Equipossible
Encyclopedia
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. For example, when considering rolling a six-sided die, why do we typically view the possible outcomes as {1,2,3,4,5,6} rather than, say, {6, not 6}? The former set contains equally possible alternatives, while the latter does not because there are five times as many alternatives inherent in 'not 6' as in 6. This is true even if the die is biased so that 6 and 'not 6' are equally likely to occur.
By the Principle of Indifference
of Laplace, equipossible alternatives may be accorded equal probabilities if nothing more is known about the underlying probability distribution
.
It is a matter of contention whether the concept of equipossibility, also called equispecificity (from equispecific), can truly be distinguished from the concept of equiprobability.
In Bayesian inference
, a widely used definition of equipossibility is "a transformation group which leaves invariant one's state of knowledge". Equiprobability is then defined by normalizing the Haar measure
of this symmetry group. This 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 possibility theory
Possibility theory
Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. D. Dubois and H. Prade further...
that is a precursor to the notion of equiprobability 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...
. It is used to distinguish what can occur in a probability experiment. For example, when considering rolling a six-sided die, why do we typically view the possible outcomes as {1,2,3,4,5,6} rather than, say, {6, not 6}? The former set contains equally possible alternatives, while the latter does not because there are five times as many alternatives inherent in 'not 6' as in 6. This is true even if the die is biased so that 6 and 'not 6' are equally likely to occur.
By the 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....
of Laplace, equipossible alternatives may be accorded equal probabilities if nothing more is known about the underlying probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
.
It is a matter of contention whether the concept of equipossibility, also called equispecificity (from equispecific), can truly be distinguished from the concept of equiprobability.
In Bayesian inference
Bayesian inference
In statistics, Bayesian inference is a method of statistical inference. It is often used in science and engineering to determine model parameters, make predictions about unknown variables, and to perform model selection...
, a widely used definition of equipossibility is "a transformation group which leaves invariant one's state of knowledge". Equiprobability is then defined by normalizing 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....
of this symmetry group. This 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....
.
External links
- Book Chapter by Henry E. Kyburg Jr. on equipossibility, with the 6/not-6 example above
- Quotes on equipossibility in classical probability