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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
. The resulting conditional probability distribution is a parametrized family of probability measures called a Markov kernel.


Definition


Conditional probability distribution

Consider two random variables X, Y : \Omega \to \mathbb. The ''conditional probability distribution'' of ''Y'' given ''X'' is a two variable function \kappa_: \mathbb \times \mathcal(\mathbb) \to ,1/math> If the random variable ''X'' is discrete :\kappa_(x, A) = P(Y \in A , X = x) = \begin \frac & \text P(X = x) > 0 \\ \text & \text. \end If the random variables ''X'', ''Y'' are continuous with density f_(x,y). :\kappa_(x, A) = \begin \frac & \text \int_\mathbb f_(x, y) \mathrmy > 0 \\ \text & \text. \end A more general definition can be given in terms of
conditional expectation In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given ...
. Consider a function e_ : \mathbb \to ,1/math> satisfying :e_(X(\omega)) = \mathbb X\omega) for almost all \omega. Then the conditional probability distribution is given by :\kappa_(x, A) = e_(x). As with conditional expectation, this can be further generalized to conditioning on a sigma algebra \mathcal. In that case the conditional distribution is a function \Omega \times \mathcal(\mathbb) \to
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math>: : \kappa_(\omega, A) = \mathbb \mathcal/math>


Regularity

For working with \kappa_, it is important that it be ''regular'', that is: # For almost all ''x'', A \mapsto \kappa_(x, A) is a probability measure # For all ''A'', x \mapsto \kappa_(x, A) is a measurable function In other words \kappa_ is a Markov kernel. The first condition holds trivially, but the proof of the second is more involved. It can be shown that if ''Y'' is a random element \Omega \to S in a Radon space ''S'', there exists a \kappa_ that satisfies the measurability condition. It is possible to construct more general spaces where a regular conditional probability distribution does not exist.


Relation to conditional expectation

For discrete and continuous random variables, the
conditional expectation In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given ...
can be expressed as : \begin \mathbb X=x&= \sum_y y \, P(Y=y, X=x)\\ \mathbb X=x&= \int y \, f_(x, y) \mathrmy \end where f_(x, y) is the conditional density of given . This result can be extended to measure theoretical conditional expectation using the regular conditional probability distribution: :\mathbb X\omega) = \int y \, \kappa_(\omega, \mathrmy) .


Formal definition

Let (\Omega, \mathcal F, P) be a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
, and let T:\Omega\rightarrow E be a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
, defined as a Borel-
measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is i ...
from \Omega to its
state space A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the t ...
(E, \mathcal E). One should think of T as a way to "disintegrate" the sample space \Omega into \_. Using the
disintegration theorem In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related ...
from the measure theory, it allows us to "disintegrate" the measure P into a collection of measures, one for each x \in E. Formally, a regular conditional probability is defined as a function \nu:E \times\mathcal F \rightarrow ,1 called a "transition probability", where: * For every x \in E, \nu(x, \cdot) is a probability measure on \mathcal F. Thus we provide one measure for each x \in E. * For all A\in\mathcal F, \nu(\cdot, A) (a mapping E \to ,1/math>) is \mathcal E-measurable, and * For all A\in\mathcal F and all B\in\mathcal ED. Leao Jr. et al. ''Regular conditional probability, disintegration of probability and Radon spaces.'' Proyecciones. Vol. 23, No. 1, pp. 15–29, May 2004, Universidad Católica del Norte, Antofagasta, Chil
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/ref> :P\big(A\cap T^(B)\big) = \int_B \nu(x,A) \,P\big(T^(d x)\big). where P\circ T^ is the
pushforward measure In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function. Definition Given mea ...
T_*P of the distribution of the random element T, x\in\mathrm\,T, i.e. the
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
of the T_* P. Specifically, if we take B=E, then A \cap T^(E) = A, and so :P(A) = \int_E \nu(x,A) \,P\big(T^(d x)\big), where \nu(x, A) can be denoted, using more familiar terms P(A\ , \ T=x).


Alternate definition

Consider a Radon space \Omega (that is a probability measure defined on a Radon space endowed with the Borel sigma-algebra) and a real-valued random variable ''T''. As discussed above, in this case there exists a regular conditional probability with respect to ''T''. Moreover, we can alternatively define the regular conditional probability for an event ''A'' given a particular value ''t'' of the random variable ''T'' in the following manner: : P (A, T=t) = \lim_ \frac , where the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
is taken over the net of
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (Y ...
neighborhoods A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural ar ...
''U'' of ''t'' as they become smaller with respect to set inclusion. This limit is defined if and only if the probability space is
Radon Radon is a chemical element with the symbol Rn and atomic number 86. It is a radioactive, colourless, odourless, tasteless noble gas. It occurs naturally in minute quantities as an intermediate step in the normal radioactive decay chains through ...
, and only in the support of ''T'', as described in the article. This is the restriction of the transition probability to the support of ''T''. To describe this limiting process rigorously: For every \epsilon > 0, there exists an open neighborhood ''U'' of the event , such that for every open ''V'' with \ \subset V \subset U, :\left, \frac {P(V)}-L\ < \epsilon, where L = P (A, T=t) is the limit.


See also

* Conditioning (probability) *
Disintegration theorem In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related ...
* Adherent point *
Limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...


References


External links


Regular Conditional Probability
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PlanetMath
Conditional probability Measure theory