Conditional expectation

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In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution. In other words, it is the expected value of one variable given the value(s) of one or more other variables. It is also known as conditional expected value or conditional mean.

The concept of conditional expectation is important in Kolmogorov's measure-theoretic definition of probability theory. The concept of conditional probability is defined in terms of conditional expectation.


Let X and Y be discrete random variables, then the conditional expectation of X given the event Y=y is a function of y over the range of Y

 \operatorname{E} (X | Y=y ) = \sum_{x \in \mathcal{X}} x \ \operatorname{P}(X=x|Y=y) = \sum_{x \in \mathcal{X}} x \ \frac{\operatorname{P}(X=x,Y=y)}{\operatorname{P}(Y=y)},

where \mathcal{X} is the range of X.

If now X is a continuous random variable, while Y remains a discrete variable, the conditional expectation is:

 \operatorname{E} (X | Y=y )= \int_{\mathcal{X}} x f_X (x |Y=y) dx

where  f_X (\,\cdot\, |Y=y) is the conditional density of X given Y=y.

A problem arises when Y is also a continuous random variable. In this case, the probability P(Y=y) = 0, and the Borel–Kolmogorov paradox demonstrates the ambiguity of attempting to define conditional probability along these lines.

However the above expression may be rearranged in the form:

 \operatorname{E} (X | Y=y) f_Y(y) = \int_{\mathcal{X}} x  \ f_{X,Y}(x,y) {\rm d}x

It should hold for any measurable subset B of the domain of Y that:

 \int_B \operatorname{E} (X | Y=y) f_Y(y)  {\rm d}y = \int_B \int_{\mathcal{X}} x  \ f_{X,Y}(x,y)\  {\rm d}x {\rm d}y .

In fact, this is a sufficient condition to define both conditional expectation and conditional probability.

Formal definition[edit]

Let \scriptstyle (\Omega, \mathcal {F}, \operatorname {P} ) be a probability space, with a random variable \scriptstyle X:\Omega \to \mathbb{R}^n and a sub-σ-algebra \scriptstyle \mathcal {H} \subseteq \mathcal {F} .

Then a conditional expectation of X given \scriptstyle \mathcal {H} (denoted as \scriptstyle \operatorname{E}\left[X|\mathcal {H} \right]) is any \scriptstyle \mathcal {H} -measurable function (\Omega \to \mathbb{R}^n) which satisfies:

 \int_H \operatorname{E}\left[X|\mathcal {H} \right] (\omega) \ \operatorname{d} \operatorname{P}(\omega) = \int_H X(\omega) \ \operatorname{d} \operatorname{P}(\omega)  \qquad \text{for each} \quad H \in \mathcal {H} .[1]

Note that \scriptstyle \operatorname{E}\left[X|\mathcal {H} \right] is simply the name of the conditional expectation function.


A couple of points worth noting about the definition:

  • This is not a constructive definition; we are merely given the required property that a conditional expectation must satisfy.
    • The required property has the same form as the last expression in the Calculation section.
    • Existence of a conditional expectation function is determined by the Radon–Nikodym theorem, a sufficient condition is that the (unconditional) expected value for X exist.
    • 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 \scriptstyle \mathcal {H} controls the "granularity" of the conditioning. A conditional expectation \scriptstyle{E}\left[X|\mathcal {H} \right] over a finer-grained σ-algebra \scriptstyle \mathcal {H} will allow us to condition on a wider variety of events.
    • To condition freely on values of a random variable Y with state space \scriptstyle (\mathcal Y, \Sigma) , it suffices to define the conditional expectation using the pre-image of Σ with respect to Y, so that \scriptstyle \operatorname{E}\left[X| Y\right] is defined to be \scriptstyle \operatorname{E}\left[X|\mathcal {H} \right], where
 \mathcal {H} = \sigma(Y):= Y^{-1}\left(\Sigma\right):= \{Y^{-1}(S) : S \in \Sigma \}
This suffices to ensure that the conditional expectation is σ(Y)-measurable. Although conditional expectation is defined to condition on events in the underlying probability space Ω, the requirement that it be σ(Y)-measurable allows us to condition on Y as in the introduction.

