33.2 Regular Conditional Distributions

Regular conditional distributions: definition, existence, examples.

[toddkemp-probability]

@21:17 There's a small problem here: with the rigid definition of "probability kernel" (that Q(x, . ) must be a probability measure for EVERY x), the definition of Q(x, B) here is NOT a probability kernel. At points x where the marginal density of X at x is 0 or infinity, Q(x, . ) is the 0 measure here.

This is not a real problem, because the set of such x is measure 0 with respect to the distribution of X. But to make sure we're matching all definitions exactly, we should be more careful here. So what we do is just pick our favorite probability measure on (S_1, B_1), and define Q(x, . ) to be *that* measure at those "bad" points x where the marginal density of X is 0 or infinity. A canonical choice might be to fix some point y_0 in S_1, and define Q(x, . ) to be the point-mass at y_0 at those "bad" x.