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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 ...
, the conditional expectation, conditional expected value, or conditional mean 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 po ...
is its
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
– the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur. If the random variable can take on only a finite number of values, the “conditions” are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete
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 ...
, the "conditions" are a
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
of this probability space. Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted E(X\mid Y) analogously to
conditional probability In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occur ...
. The function form is either denoted E(X\mid Y=y) or a separate function symbol such as f(y) is introduced with the meaning E(X\mid Y) = f(Y).


Examples


Example 1: Dice rolling

Consider the roll of a fair and let ''A'' = 1 if the number is even (i.e., 2, 4, or 6) and ''A'' = 0 otherwise. Furthermore, let ''B'' = 1 if the number is prime (i.e., 2, 3, or 5) and ''B'' = 0 otherwise. The unconditional expectation of A is E = (0+1+0+1+0+1)/6 = 1/2, but the expectation of A ''conditional'' on B = 1 (i.e., conditional on the die roll being 2, 3, or 5) is E \mid B=1(1+0+0)/3=1/3, and the expectation of A conditional on B = 0 (i.e., conditional on the die roll being 1, 4, or 6) is E \mid B=0(0+1+1)/3=2/3. Likewise, the expectation of B conditional on A = 1 is E \mid A=1 (1+0+0)/3=1/3, and the expectation of B conditional on A = 0 is E \mid A=0(0+1+1)/3=2/3.


Example 2: Rainfall data

Suppose we have daily rainfall data (mm of rain each day) collected by a weather station on every day of the ten–year (3652–day) period from January 1, 1990 to December 31, 1999. The unconditional expectation of rainfall for an unspecified day is the average of the rainfall amounts for those 3652 days. The ''conditional'' expectation of rainfall for an otherwise unspecified day known to be (conditional on being) in the month of March, is the average of daily rainfall over all 310 days of the ten–year period that falls in March. And the conditional expectation of rainfall conditional on days dated March 2 is the average of the rainfall amounts that occurred on the ten days with that specific date.


History

The related concept of
conditional probability In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occur ...
dates back at least to Laplace, who calculated conditional distributions. It was
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
who, in 1933, formalized it using the
Radon–Nikodym theorem In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurab ...
. In works of
Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
and Joseph L. Doob from 1953, conditional expectation was generalized to its modern definition using sub-σ-algebras.


Definitions


Conditioning on an event

If is an event in \mathcal with nonzero probability, and is a
discrete 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 po ...
, the conditional expectation of given is : \begin \operatorname (X \mid A) &= \sum_x x P(X = x \mid A) \\ & =\sum_x x \frac \end where the sum is taken over all possible outcomes of . Note that if P(A) = 0, the conditional expectation is undefined due to the division by zero.


Discrete random variables

If and are discrete random variables, the conditional expectation of given is : \begin \operatorname (X \mid Y=y) &= \sum_x x P(X = x \mid Y = y) \\ &= \sum_x x \frac \end where P(X = x, Y = y) is the
joint probability mass function Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
of and . The sum is taken over all possible outcomes of . Note that conditioning on a discrete random variable is the same as conditioning on the corresponding event: :\operatorname (X \mid Y=y) = \operatorname (X \mid A) where is the set \.


Continuous random variables

Let X and Y be continuous random variables with joint density f_(x,y), Y's density f_(y), and conditional density \textstyle f_(x, y) = \frac of X given the event Y=y. The conditional expectation of X given Y=y is : \begin \operatorname (X \mid Y=y) &= \int_^\infty x f_(x\mid y) \, \mathrmx \\ &= \frac\int_^\infty x f_(x,y) \, \mathrmx. \end When the denominator is zero, the expression is undefined. Note that conditioning on a continuous random variable is not the same as conditioning on the event \ as it was in the discrete case. For a discussion, see Conditioning on an event of probability zero. Not respecting this distinction can lead to contradictory conclusions as illustrated by the Borel-Kolmogorov paradox.


