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 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, 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. The function form is either denoted or a separate function symbol such as is introduced with the meaning .

ExamplesEdit

Example 1: Dice rollingEdit

Consider the roll of a fair die 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.

1 2 3 4 5 6
A 0 1 0 1 0 1
B 0 1 1 0 1 0

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 dataEdit

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.

HistoryEdit

The related concept of conditional probability dates back at least to Laplace, who calculated conditional distributions. It was Andrey Kolmogorov who, in 1933, formalized it using the Radon–Nikodym theorem.[1] In works of Paul Halmos[2] and Joseph L. Doob[3] from 1953, conditional expectation was generalized to its modern definition using sub-σ-algebras.[4]

DefinitionsEdit

Conditioning on an eventEdit

If A is an event in   with nonzero probability, and X is a discrete random variable, the conditional expectation of X given A is

 

where the sum is taken over all possible outcomes of X.

Note that if  , the conditional expectation is undefined due to the division by zero.

Discrete random variablesEdit

If X and Y are discrete random variables, the conditional expectation of X given Y is

 

where   is the joint probability mass function of X and Y. The sum is taken over all possible outcomes of X.

Note that conditioning on a discrete random variable is the same as conditioning on the corresponding event:

 

where A is the set  .

Continuous random variablesEdit

If X and Y are continuous random variables with joint density  , the conditional expectation of X given Y 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 variablesEdit

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[5] and admits important generalizations. In the context of   random variables, conditional expectation is also called regression.

In what follows let   be a probability space, and   in   with mean   and variance  . The expectation   minimizes the mean squared error:

 .

The conditional expectation of X is defined analogously, except instead of a single number  , the result will be a function  . Let   be a random vector also in  . 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.

UniquenessEdit

Example 1: Consider the case where Y 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 Y 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, 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 Y.

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.

ExistenceEdit

The existence of a minimizer for   is non-trivial. It can be shown that

 

is a closed subspace of the Hilbert space  .[6] By the Hilbert projection theorem, the necessary and sufficient condition for   to be a minimizer is that for all   in M we have

 .

In words, this equation says that the residual   is orthogonal to the space M of all functions of Y. This orthogonality condition, applied to the indicator functions  , is used below to extend conditional expectation to the case that X and Y are not necessarily in  .

Connections to regressionEdit

The conditional expectation is often approximated in applied mathematics and statistics due to the difficulties in analytically calculating it, and for interpolation.[7]

The Hilbert subspace

 

defined above is replaced with subsets thereof by restricting the functional form of g, rather than allowing any measureable function. Examples of this are decision tree regression when g is required to be a simple function, linear regression when g is required to be affine, etc.

These generalizations of conditional expectation come at the cost of many of its properties no longer holding. For example, let M be the space of all linear functions of Y and let   denote this generalized conditional expectation/  projection. If   does not contain the constant functions, the tower property   will not hold.

An important special case is when X and Y 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-algebraEdit

 
Conditional expectation with respect to a σ-algebra: in this example the probability space   is the [0,1] interval with the Lebesgue measure. We define the following σ-algebras:  ;   is the σ-algebra generated by the intervals with end-points 0, ¼, ½, ¾, 1; and   is the σ-algebra generated by the intervals with end-points 0, ½, 1. Here the conditional expectation is effectively the average over the minimal sets of the σ-algebra.

Consider the following:

  •   is a probability space.
  •   is a random variable 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   which satisfies:

 

for each  .[8]

As noted in the   discussion, this is condition equivalent to saying that the residual   be orthogonal to the indicator functions  :

 

ExistenceEdit

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 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 variableEdit

Consider, in addition to the above,

  • A measurable space  , and
  • A random variable  .

The conditional expectation of X given Y is defined by applying the above construction on the σ-algebra generated by Y:

 .

By the Doob-Dynkin lemma, there exists a function   such that

 .

DiscussionEdit

  • This is not a constructive definition; we are merely given the required property that a conditional expectation must satisfy.
    • The definition of   may resemble that of   for an event   but these are very different objects. The former is a  -measurable function  , while the latter is an element of   and   for  .
    • 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   controls the "granularity" of the conditioning. A conditional expectation   over a finer (larger) σ-algebra   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 probabilityEdit

For a Borel subset B in  , one can consider the collection of random variables

 .

It can be shown that they form a Markov kernel, that is, for almost all  ,   is a probability measure.[9]

The Law of the unconscious statistician is then

 .

