly N mutually exclusive
events can occur, we are justified in assigning each the probability 1/N. This subjective assignment of probabilities is especially justified for situations such as rolling dice and lotteries
since these experiments carry a symmetry
structure, and one's state of knowledge must clearly be invariant under this symmetry.
A similar argument could lead to the seemingly absurd conclusion that the sun is as likely to rise as to not rise tomorrow morning. However, the conclusion that the sun is equally likely to rise as it is to not rise is only absurd when additional information is known, such as the laws of gravity and the sun's past history. Similar applications of the concept are effectively instances of circular reasoning
, with "equally likely" events being assigned equal probabilities, which means in turn that they are equally likely. Despite this, the notion remains useful in probabilistic and statistical
modeling
.
In Bayesian probability
, one needs to establish prior probabilities for the various hypotheses before applying Bayes' theorem
. One procedure is to assume that these prior probabilities have some symmetry which is typical of the experiment, and then assign a prior which is proportional to the Haar measure
for the symmetry group: this generalization of equiprobability is known as the principle of transformation groups
.
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
concept in probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
that allows one to assign equal probabilities to outcomes when they are judged to be equipossible
Equipossible
Equipossibility is a philosophical concept in possibility theory that is a precursor to the notion of equiprobability in probability theory. It is used to distinguish what can occur in a probability experiment...
or to be "equally likely" in some sense. The best-known formulation of the rule is Laplace's principle of indifference
Principle of indifference
The principle of indifference is a rule for assigning epistemic probabilities.Suppose that there are n > 1 mutually exclusive and collectively exhaustive possibilities....
(or principle of insufficient reason), which states that, when "we have no other information than" that exactly N mutually exclusive
Mutually exclusive
In layman's terms, two events are mutually exclusive if they cannot occur at the same time. An example is tossing a coin once, which can result in either heads or tails, but not both....
events can occur, we are justified in assigning each the probability 1/N. This subjective assignment of probabilities is especially justified for situations such as rolling dice and lotteries
Lottery
A lottery is a form of gambling which involves the drawing of lots for a prize.Lottery is outlawed by some governments, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of regulation of lottery by governments...
since these experiments carry a symmetry
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
structure, and one's state of knowledge must clearly be invariant under this symmetry.
A similar argument could lead to the seemingly absurd conclusion that the sun is as likely to rise as to not rise tomorrow morning. However, the conclusion that the sun is equally likely to rise as it is to not rise is only absurd when additional information is known, such as the laws of gravity and the sun's past history. Similar applications of the concept are effectively instances of circular reasoning
Circular reasoning
Circular reasoning, or in other words, paradoxical thinking, is a type of formal logical fallacy in which the proposition to be proved is assumed implicitly or explicitly in one of the premises. For example:"Only an untrustworthy person would run for office...
, with "equally likely" events being assigned equal probabilities, which means in turn that they are equally likely. Despite this, the notion remains useful in probabilistic and statistical
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
modeling
Scientific modelling
Scientific modelling is the process of generating abstract, conceptual, graphical and/or mathematical models. Science offers a growing collection of methods, techniques and theory about all kinds of specialized scientific modelling...
.
In Bayesian probability
Bayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...
, one needs to establish prior probabilities for the various hypotheses before applying Bayes' theorem
Bayes' theorem
In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....
. One procedure is to assume that these prior probabilities have some symmetry which is typical of the experiment, and then assign a prior which is proportional to the Haar measure
Haar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
for the symmetry group: this generalization of equiprobability is known as the principle of transformation groups
Principle of transformation groups
The principle of transformation groups is a rule for assigning epistemic probabilities in a statistical inference problem. It was first suggested by Edwin T Jaynes and can be seen as a generalisation of the principle of indifference....
.
See also
- Principle of indifferencePrinciple of indifferenceThe principle of indifference is a rule for assigning epistemic probabilities.Suppose that there are n > 1 mutually exclusive and collectively exhaustive possibilities....
- Laplacian smoothingLaplacian smoothingLaplacian smoothing is an algorithm to smooth a polygonal mesh. For each vertex in a mesh, a new position is chosen based on local information and the vertex is moved there...
- Uninformative prior
- A priori probabilityA priori probabilityThe term a priori probability is used in distinguishing the ways in which values for probabilities can be obtained. In particular, an "a priori probability" is derived purely by deductive reasoning...
- AequiprobabilismAequiprobabilismAequiprobabilism, also spelled Æquiprobabilism, is one of several doctrines in moral theology opposed to Probabilism.-Teaching:This system can be expressed in the three following propositions:...
External links
- Quotes on equiprobability in classical probability