L2 random variables

All random variables in this section are assumed to be in L^2, that is
square integrable In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
. In its full generality, conditional expectation is developed without this assumption, see below under Conditional expectation with respect to a sub-σ-algebra. The L^2 theory is, however, considered more intuitive and admits important generalizations. In the context of L^2 random variables, conditional expectation is also called regression. In what follows let (\Omega, \mathcal, P) be a probability space, and X: \Omega \to \mathbb in L^2 with mean \mu_X and
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
\sigma_X^2. The expectation \mu_X minimizes the
mean squared error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
: : \min_ \operatorname\left((X - x)^2\right) = \operatorname\left((X - \mu_X)^2\right) = \sigma_X^2 . The conditional expectation of is defined analogously, except instead of a single number \mu_X, the result will be a function e_X(y). Let Y: \Omega \to \mathbb^n be a
random vector In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. ...
. The conditional expectation e_X: \mathbb^n \to \mathbb is a measurable function such that : \min_ \operatorname\left((X - g(Y))^2\right) = \operatorname\left((X - e_X(Y))^2\right) . Note that unlike \mu_X, the conditional expectation e_X is not generally unique: there may be multiple minimizers of the mean squared error.


Uniqueness

Example 1: Consider the case where is the constant random variable that's always 1. Then the mean squared error is minimized by any function of the form : e_X(y) = \begin \mu_X & \text y = 1 \\ \text & \text \end Example 2: Consider the case where is the 2-dimensional random vector (X, 2X). Then clearly :\operatorname(X \mid Y) = X but in terms of functions it can be expressed as e_X(y_1, y_2) = 3y_1-y_2 or e'_X(y_1, y_2) = y_2 - y_1 or infinitely many other ways. In the context of
linear regression In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is call ...
, this lack of uniqueness is called
multicollinearity In statistics, multicollinearity (also collinearity) is a phenomenon in which one predictor variable in a multiple regression model can be linearly predicted from the others with a substantial degree of accuracy. In this situation, the coeffic ...
. Conditional expectation is unique up to a set of measure zero in \mathbb^n. The measure used 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 measu ...
induced by . In the first example, the pushforward measure is a
Dirac distribution In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
at 1. In the second it is concentrated on the "diagonal" \, so that any set not intersecting it has measure 0.


Existence

The existence of a minimizer for \min_g \operatorname\left((X - g(Y))^2\right) is non-trivial. It can be shown that : M := \ = L^2(\Omega, \sigma(Y)) is a closed subspace of the Hilbert space L^2(\Omega). By the
Hilbert projection theorem In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector x in a Hilbert space H and every nonempty closed convex C \subseteq H, there exists a unique vector m \in C for which \, c - x\, ...
, the necessary and sufficient condition for e_X to be a minimizer is that for all f(Y) in we have : \langle X - e_X(Y), f(Y) \rangle = 0. In words, this equation says that the residual X - e_X(Y) is orthogonal to the space of all functions of . This orthogonality condition, applied to the indicator functions f(Y) = 1_, is used below to extend conditional expectation to the case that and are not necessarily in L^2.


Connections to regression

The conditional expectation is often approximated in
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
and
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
due to the difficulties in analytically calculating it, and for interpolation. The Hilbert subspace : M = \ defined above is replaced with subsets thereof by restricting the functional form of , rather than allowing any measureable function. Examples of this are decision tree regression when is required to be a
simple function In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, ...
,
linear regression In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is call ...
when is required to be
affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...
, etc. These generalizations of conditional expectation come at the cost of many of its properties no longer holding. For example, let be the space of all linear functions of and let \mathcal_ denote this generalized conditional expectation/L^2 projection. If M does not contain the constant functions, the tower property \operatorname(\mathcal_M(X)) = \operatorname(X) will not hold. An important special case is when and are jointly normally distributed. In this case it can be shown that the conditional expectation is equivalent to linear regression: : e_X(Y) = \alpha_0 + \sum_i \alpha_i Y_i for coefficients \_ described in Multivariate normal distribution#Conditional distributions.