This shows that conditional expectations are, like their unconditional counterparts, integrations, against a conditional measure.

Basic propertiesEdit

All the following formulas are to be understood in an almost sure sense. The σ-algebra   could be replaced by a random variable  .

  • Pulling out independent factors:
    • If   is independent of  , then  .
Proof

Let  . Then   is independent of  , so we get that

 

Thus the definition of conditional expectation is satisfied by the constant random variable  , as desired.

    • If   is independent of  , then  . Note that this is not necessarily the case if   is only independent of   and of  .
    • If   are independent,   are independent,   is independent of   and   is independent of  , then  .
  • Stability:
    • If   is  -measurable, then  .
    • If Z is a random variable, then  . In its simplest form, this says  .
  • Pulling out known factors:
    • If   is  -measurable, then  .
    • If Z is a random variable, then  .
  • Law of total expectation:  .[10]
  • Tower property:
    • For sub-σ-algebras   we have  .
      • A special case is when Z is a  -measurable random variable. Then   and thus  .
      • Doob martingale property: the above with   (which is  -measurable), and using also  , gives  .
    • For random variables   we have  .
    • For random variables   we have  .
  • Linearity: we have   and   for  .
  • Positivity: If   then  .
  • Monotonicity: If   then  .
  • Monotone convergence: If   then  .
  • Dominated convergence: If   and   with  , then  .
  • Fatou's lemma: If   then  .
  • Jensen's inequality: If   is a convex function, then  .
  • Conditional variance: Using the conditional expectation we can define, by analogy with the definition of the variance as the mean square deviation from the average, the conditional variance
    • Definition:  
    • Algebraic formula for the variance:  
    • Law of total variance:  .
  • Martingale convergence: For a random variable  , that has finite expectation, we have  , if either   is an increasing series of sub-σ-algebras and   or if   is a decreasing series of sub-σ-algebras and  .
  • Conditional expectation as  -projection: If   are in the Hilbert space of square-integrable real random variables (real random variables with finite second moment) then
    • for  -measurable  , we have  , i.e. the conditional expectation   is in the sense of the L2(P) scalar product the orthogonal projection from   to the linear subspace of  -measurable functions. (This allows to define and prove the existence of the conditional expectation based on the Hilbert projection theorem.)
    • the mapping   is self-adjoint:  
  • Conditioning is a contractive projection of Lp spaces  . I.e.,   for any p ≥ 1.
  • Doob's conditional independence property:[11] If   are conditionally independent given  , then   (equivalently,  ).

See alsoEdit

Probability lawsEdit

NotesEdit

  1. ^ Kolmogorov, Andrey (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung (in German). Berlin: Julius Springer. p. 46.
  2. ^ Oxtoby, J. C. (1953). "Review: Measure theory, by P. R. Halmos" (PDF). Bull. Amer. Math. Soc. 59 (1): 89–91. doi:10.1090/s0002-9904-1953-09662-8.
  3. ^ J. L. Doob (1953). Stochastic Processes. John Wiley & Sons. ISBN 0-471-52369-0.
  4. ^ Olav Kallenberg: Foundations of Modern Probability. 2. edition. Springer, New York 2002, ISBN 0-387-95313-2, p. 573.
  5. ^ "probability - Intuition behind Conditional Expectation". Mathematics Stack Exchange.
  6. ^ Brockwell, Peter J. (1991). Time series : theory and methods (2nd ed.). New York: Springer-Verlag. ISBN 978-1-4419-0320-4.
  7. ^ Hastie, Trevor. The elements of statistical learning : data mining, inference, and prediction (PDF) (Second, corrected 7th printing ed.). New York. ISBN 978-0-387-84858-7.
  8. ^ Billingsley, Patrick (1995). "Section 34. Conditional Expectation". Probability and Measure (3rd ed.). John Wiley & Sons. p. 445. ISBN 0-471-00710-2.
  9. ^ Klenke, Achim. Probability theory : a comprehensive course (Second ed.). London. ISBN 978-1-4471-5361-0.
  10. ^ "Conditional expectation". www.statlect.com. Retrieved 2020-09-11.
  11. ^ Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). York, PA, USA: Springer. p. 110. ISBN 0-387-95313-2.

ReferencesEdit

  • William Feller, 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
  • Grimmett, Geoffrey; Stirzaker, David (2001). Probability and Random Processes (3rd ed.). Oxford University Press. ISBN 0-19-857222-0., pages 67–69

External linksEdit