Conditional expectation

In probability theory, the conditional expectation of a random variable is another random variable equal to the average of the former over each possible "condition". In the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space. This definition is then generalized to any probability space using measure theory.

Conditional expectation is also known as conditional expected value or conditional mean.

In modern probability theory the concept of conditional probability is defined in terms of conditional expectation.

Concept

The concept of conditional expectation can be nicely illustrated through the following example. Suppose we have daily rainfall data (mm of rain each day) collected by a weather station on every day of the ten year period from Jan 1, 1990 to Dec 31, 1999. The conditional expectation of daily rainfall knowing the month of the year is the average of daily rainfall over all days of the ten year period that fall in a given month. These data then may be considered either as a function of each day (so for example its value for Mar 3, 1992, would be the sum of daily rainfalls on all days that are in the month of March during the ten years, divided by the number of these days, which is 310) or as a function of just the month (so for example the value for March would be equal to the value of the previous example).

It is important to note the following.

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.[1] In works of Paul Halmos[2] and Joseph L. Doob[3] from 1953, conditional expectation was generalized to its modern definition using sub-sigma-algebras.[4]

Classical definition

Conditional expectation with respect to an event

In classical probability theory the conditional expectation of X given an event H (which may be the event Y=y for a random variable Y) is the average of X over all outcomes in H, that is

 \operatorname{E} (X \mid H ) = \frac{\sum_{\omega \in H} X(\omega)}{|H|}

The sum above can be grouped by different values of \scriptstyle X(\omega), to get a sum over the range \scriptstyle \mathcal{X} of X

 \operatorname{E} (X \mid H ) = \sum_{x\in\mathcal{X}} x \, \frac{|\{\omega \in H \mid X(\omega) = x\}|}{|H|}

In modern probability theory, when H 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 Y and for y in the range of Y if the event H is Y=y. Let \scriptstyle (\Omega, \mathcal{F}, P) be a probability space, X a random variable on that probability space, and \scriptstyle H \in \mathcal{F} an event with strictly positive probability \scriptstyle P(H) > 0. Then the conditional expectation of X given the event H is

 \operatorname{E} (X \mid H) = \frac{\operatorname{E}(1_H X)}{\operatorname{P}(H)} = \int_{x\in\mathcal{X}} x \operatorname{P}(dx \mid H),

where \scriptstyle \mathcal{X} is the range of X and \operatorname{P}(A \mid H) = \frac{\operatorname{P}(A \cap H)}{\operatorname{P}(H)} is the conditional probability of A knowing H.

When P(H) = 0 (for instance if Y is a continuous random variable and H is the event Y=y, this is in general the case), the Borel–Kolmogorov paradox demonstrates the ambiguity of attempting to define the conditional probability knowing the event H. The above formula shows that this problem transposes to the conditional expectation. So instead one only defines the conditional expectation with respect to a sigma-algebra or a random variable.

Conditional expectation with respect to a random variable

If Y is a discrete random variable with range \scriptstyle \mathcal{Y} , then we can define on \scriptstyle \mathcal{Y} the function

 g: y \mapsto \operatorname{E} (X \mid Y=y ).

Sometimes this function is called the conditional expectation of X with respect to Y. In fact, according to the modern definition, it is \scriptstyle g \circ Y that is called the conditional expectation of X with respect to Y, so that we have

 \operatorname{E} (X \mid Y): \omega \mapsto \operatorname{E} (X \mid Y=Y(\omega)).

which is a random variable.

As mentioned above, if Y is a continuous random variable, it is not possible to define \scriptstyle \operatorname{E} (X \mid Y) by this method. As explained in the Borel–Kolmogorov paradox, we have to specify what limiting procedure produces the set Y = y. This can be naturally done by defining the set \scriptstyle H_y^\epsilon = \{ \omega \mid \|Y(\omega)-y\| < \epsilon \} , and taking the limit \scriptstyle \epsilon \to 0 , so that if \scriptstyle P(H_y^\epsilon) > 0 for all \scriptstyle \epsilon > 0 , then

 g: y \mapsto \lim_{\epsilon \to 0}\operatorname{E} (X \mid H_y^\varepsilon ) .

The modern definition is analogous to the above except that the above limiting process is replaced by the Radon–Nikodym derivative, so the result holds more generally.

Formal definition

Conditional expectation with respect to a σ-algebra

Conditional expectation with respect to a sigma-algebra: in this example the probability space \scriptstyle (\Omega, \mathcal {F}, \operatorname {P} ) is the [0,1] interval with the Lebesgue measure. We define the following σ-algebras: \scriptstyle \mathcal{A} = \mathcal{F} while \scriptstyle \mathcal{B} is the σ-algebra generated by the intervals with end-points 0, ¼, ½, ¾, 1 and \scriptstyle \mathcal{C} 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

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

 \int_H \operatorname{E}(X \mid \mathcal{H}) \; dP = \int_H X \; dP \qquad \text{for each} \quad H \in \mathcal{H} .[5]