Conditional expectation with respect to a sub-σ-algebra

Consider the following: * (\Omega, \mathcal, P) is 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 ...
. * X\colon\Omega \to \mathbb^n is 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 po ...
on that probability space with finite expectation. * \mathcal \subseteq \mathcal is a sub- σ-algebra of \mathcal. Since \mathcal is a sub \sigma-algebra of \mathcal, the function X\colon\Omega \to \mathbb^n is usually not \mathcal-measurable, thus the existence of the integrals of the form \int_H X \,dP, _\mathcal, where H\in\mathcal and P, _\mathcal is the restriction of P to \mathcal, cannot be stated in general. However, the local averages \int_H X\,dP can be recovered in (\Omega, \mathcal, P, _\mathcal) with the help of the conditional expectation. A conditional expectation of ''X'' given \mathcal, denoted as \operatorname(X\mid\mathcal), is any \mathcal-
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 in di ...
\Omega \to \mathbb^n which satisfies: :\int_H \operatorname(X \mid \mathcal) \,\mathrmP = \int_H X \,\mathrmP for each H \in \mathcal. As noted in the L^2 discussion, this condition is equivalent to saying that the residual X - \operatorname(X \mid \mathcal) is orthogonal to the indicator functions 1_H: : \langle X - \operatorname(X \mid \mathcal), 1_H \rangle = 0


Existence

The existence of \operatorname(X\mid\mathcal) can be established by noting that \mu^X\colon F \mapsto \int_F X \, \mathrmP for F \in \mathcal is a finite measure on (\Omega, \mathcal) that is
absolutely continuous In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central oper ...
with respect to P. If h is the natural injection from \mathcal to \mathcal, then \mu^X \circ h = \mu^X, _\mathcal is the restriction of \mu^X to \mathcal and P \circ h = P, _\mathcal is the restriction of P to \mathcal. Furthermore, \mu^X \circ h is absolutely continuous with respect to P \circ h, because the condition :P \circ h (H) = 0 \iff P(h(H)) = 0 implies :\mu^X(h(H)) = 0 \iff \mu^X \circ h(H) = 0. Thus, we have :\operatorname(X\mid\mathcal) = \frac = \frac, where the derivatives are Radon–Nikodym derivatives of measures.


Conditional expectation with respect to a random variable

Consider, in addition to the above, * A
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...
(U, \Sigma), and * A random variable Y\colon\Omega \to U. The conditional expectation of given is defined by applying the above construction on the σ-algebra generated by : :\operatorname Y:= \operatorname \sigma(Y)/math>. By the Doob-Dynkin lemma, there exists a function e_X \colon U \to \mathbb^n such that :\operatorname Y= e_X(Y).


Discussion

* This is not a constructive definition; we are merely given the required property that a conditional expectation must satisfy. ** The definition of \operatorname(X \mid \mathcal) may resemble that of \operatorname(X \mid H) for an event H but these are very different objects. The former is a \mathcal-measurable function \Omega \to \mathbb^n, while the latter is an element of \mathbb^n and \operatorname(X \mid H)\ P(H)= \int_H X \,\mathrmP= \int_H \operatorname (X\mid\mathcal)\,\mathrmP for H\in\mathcal. ** Uniqueness can be shown to be
almost sure In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
: that is, versions of the same conditional expectation will only differ on a set of probability zero. * The σ-algebra \mathcal controls the "granularity" of the conditioning. A conditional expectation E(X\mid\mathcal) over a finer (larger) σ-algebra \mathcal retains information about the probabilities of a larger class of events. A conditional expectation over a coarser (smaller) σ-algebra averages over more events.


Conditional probability

For a Borel subset in \mathcal(\mathbb^n), one can consider the collection of random variables : \kappa_\mathcal(\omega, B) := \operatorname(1_, \mathcal)(\omega) . It can be shown that they form a
Markov kernel In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite ...
, that is, for almost all \omega, \kappa_\mathcal(\omega, -) is a probability measure. The
Law of the unconscious statistician In probability theory and statistics, the law of the unconscious statistician, or LOTUS, is a theorem used to calculate the expected value of a function ''g''(''X'') of a random variable ''X'' when one knows the probability distribution of ''X'' but ...
is then : \operatorname \mathcal= \int f(x) \kappa_\mathcal(-, \mathrmx) . This shows that conditional expectations are, like their unconditional counterparts, integrations, against a conditional measure.