Definition of conditional probability[edit]

For any event A \in \mathcal{A} \supseteq \mathcal B, define the indicator function:

\mathbf{1}_A (\omega) = \begin{cases} 1 \; &\text{if } \omega \in A, \\ 0 \; &\text{if } \omega \notin A, \end{cases}

which is a random variable with respect to the Borel σ-algebra on (0,1). Note that the expectation of this random variable is equal to the probability of A itself:

\operatorname{E}(\mathbf{1}_A) = \operatorname{P}(A). \;

Then the conditional probability given \scriptstyle \mathcal B is a function \scriptstyle \operatorname{P}(\cdot|\mathcal{B}):\mathcal{A} \times \Omega \to (0,1) such that \scriptstyle \operatorname{P}(A|\mathcal{B}) is the conditional expectation of the indicator function for A:

\operatorname{P}(A|\mathcal{B}) = \operatorname{E}(\mathbf{1}_A|\mathcal{B}) \;

In other words, \scriptstyle \operatorname{P}(A|\mathcal{B}) is a \scriptstyle \mathcal B-measurable function satisfying

\int_B \operatorname{P}(A|\mathcal{B}) (\omega) \, \operatorname{d} \operatorname{P}(\omega) = \operatorname{P} (A \cap B) \qquad \text{for all} \quad A \in \mathcal{A}, B \in  \mathcal{B}.

A conditional probability is regular if \scriptstyle \operatorname{P}(\cdot|\mathcal{B})(\omega) is also a probability measure for all ω ∈ Ω. An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.

  • For the trivial sigma algebra \mathcal B= \{\emptyset,\Omega\} the conditional probability is a constant function, \operatorname{P}\!\left( A| \{\emptyset,\Omega\} \right) \equiv\operatorname{P}(A).
  • For A\in \mathcal{B}, as outlined above, \operatorname{P}(A|\mathcal{B})=1_A..

See also conditional probability distribution.

Conditioning as factorization[edit]

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 \Sigma of subsets. A U-valued random variable is a function Y\colon (\Omega,\mathcal A) \to (U,\Sigma) such that Y^{-1}(B)\in \mathcal A for any measurable subset B\in \Sigma 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. Then Q is a probability measure on the measurable space U defined on its σ-algebra of measurable sets.

Theorem. If X is an integrable random variable on Ω then there is one and, up to equivalence a.e. relative to Q, only one integrable function g on U, which is written g= \operatorname{E}(X \mid Y) or g(u)= \operatorname{E}(X \mid Y=u), 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 signed measure μ on the measurable subsets of U. Moreover, this measure μ 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. Then the Radon–Nikodym theorem provides the function g, equal to the density of μ with respect to Q.

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 \mid Y=u) \ d \operatorname{Q} (u),

and it holds that

\operatorname{E}(X \mid Y) \circ Y= \operatorname{E}\left(X \mid Y^{-1} \left(\Sigma\right)\right).

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 \mid Y) \circ Y)(\omega) \ d \operatorname{P} (\omega).

The equation means that the integrals of X and the composition \operatorname{E}(X \mid Y=\ \cdot)\circ Y over sets of the form Y−1(B), for B a measurable subset of U, are identical.

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

A diagram, commutative in an average sense.

Conditioning relative to a subalgebra[edit]

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 Hilbert space of square-integrable real random variables (in other words, real random variables with finite second moment) the mapping X → E(X | N) is self-adjoint

\operatorname E(X\cdot\operatorname E(Y\mid N)) = \operatorname E\left(\operatorname E(X\mid N)\cdot \operatorname E(Y\mid N)\right) = \operatorname E(\operatorname E(X\mid N)\cdot Y)

and a projection (i.e. idempotent)

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

Basic properties[edit]

Let (Ω, M, P) be a probability space, and let N be a σ-subalgebra of M.

  • Conditioning with respect to N  is linear on the space of integrable real random variables.
  • \operatorname{E}(1\mid N) = 1. More generally, \operatorname{E} (Y\mid N)= Y for every integrable N–measurable random variable Y on Ω.
  • \operatorname{E}(1_B \,\operatorname{E} (X\mid N))= \operatorname{E}(1_B \, X)   for all B ∈ N and every integrable random variable X on Ω.
 f(\operatorname{E}(X \mid N) ) \leq  \operatorname{E}(f \circ X \mid N).
 L^s_P(\Omega; M) \rightarrow L^s_P(\Omega; N), \text{ i.e. } \operatorname{E}|\operatorname{E}(X\mid N)|^s \le \operatorname{E}|X|^s
for any s ≥ 1.
  • Linearity property: the usual linearity property of the (unconditional) expectation holds also for the conditional expectation:
\operatorname{E}(aX+bZ+c\mid Y)=a \operatorname{E}(X\mid Y)+b \operatorname{E}(Z\mid Y)+c .
\operatorname{E}(X\mid Y)=\operatorname{E}(X).
  • \operatorname{E}(Xf(Y)\mid Y)=f(Y)\operatorname{E}(X\mid Y). In particular: \operatorname{E}(f(Y)\mid Y)=f(Y), and in its simplest form: \operatorname{E}(Y\mid Y)=Y

See also[edit]



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