Conditional expectation

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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 (unconditional) 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.

This concept generalizes to any probability space using measure theory.

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

Examples[edit]

Example 1. Consider the roll of a fair die and let A = 1 if the number is even (i.e. 2, 4, or 6) and A = 0 otherwise. Furthermore, let B = 1 if the number is prime (i.e. 2, 3, or 5) and B = 0 otherwise.

1 2 3 4 5 6
A 0 1 0 1 0 1
B 0 1 1 0 1 0

The unconditional expectation of A is . But the expectation of A conditional on B = 1 (i.e., conditional on the die roll being 2, 3, or 5) is , and the expectation of A conditional on B = 0 (i.e., conditional on the die roll being 1, 4, or 6) 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 (mm of rain each day) collected by a weather station on every day of the ten–year (3652–day) period from Jan 1, 1990 to Dec 31, 1999. The unconditional expectation of rainfall for an unspecified day is the average of the rainfall amounts of those 3652 days. The conditional expectation of rainfall for an otherwise unspecified day known to be (conditional on being) 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 2 is the average of the rainfall amounts that occurred on the ten days with that specific date.

History[edit]

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-σ-algebras.[4]

Classical definition[edit]

Conditional expectation with respect to an event[edit]

In classical probability theory the conditional expectation of given an event (which may be the event for a random variable ) 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[clarification needed] 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 (for instance if is a continuous random variable and is the event , this is in general the case), 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[edit]

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 random variable on 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

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[edit]

Conditional expectation with respect to a sub-σ-algebra[edit]

Conditional expectation with respect to a σ-algebra: in this example the probability space is the [0,1] interval with the Lebesgue measure. We define the following σ-algebras: ; is the σ-algebra generated by the intervals with end-points 0, ¼, ½, ¾, 1; and 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:

  • is a probability space.
  • is a random variable on that probability space with finite expectation.
  • is a sub-σ-algebra of .

Since is a subalgebra 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 .[5]

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[edit]

Consider, in addition to the above,

  • A measurable space , and
  • A random variable .

Let be a -measurable function such that, for every -measurable function ,

Then the random variable , 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 (see above) defined by the pre-image of Σ by Y. If we define

then

.

Discussion[edit]

  • 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 . Evaluating the former at yields the latter.
    • Existence of a conditional expectation function may be proven by the Radon–Nikodym theorem. A sufficient condition is that the (unconditional) expected value for X exists.
    • 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 controls the "granularity" of the conditioning. A conditional expectation over a finer (larger) σ-algebra retains information about the probabilities of a larger class of events. A conditional expectation over a coarser (smaller) σ-algebra averages over more events.

Conditioning as factorization[edit]

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.

A diagram, commutative in an average sense.

Computation[edit]

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 (where fX,Y(x, y) gives the joint density of X and Y) being the conditional density of X given Y = y.

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

where (where fY(y) gives the density of Y).

Basic properties[edit]

All the following formulas are to be understood in an almost sure sense. The σ-algebra could be replaced by a random variable .

  • Pulling out independent factors:
    • If is independent of , then .
Proof

Let . Then is independent of , so we get that

Thus the definition of conditional expectation is satisfied by the constant random variable , as desired.

    • If is independent of , then . Note that this is not necessarily the case if is only independent of and of .
    • If are independent, are independent, is independent of and is independent of , then .
  • Stability:
    • If is -measurable, then .
    • If Z is a random variable, then . In its simplest form, this says .
  • Pulling out known factors:
    • If is -measurable, then .
    • If Z is a random variable, then .
  • Law of total expectation: .
  • Tower property:
    • For sub-σ-algebras we have .
      • A special case is when Z is a -measurable random variable. Then and thus .
      • Doob martingale property: the above with (which is -measurable), and using also , gives .
    • For random variables we have .
    • For random variables we have .
  • Linearity: we have and for .
  • Positivity: If then .
  • Monotonicity: If then .
  • Monotone convergence: If then .
  • Dominated convergence: If and with , then .
  • Fatou's lemma: If then .
  • Jensen's inequality: If is a convex function, then .
  • Conditional variance: Using the conditional expectation we can define, by analogy with the definition of the variance as the mean square deviation from the average, the conditional variance
    • Definition:
    • Algebraic formula for the variance:
    • Law of total variance: .
  • Martingale convergence: For a random variable , that has finite expectation, we have , if either is an increasing series of sub-σ-algebras and or if is a decreasing series of sub-σ-algebras and .
  • Conditional expectation as -projection: If are in the Hilbert space of square-integrable real random variables (real random variables with finite second moment) then
    • for -measurable , we have , i.e. the conditional expectation is in the sense of the L2(P) scalar product the orthogonal projection from to the linear subspace of -measurable functions. (This allows to define and prove the existence of the conditional expectation based on the Hilbert projection theorem.)
    • the mapping is self-adjoint:
  • Conditioning is a contractive projection of Lp spaces . I.e., for any p ≥ 1.
  • Doob's conditional independence property:[6] If are conditionally independent given , then (equivalently, ).

See also[edit]

Notes[edit]

  1. ^ Kolmogorov, Andrey (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung (in German). Berlin: Julius Springer. p. 46. [page needed]
  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. 
  6. ^ Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). York, PA, USA: Springer. p. 110. ISBN 0-387-95313-2. 

References[edit]

  • 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, page 28[page needed]
  • Grimmett, Geoffrey; Stirzaker, David (2001). Probability and Random Processes (3rd ed.). Oxford University Press. ISBN 0-19-857222-0. , pages 67–69

External links[edit]