Basic properties

All the following formulas are to be understood in an almost sure sense. The σ-algebra \mathcal could be replaced by a random variable Z, i.e. \mathcal=\sigma(Z). * Pulling out independent factors: ** If X is
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...
of \mathcal, then E(X\mid\mathcal) = E(X). Let B \in \mathcal. Then X is independent of 1_B, so we get that :\int_B X\,dP = E(X1_B) = E(X)E(1_B) = E(X)P(B) = \int_B E(X)\,dP. Thus the definition of conditional expectation is satisfied by the constant random variable E(X), as desired. ** If X is independent of \sigma(Y, \mathcal), then E(XY\mid \mathcal) = E(X) \, E(Y\mid\mathcal). Note that this is not necessarily the case if X is only independent of \mathcal and of Y. ** If X,Y are independent, \mathcal,\mathcal are independent, X is independent of \mathcal and Y is independent of \mathcal, then E(E(XY\mid\mathcal)\mid\mathcal) = E(X) E(Y) = E(E(XY\mid\mathcal)\mid\mathcal). * Stability: ** If X is \mathcal-measurable, then E(X\mid\mathcal) = X. ** In particular, for sub-σ-algebras \mathcal_1\subset\mathcal_2 \subset\mathcal we have E(E(X\mid\mathcal_2)\mid\mathcal_1) = E(X\mid\mathcal_1). ** If ''Z'' is a random variable, then \operatorname(f(Z) \mid Z)=f(Z). In its simplest form, this says \operatorname(Z \mid Z)=Z. * Pulling out known factors: ** If X is \mathcal-measurable, then E(XY\mid\mathcal) = X \, E(Y\mid\mathcal). ** If ''Z'' is a random variable, then \operatorname(f(Z) Y \mid Z)=f(Z)\operatorname(Y \mid Z). *
Law of total expectation The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem, among other names, states that if X is a random variable whose expected v ...
: E(E(X \mid \mathcal)) = E(X). * Tower property: ** For sub-σ-algebras \mathcal_1\subset\mathcal_2 \subset\mathcal we have E(E(X\mid\mathcal_2)\mid\mathcal_1) = E(X\mid\mathcal_1). *** A special case \mathcal_1=\ recovers the Law of total expectation: E(E(X\mid\mathcal_1) ) = E(X ). *** A special case is when ''Z'' is a \mathcal-measurable random variable. Then \sigma(Z) \subset \mathcal and thus E(E(X \mid \mathcal) \mid Z) = E(X \mid Z). ***
Doob martingale In the probability theory, mathematical theory of probability, a Doob martingale (named after Joseph L. Doob, also known as a Levy martingale) is a stochastic process that approximates a given random variable and has the Martingale (probability t ...
property: the above with Z = E(X \mid \mathcal) (which is \mathcal-measurable), and using also \operatorname(Z \mid Z)=Z, gives E(X \mid E(X \mid \mathcal)) = E(X \mid \mathcal). ** For random variables X,Y we have E(E(X\mid Y)\mid f(Y)) = E(X\mid f(Y)). ** For random variables X,Y,Z we have E(E(X\mid Y,Z)\mid Y) = E(X\mid Y). * Linearity: we have E(X_1 + X_2 \mid \mathcal) = E(X_1 \mid \mathcal) + E(X_2 \mid \mathcal) and E(a X \mid \mathcal) = a\,E(X \mid \mathcal) for a\in\R. * Positivity: If X \ge 0 then E(X \mid \mathcal) \ge 0. * Monotonicity: If X_1 \le X_2 then E(X_1 \mid \mathcal) \le E(X_2 \mid \mathcal). * Monotone convergence: If 0\leq X_n \uparrow X then E(X_n \mid \mathcal) \uparrow E(X \mid \mathcal). *
Dominated convergence In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
: If X_n \to X and , X_n, \le Y with Y \in L^1, then E(X_n \mid \mathcal) \to E(X \mid \mathcal). *
Fatou's lemma In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's lemma ...
: If \textstyle E(\inf_n X_n \mid \mathcal) > -\infty then \textstyle E(\liminf_ X_n \mid \mathcal) \le \liminf_ E(X_n \mid \mathcal). *
Jensen's inequality In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier pr ...
: If f \colon \mathbb \rightarrow \mathbb is a
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
, then f(E(X\mid \mathcal)) \le E(f(X)\mid\mathcal). *
Conditional variance In probability theory and statistics, a conditional variance is the variance of a random variable given the value(s) of one or more other variables. Particularly in econometrics, the conditional variance is also known as the scedastic function or ...
: Using the conditional expectation we can define, by analogy with the definition of the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
as the mean square deviation from the average, the conditional variance ** Definition: \operatorname(X \mid \mathcal) = \operatorname\bigl( (X - \operatorname(X \mid \mathcal))^2 \mid \mathcal \bigr) **Algebraic formula for the variance: \operatorname(X \mid \mathcal) = \operatorname(X^2 \mid \mathcal) - \bigl(\operatorname(X \mid \mathcal)\bigr)^2 **
Law of total variance In probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if X and Y are random variables on the same probability space, and ...
: \operatorname(X) = \operatorname(\operatorname(X \mid \mathcal)) + \operatorname(\operatorname(X \mid \mathcal)). *
Martingale convergence In mathematicsspecifically, in the theory of stochastic processesDoob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob. Informally, the martingal ...
: For a random variable X, that has finite expectation, we have E(X\mid\mathcal_n) \to E(X\mid\mathcal), if either \mathcal_1 \subset \mathcal_2 \subset \dotsb is an increasing series of sub-σ-algebras and \textstyle \mathcal = \sigma(\bigcup_^\infty \mathcal_n) or if \mathcal_1 \supset \mathcal_2 \supset \dotsb is a decreasing series of sub-σ-algebras and \textstyle \mathcal = \bigcap_^\infty \mathcal_n. * Conditional expectation as L^2-projection: If X,Y are in the
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
of
square-integrable In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real number, real- or complex number, complex-valued measurable function for which the integral of the s ...
real random variables (real random variables with finite second moment) then ** for \mathcal-measurable Y, we have E(Y(X - E(X\mid\mathcal))) = 0, i.e. the conditional expectation E(X\mid\mathcal) is in the sense of the ''L''2(''P'') scalar product the
orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
from X to the
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
of \mathcal-measurable functions. (This allows to define and prove the existence of the conditional expectation based on the
Hilbert projection theorem In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector x in a Hilbert space H and every nonempty closed convex C \subseteq H, there exists a unique vector m \in C for which \, c - x\, ...
.) ** the mapping X \mapsto \operatorname(X\mid\mathcal) is
self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a sta ...
: \operatorname E(X \operatorname E(Y \mid \mathcal)) = \operatorname E\left(\operatorname E(X \mid \mathcal) \operatorname E(Y \mid \mathcal)\right) = \operatorname E(\operatorname E(X \mid \mathcal) Y) * Conditioning is a contractive projection of ''L''p spaces L^p(\Omega, \mathcal, P) \rightarrow L^p(\Omega, \mathcal, P). I.e., \operatorname\big(, \operatorname(X \mid\mathcal), ^p \big) \le \operatorname\big(, X, ^p\big) for any ''p'' ≥ 1. * Doob's conditional independence property: If X,Y are
conditionally independent In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of conditional probabil ...
given Z, then P(X \in B\mid Y,Z) = P(X \in B\mid Z) (equivalently, E(1_\mid Y,Z) = E(1_ \mid Z)).


