Conditional expectation

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In probability theory, a conditional expectation (also known as conditional expected value or conditional mean) is the expected value of a real random variable with respect to a conditional probability distribution.

Thus if X is a random variable, and A is an event whose probability is not 0, then the conditional probability distribution of X given A assigns a probability P(X ≤ x | A) to the interval (−∞, x], and we have a conditional probability distribution, which may have a first moment, called E(X | A), the conditional expectation of X given the event A.

If Y is another random variable, then the conditional expectation E(X | Y = y) of X given the value Y = y is a function of y, which let us call g(y). (Advanced arguments are needed because the event that Y = y may have probability zero.) The conditional expectation of X given random variable Y, denoted by E(X | Y), is g(Y), another random variable whose value depends on that of Y. (Reminder for those less-than-accustomed to the conventional language and notation of probability theory: this paragraph is an example of why case-sensitivity of notation must not be neglected, since capital Y and lower-case y refer to different things.)

It turns out that the conditional expectation E(X | Y) can be expressed only in terms of the sigma-algebra, say G, generated by the events Y ≤ y, rather than the particular values of Y. For a sigma-algebra G, the conditional expectation E(X | G) of X given sigma-algebra G, is a random variable that is G-measurable and whose integral over any G-measurable set is the same as the integral of X over the same set. The existence of this conditional expectation follows from the Radon-Nikodym theorem. If X happens to be G-measurable, then E(X | G) = X.

If X has an expected value, or -- what is the same thing -- E(|X|) < ∞, then the conditional expectation E(X | Y) also has an expected value, which is the same as that of X. That fact is the law of total expectation.

1 Special cases
2 Mathematical formalism
3 Conditioning as factorization
4 Conditioning relative to a subalgebra
5 Basic properties
6 See also
7 References

In the simplest case, if A is an event whose probability is not 0, then

$\operatorname{P}(S \mid A) = \frac{\operatorname{P}( A \cap S)}{\operatorname{P}(A)},$

as a function of S, is a probability measure on A and E(X | A) is the expectation of X with respect to this probability P_A. In case X is a discrete random variable, and with finite first moment, the expectation is explicitly given by

$\operatorname{E}(X | A) = \sum_r r \cdot \operatorname{P}_A\{X = r\} = \sum_r r \cdot \frac{\operatorname{P}(A \cap \{X = r\})}{\operatorname{P}(A)}$

where {X = r} is the event that X takes on the value r. Since X has finite first moment, it can be shown that this sum converges absolutely. The sum is countable since {X = r} has probability 0 for all but countable many values of r.

If X is the indicator function of an event S, then E(X | A) is just the conditional probability P_A(S).

If Y is another real random variable, then for each value of y we consider the event {Y = y}. The conditional expectation E(X | Y = y) is shorthand for E(X | {Y = y}). In general, this may not be defined, since {Y = y} may have zero probability.

The way out of this limitation is as follows: If both X and Y are discrete random variables then for any subset B of Y

$\operatorname{E}(X \ \mathbf{1}_{Y \in B}) = \sum_{r \in B} \operatorname{E}(X| Y = r) \operatorname{P}\{Y=r\}$

where 1 is the indicator function. For general random variables Y, P{Y = r} is zero for almost every r. As a first step in dealing with this problem, let us consider the case Y has a continuous distribution function. This means there is a non-negative integrable function φ_Y on R which is the density of Y. This means

$\operatorname{P}\{Y \leq a\} = \int_{-\infty}^a \phi_Y(s) \, ds$

for any a in R. We can then show the following: for any integrable random variable X, there is a function g on R such that

$\operatorname{E}(X \, \mathbf{1}_{Y \leq a}) = \int_{-\infty}^a g(t) \phi_Y(t) \, dt.$

This function g is a suitable candidate for the conditional expectation.

Let X, Y be real random variables on some probability space (Ω, M, P) where M is the σ-algebra of measurable sets on which P is defined. We consider two measures on R:

Q is the law of Y defined by Q(B) = P(Y⁻¹(B)) for every Borel subset B of R is a probability measure on the real line R. Now
P_X given by

$\operatorname{P}_X(B) = \operatorname{E}_P ( X 1_{Y \in B } ) = \int_{Y^{-1}(B)} X(\omega) \ d \operatorname{P}(\omega).$

If X is an integrable random variable, then P_X is absolutely continuous with respect to Q. In this case, it can be shown the Radon-Nikodym derivative of P_X with respect to Q exists; moreover it is uniquely determined almost everywhere with respect to Q. This random variable is the conditional expectation of X given Y, or more accurately a version of the conditional expectation of X given Y.

