Conditional Expectation - Conditioning As Factorization

Conditioning As Factorization

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 such that for any measurable subset 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 ) such that for any measurable subset B of 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

and 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:

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

E(X|Y)= goY Ω ───────────────────────────> R Y g=E(X|Y= ·) Ω ──────────> R ───────────> R ω ──────────> Y(ω) ───────────> g(Y(ω)) = E(X|Y=Y(ω)) y ───────────> g( y ) = E(X|Y= y )

The equation means that the integrals of X and the composition over sets of the form Y−1(B), for B a measurable subset of U, are identical.