Conditional expectation with respect to a conditional‐probability measure

This chapter introduces the concept of a 𝒸-conditional expectation EB(Y | 𝒸) of Y with respect to the conditional-probability measure PB on (Ω, 𝒜). In empirical applications, the conditional expectation EZ=z(Y | X) can be used to describe how the conditional expectation values of Y depend on the values x of X given that Z takes on the value z. The dependency of Y on X described by EZ=z(Y | X) may not only differ for different values z1 and z2 of Z, but also differ from the dependency described by the X-conditional expectation E(Y | X) of Y with respect to P. The chapter shows how the concept of partial conditional expectation is related to a conditional expectation with respect to a conditional-probability measure, and presents a necessary and sufficient condition for uniqueness of a conditional expectation. It also explains an implication of conditional mean independence on conditional expectations with respect to PZ=z..