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 p ...
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 ...
– 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 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
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 occu ...
. The function form is either denoted
or a separate function symbol such as
is introduced with the meaning
.
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
, but the expectation of A ''conditional'' on B = 1 (i.e., conditional on the die roll being 2, 3, or 5) is
, and the expectation of A conditional on B = 0 (i.e., conditional on the die roll being 1, 4, or 6) is
. Likewise, the expectation of B conditional on A = 1 is
, and the expectation of B conditional on A = 0 is
.
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 occu ...
dates back at least to
Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summariz ...
, 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 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, operat ...
[ 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 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
:
where the sum is taken over all possible outcomes of .
Note that if , 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
:
where is the joint probability mass function 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:
:
where is the set .
Continuous random variables
Let and be continuous random variables
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
with joint density
's density
and conditional density of given the event
The conditional expectation of given is
:
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 , that is square integrable.
In its full generality, conditional expectation is developed without this assumption, see below under Conditional expectation with respect to a sub-σ-algebra. The theory is, however, considered more intuitive and admits important generalizations.
In the context of random variables, conditional expectation is also called regression
Regression or regressions may refer to:
Science
* Marine regression, coastal advance due to falling sea level, the opposite of marine transgression
* Regression (medicine), a characteristic of diseases to express lighter symptoms or less extent ( ...
.
In what follows let be a probability space, and in
with mean 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 number ...
.
The expectation 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 betwe ...
:
:.
The conditional expectation of is defined analogously, except instead of a single number
, the result will be a function . Let 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 valu ...
. The conditional expectation is a measurable function such that
:.
Note that unlike , the conditional expectation 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
:
Example 2: Consider the case where is the 2-dimensional random vector . Then clearly
:
but in terms of functions it can be expressed as or 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 ...
, this lack of uniqueness is called multicollinearity.
Conditional expectation is unique up to a set of measure zero in . The measure used is the pushforward measure induced by .
In the first example, the pushforward measure is a Dirac distribution 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 is non-trivial. It can be shown that
:
is a closed subspace of the Hilbert space .
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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear a ...
, the necessary and sufficient condition for
to be a minimizer is that for all in we have
:.
In words, this equation says that the residual is orthogonal to the space of all functions of .
This orthogonality condition, applied to the indicator functions ,
is used below to extend conditional expectation to the case that and are not necessarily in .
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 mathemat ...
and statistics due to the difficulties in analytically calculating it, and for interpolation.
The Hilbert subspace
:
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, 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 ...
when is required to be affine
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substi ...
, 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 denote this generalized conditional expectation/ projection. If does not contain the constant functions, the tower property
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 ...
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:
:
for coefficients described in Multivariate normal distribution#Conditional distributions.
Conditional expectation with respect to a sub-σ-algebra
Consider the following:
* 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 ...
.
* 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 p ...
on that probability space with finite expectation.
* is a sub- σ-algebra of .
Since is a sub -algebra of , the function is usually not -measurable, thus the existence of the integrals of the form , where and is the restriction of to , cannot be stated in general. However, the local averages can be recovered in with the help of the conditional expectation.
A conditional expectation of ''X'' given , denoted as , is any -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 ...
which satisfies:
:
for each .[
As noted in the discussion, this condition is equivalent to saying that the residual is orthogonal to the indicator functions :
:
]
Existence
The existence of can be established by noting that for is a finite measure on that is absolutely continuous with respect to . If is the natural injection
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iota ...
from to , then is the restriction of to and is the restriction of to . Furthermore, is absolutely continuous with respect to , because the condition
:
implies
:
Thus, we have
:
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 ...
, and
* A random variable .
The conditional expectation of given is defined by applying the above construction on the σ-algebra generated by :
:
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: 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, that is, for almost all \omega,
\kappa_\mathcal(\omega, -) is a probability measure.
The Law of the unconscious statistician 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
* Independe ...
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 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: 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: 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 ...
: 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 the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poin ...
, 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 number ...
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: \operatorname(X) = \operatorname(\operatorname(X \mid \mathcal)) + \operatorname(\operatorname(X \mid \mathcal)).
* Martingale convergence: 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 natu ...
of square-integrable 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 i ...
from X to the linear subspace 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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear a ...
.)
** 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 In mathematics, a contraction mapping, or contraction or contractor, on a metric space (''M'', ''d'') is a function ''f'' from ''M'' to itself, with the property that there is some real number 0 \leq k < 1 such that for all ''x'' and ...
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)
*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 ...
*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- ...
* Factorization lemma
*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 considere ...
*Non-commutative conditional expectation
In mathematics, non-commutative conditional expectation is a generalization of the notion of conditional expectation in classical probability. The space of essentially bounded measurable functions on a \sigma-finite measure space (X, \mu) is the c ...
Probability laws
* Law of total cumulance (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 ...
* Law of total probability
* Law of total variance
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