Conditional expectation


In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. 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.

Examples

Example 1: Dice rolling

Consider the roll of a fair dice and let A = 1 if the number is even and A = 0 otherwise. Furthermore, let B = 1 if the number is prime and B = 0 otherwise.
123456
A010101
B011010

The unconditional expectation of A is, but the expectation of A conditional on B = 1 is, and the expectation of A conditional on B = 0 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 collected by a weather station on every day of the ten-year 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 in the month of March, is the average of daily rainfall over all 310 days of the ten–year period that fall in March. Similarly, 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 dates back at least to Laplace, who calculated conditional distributions. It was Andrey Kolmogorov who, in 1933, formalized it using the Radon–Nikodym theorem. In works of Paul Halmos 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, the conditional expectation
of given is
where the sum is taken over all possible outcomes of.
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.
As above, the expression is undefined if.
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 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.
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.
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 is defined analogously, except instead of a single number
, the result will be a function. Let be a random vector. 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 is 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, 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, 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 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 measurable function. Examples of this are decision tree regression when is required to be a simple function, linear regression when 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
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
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.
  • 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.
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 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,
The conditional expectation of given is defined by applying the above construction on the σ-algebra generated by :
By the Doob–Dynkin lemma, there exists a measurable function such that

Discussion

  • 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.
  • ** Often, one would like to think of as a measure on for fixed H. For example, it is extremely useful to claim that is additive for almost all H. However, this does not immediately follow because each may have a different null set. Because countable unions of null sets are null sets, for a countable set of, one can choose "versions" of each with aligned null sets as to maintain additivity for almost all H. However, to align the "null sets of dysfunction" of over all possible, and thus treat as an almost surely unique measure over , we need further regularity conditions. Intuitively, to do this, we need to be able to approximate all possible with a countable set of them. This directly corresponds to the conditions for creating a regular probability measure, which are separability and completeness.
  • The σ-algebra controls the "granularity" of the conditioning. A conditional expectation over a finer σ-algebra retains information about the probabilities of a larger class of events. A conditional expectation over a coarser σ-algebra averages over more events.