The existence of \scriptstyle \operatorname{E}(X\mid\mathcal {H}) can be established by noting that \scriptstyle \mu^X: F \mapsto \int_F X for \scriptstyle F \in \mathcal{F} is a measure on \scriptstyle (\Omega, \mathcal {F}) that is absolutely continuous with respect to \scriptstyle P. Furthermore, if \scriptstyle h is the natural injection from \scriptstyle \mathcal {H} to \scriptstyle \mathcal {F} then \scriptstyle \mu^X \circ h = \mu^X_{|\mathcal{H}} is the restriction of \scriptstyle \mu^X to \scriptstyle \mathcal{H} and \scriptstyle P \circ h = P_{|\mathcal{H}} is the restriction of \scriptstyle P to \scriptstyle \mathcal {H} and \scriptstyle \mu^X \circ h is absolutely continuous with respect to \scriptstyle P \circ h since \scriptstyle P \circ h (H) = 0 \Leftrightarrow P(h(H)) = 0 \Rightarrow \mu^X(h(H)) = 0 \Leftrightarrow \mu^X \circ h(H) = 0. Thus, we have

 \operatorname{E}(X\mid\mathcal {H}) = \frac{d\mu^X_{|\mathcal{H}}}{dP_{|\mathcal{H}}} = \frac{d(\mu^X \circ h)}{d(P \circ h)}

where the derivatives are Radon–Nikodym derivatives of measures.

Conditional expectation with respect to a random variable

Consider further to the above

Then for any \scriptstyle \Sigma -measurable function \scriptstyle g: U \to \mathbb{R}^n which satisfies:

 \int g(Y) f(Y) dP = \int X f(Y) dP \qquad \text{for every }\Sigma\text{-measurable function} \quad f:U \to  \mathbb{R}^n .

the random variable \scriptstyle g(Y), denoted as \scriptstyle \operatorname{E}(X\mid Y), is a conditional expectation of X given \scriptstyle Y .

This definition is equivalent to defining the conditional expectation using the pre-image of Σ with respect to Y. If we define

 \mathcal {H} = \sigma(Y) := Y^{-1}\left(\Sigma\right) := \{Y^{-1}(B) : B \in \Sigma \}

then

 \operatorname{E}(X\mid Y) = \operatorname{E}(X\mid\mathcal {H}) = \frac{d(\mu^X \circ Y^{-1})}{d(P \circ Y^{-1})} \circ Y .

Discussion

A couple of points worth noting about the definition:

Conditioning as factorization

In the definition of conditional expectation that we provided above, the fact that Y is a real random element is irrelevant. Let (U, \Sigma) be a measurable space, where \Sigma \subset \mathcal{P}(U) is a σ-algebra in U. A U-valued random element is a measurable function Y\colon \Omega \to U, i.e. Y^{-1}(B)\in \mathcal F for all B\in \Sigma. The distribution of Y is the probability measure \mathbb{P}_Y : \Sigma \to \mathbb{R} such that \mathbb{P}_Y(B) = \mathbb{P}(Y^{-1}(B)).

Theorem. If X : \Omega \to \mathbb{R} is an integrable random variable, then there exists a \mathbb{P}_Y-unique integrable random element \operatorname{E}(X \mid Y) : U \to \mathbb{R}, such that

 \int_{Y^{-1}(B)} X \; d \operatorname{\mathbb{P}} = \int_{B} \operatorname{E}(X \mid Y) \; d \operatorname{\mathbb{P}_Y},

for all B \in \Sigma.

Proof sketch

Let \mu : \Sigma \to \mathbb{R} be such that \mu(B) = \int_{Y^{-1}(B)} X \; d \operatorname{\mathbb{P}}. Then \mu is a signed measure which is absolutely continuous with respect to \mathbb{P}_Y. Indeed \mathbb{P}_Y(B) = 0 means exactly that \mathbb{P}(Y^{-1}(B)) = 0. 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 \mu with respect to \mathbb{P}_Y, which we denote by \operatorname{E}(X \mid Y). \square

Comparing with conditional expectation with respect to sub-sigma algebras, 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 \ d \operatorname{\mathbb{P}} = \int_{Y^{-1}(B)} (\operatorname{E}(X \mid Y) \circ Y) \ d \operatorname{\mathbb{P}}.

The equation means that the integrals of X and the composition \operatorname{E}(X \mid Y) \circ Y over sets of the form Y^{-1}(B), for B \in \Sigma, are identical.

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

 \operatorname{E} (X \mid Y=y ) = \sum_{x \in \mathcal{X}} x \ \operatorname{P}(X=x\mid 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 X is a continuous random variable, while Y remains a discrete variable, the conditional expectation is:

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

with  f_X (x \mid Y=y) = \frac{f_{X, Y}(x, y)}{\operatorname{P}(Y=y)} (where fX,Y(x, y) gives the joint density of X and Y) is the conditional density of X given Y=y.

If both X and Y are continuous random variables, then the conditional expectation is:

 \operatorname{E} (X \mid Y=y )= \int_{\mathcal{X}} x f_{X\mid Y} (x\mid y) \, dx

where  f_{X\mid Y} (x\mid y) = \frac{f_{X, Y}(x, y)}{f_Y(y)} (where fY(y) gives the density of Y).

Basic properties

All the following formulas are to be understood in an almost sure sense. The sigma-algebra \scriptstyle \mathcal{H} could be replaced by a random variable \scriptstyle Z

See also

Notes

  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, S. 573.
  5. Billingsley, Patrick (1995). "Section 34. Conditional Expectation". Probability and Measure (3rd ed.). John Wiley & Sons. p. 445. ISBN 0-471-00710-2.

References

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

External links

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