Homework Assignment 10: Use and Abuse of Conditioning

where we would need to substitute consistent estimates of the conditional probability Pr(Y |X = x, S = s) and the marginal probability Pr(S = s). It is sometimes useful to short-cut having to estimate the marginal probability by using the law of large numbers. Notice that, with respect to the sum over s, Pr(Y |X = x, S = s) is just some function of s. If we have an IID sample of realizations of S, say s1, s2, . . . sn, then the law of large numbers says that