CHAPTER 5 - CONDITIONING

Summary

1. SECTION 1 considers the elementary case of conditioning on a map that takes only finitely many different values, as motivation for the general definition.

2. SECTION 2 defines conditional probability distributions for conditioning on the value of a general measurable map.

3. SECTION 3 discusses existence of conditional distributions by means of a slightly more general concept, disintegration, which is essential for the understanding of general conditional densities.

4. SECTION 4 defines conditional densities. It develops the general analog of the elementary formula for a conditional density: (joint density)/(marginal density).

5. SECTION *5 illustrates how conditional distributions can be identified by symmetry considerations. The classical Borel paradox is presented as a warning against the misuse of symmetry.

6. SECTION 6 discusses the abstract Kolmogorov conditional expectation, explaining why it is natural to take the conditioning information to be a sub-sigma-field.

7. SECTION *7 discusses the statistical concept of sufficiency.

Conditional distributions: the elementary case

In introductory probability courses, conditional probabilities of events are defined as ratios, ℙ(A∣B) = ℙAB/ℙB, provided ℙB ≠ 0. The division by ℙB ensures that ℙ(· ∣ B) is also a probability measure, which puts zero mass outside the set B, that is, ℙ(B^c ∣ B) = 0. The conditional expectation of a random variable X is defined as its expectation with respect to ℙ(· ∣ B), or, more succinctly, ℙ(X ∣ B) = ℙ(XB)/ℙB. If ℙB = 0, the conditional probabilities and conditional expectations are either left undefined or are extracted by some heuristic limiting argument.