TABLE OF CONTENTS Discrete conditional probabilities were seen to provide a method for guessing an unknown class from an observation: if all incorrect choices have equal costs so that the overall optimality criterion is to minimize the probability of error, then the optimal classification rule is to guess that the class X = k, where p XY ( k y) = max z p XY ( x y), the maximum a-posteriori or MAP decision rule. There is an analogous problem and solution in the continuous case, but the result does not have as strong an interpretation as in the discrete case. A more complete analogy will be derived in the next chapter.
As in the discrete case, suppose that a random variable Y is observed and the goal is to make a good guess 
( Y) of another random variable X that is jointly distributed with Y. Unfortunately in the continuous case it does not make sense to measure the quality of such a guess by the probability of its being correct because now that probability is usually zero. For example, if Y is formed by adding a Gaussian signal X to an independent Gaussian noise W to form an observation Y = X + W as in the previous section, then no rule is going to recover X perfectly from Y. Nonetheless, intuitively there should be reasonable ways to...
