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.
With multiple random variables, for one random variable to be mean independent of all others both individually and collectively means that each conditional expectation equals the random variable's expected value. This always holds if the variables are independent, but mean independence is a weaker condition.
Depending on the nature of the conditioning, the conditional expectation can be either a random variable itself or a fixed value. With two random variables, if the expectation of a random variable is expressed conditional on another random variable without a particular value of being specified, then the expectation of conditional on, denoted, is a function of the random variable and hence is itself a random variable. Alternatively, if the expectation of is expressed conditional on the occurrence of a particular value of, denoted, then the conditional expectation is a fixed value.

Examples

Example 1. Consider the roll of a fair 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. Suppose we have daily rainfall data collected by a weather station on every day of the ten–year period from January 1st, 1990 to December 31st, 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 falls in March. And the conditional expectation of rainfall conditional on days dated March 2nd 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.

Classical definition

Conditional expectation with respect to an event

In classical probability theory the conditional expectation of given an event is the average of over all outcomes in, that is
where is the cardinality of.
The sum above can be grouped by different values of, to get a sum over the range of
In modern probability theory, when is an event with strictly positive probability, it is possible to give a similar formula. This is notably the case for a discrete random variable and for in the range of if the event is. Let be a probability space, is a random variable on that probability space, and an event with strictly positive probability. Then the conditional expectation of given the event is
where is the range of and is the probability measure defined, for each set, as, the conditional probability of given.
When , the Borel–Kolmogorov paradox demonstrates the ambiguity of attempting to define the conditional probability knowing the event. The above formula shows that this problem transposes to the conditional expectation. So instead one only defines the conditional expectation with respect to a σ-algebra or a random variable.

Conditional expectation with respect to a random variable

If Y is a discrete random variable on the same probability space having range, then the conditional expectation of X with respect to Y is the function of the variable defined by
There is a closely related function from to defined by
This function, which is different from the previous one, is the conditional expectation of X with respect to the σ-algebra generated by Y. The two are related by
where stands for Function composition.
As mentioned above, if Y is a continuous random variable, it is not possible to define by this method. As explained in the Borel–Kolmogorov paradox, we have to specify what limiting procedure produces the set Y = y. If the event space has a distance function, then one procedure for doing so is as follows. Define the set. Assume that each is P-measurable and that for all. Then conditional expectation with respect to is well-defined. Take the limit as tends to 0 and define
Replacing this limiting process by the Radon–Nikodym derivative yields an analogous definition that works more generally.

Formal definition

Conditional expectation with respect to a sub-σ-algebra

Consider the following:
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.
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,
Let be a -measurable function such that, for every -measurable function,
Then the measurable function, denoted as, is a conditional expectation of X given.
This definition is equivalent to defining the conditional expectation with respect to the sub--field of defined by the pre-image of Σ by Y. If we define
then

Discussion

In the definition of conditional expectation that we provided above, the fact that is a real random element is irrelevant. Let be a measurable space, where is a σ-algebra on. A -valued random element is a measurable function, i.e. for all. The distribution of is the probability measure defined as the pushforward measure, that is, such that.
Theorem. If is an integrable random variable, then there exists a unique integrable random element, defined almost surely, such that
for all.
Proof sketch. Let be such that. Then is a signed measure which is absolutely continuous with respect to. Indeed means exactly that, and since the integral of an integrable function on a set of probability 0 is 0, this proves absolute continuity. The Radon–Nikodym theorem then proves the existence of a density of with respect to. This density is.
Comparing with conditional expectation with respect to sub-σ-algebras, it holds that
We can further interpret this equality by considering the abstract change of variables formula to transport the integral on the right hand side to an integral over Ω:
The equation means that the integrals of and the composition over sets of the form, for, are identical.
This equation can be interpreted to say that the following diagram is commutative on average.

Computation

When X and Y are both discrete random variables, then the conditional expectation of X given the event Y = y can be considered as function of y for y in the range of Y:
where is the range of X.
If X is a continuous random variable, while Y remains a discrete variable, the conditional expectation is
with being the conditional density of X given Y = y.
If both X and Y are continuous random variables, then the conditional expectation is
where .

Basic properties

All the following formulas are to be understood in an almost sure sense. The σ-algebra could be replaced by a random variable.
Let. Then is independent of, so we get that
Thus the definition of conditional expectation is satisfied by the constant random variable, as desired.