See also

*
Conditioning (probability) Beliefs depend on the available information. This idea is formalized in probability theory by conditioning. Conditional probabilities, conditional expectations, and conditional probability distributions are treated on three levels: discrete prob ...
*
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 relate ...
*
Doob–Dynkin lemma In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin (also known as the factorization lemma), characterizes the situation when one random variable is a function of another by the inclusion of the \sigma-a ...
*
Factorization lemma In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kin ...
*
Joint probability distribution Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
* Non-commutative conditional expectation


Probability laws

*
Law of total cumulance In probability theory and mathematical statistics, the law of total cumulance is a generalization to cumulants of the law of total probability, the law of total expectation, and the law of total variance. It has applications in the analysis of t ...
(generalizes the other three) *
Law of total expectation The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem, among other names, states that if X is a random variable whose expected v ...
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Law of total probability In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct even ...
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Law of total variance In probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if X and Y are random variables on the same probability space, and ...


Notes


References

*
William Feller William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian-American mathematician specializing in probability theory. Early life and education Feller was born in Zagreb to Ida Oemichen-Perc, a Croa ...
, ''An Introduction to Probability Theory and its Applications'', vol 1, 1950, page 223 * Paul A. Meyer, ''Probability and Potentials'', Blaisdell Publishing Co., 1966, page 28 * , pages 67–69


External links

* {{DEFAULTSORT:Conditional Expectation Conditional probability Statistical theory