It follows that the conditional expectation satisfies

$\int_{Y^{-1}(B)} X(\omega) \ d \operatorname{P}(\omega) = \int_{B} \operatorname{E}(X|Y)(\theta) \ d \operatorname{Q}(\theta)$

for any Borel subset B of R.

In the definition of conditional expectation that we provided above, the fact that Y is a real random variable is irrelevant: Let U be a measurable space, that is a set equipped with a σ-algebra of subsets. A U-valued random variable is a function Y: Ω → U such that Y⁻¹(B) is an element of M for any measurable subset B of U.

We consider the measure Q on U given as above: Q(B) = P(Y⁻¹(B)) for every measurable subset B of U. Q is a probability measure on the measurable space U defined on its σ-algebra of measurable sets.

Theorem. If X is an integrable, U-valued random variable on Ω then there is one and, up to equivalence a.e. relative to Q, only one integrable function g such that for any measurable subset B of U:

$\int_{Y^{-1}(B)} X(\omega) \ d \operatorname{P}(\omega) = \int_{B} g(u) \ d \operatorname{Q} (u).$

There are a number of ways of proving this; one as suggested above, is to note that the expression on the left hand side defines as a function of the set B a countably additive probability measure on the measurable subsets of U. Moreover, this measure is absolutely continuous relative to Q. Indeed Q(B) = 0 means exactly that Y⁻¹(B) has probability 0. The integral of an integrable function on a set of probability 0 is itself 0. This proves absolute continuity.

The defining condition of conditional expectation then is the equation

$\int_{Y^{-1}(B)} X(\omega) \ d \operatorname{P}(\omega) = \int_{B} \operatorname{E}(X|Y)(u) \ d \operatorname{Q} (u).$

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:

$\int_{Y^{-1}(B)} X(\omega) \ d \operatorname{P}(\omega) = \int_{Y^{-1}(B)} [\operatorname{E}(X|Y) \circ Y](\omega) \ d \operatorname{P} (\omega).$

This equation can be interpreted to say that the following diagram is commutative in the average.

The equation means that the integrals of X and the composition $\operatorname{E}(X|Y)\circ Y$ over sets of the form $Y - 1 (B)$ for $B$ measurable are identical.

There is another viewpoint for conditioning involving σ-subalgebras N of the σ-algebra M. This version is a trivial specialization of the preceding: we simply take U to be the space Ω with the σ-algebra N and Y the identity map. We state the result:

Theorem. If X is an integrable real random variable on Ω then there is one and, up to equivalence a.e. relative to P, only one integrable function g such that for any set B belonging to the subalgebra N

$\int_{B} X(\omega) \ d \operatorname{P}(\omega) = \int_{B} g(\omega) \ d \operatorname{P} (\omega)$

where g is measurable with respect to N (a stricter condition than the measurability with respect to M required of X). This form of conditional expectation is usually written: E(X|N). This version is preferred by probabilists. One reason is that on the space of square-integrable real random variables (in other words, real random variables with finite second moment) the mapping X → E(X|N) is the self-adjoint orthogonal projection

$L^2_{\operatorname{P}}(X;M) \rightarrow L^2_{\operatorname{P}}(X;N).$

Let (Ω,M,P) be a probability space.

Conditioning with respect to a σ-subalgebra N is linear on the space of integrable real random variables.
E(1|N) = 1
Jensen's inequality holds: If f is a convex function,then

$f(\operatorname{E}(X|N) ) \leq \operatorname{E}(f \circ X |N).$

Conditioning is a contractive projection

$L^s_P(X;M) \rightarrow L^s_P(X;N)$

for any s ≥ 1.

Law of total probability
Law of total expectation
Law of total variance
Law of total cumulance (generalizes the other three)

William Feller, An Introduction to Probability Theory and its Applications, vol 1, 1950
Paul A. Meyer, Probability and Potentials, Blaisdell Publishing Co., 1966
G. R. Grimmett & D. R. Stirzaker, Probability and Random Processes, 1995

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Categories: Articles to be merged since March 2007 | Probability theory

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