We consider the problem of predicting a random variable X from observations, denoted by a random variable Z. It is well known that the conditional expectation E[X|Z] is the optimal L/sup 2/ predictor (also known as "the least-mean-square error" predictor) of X, among all (Borel measurable) functions of Z. In this orrespondence, we provide necessary and sufficient conditions for the general loss functions under which the conditional expectation is the unique optimal predictor. We show that E[X|Z] is the optimal predictor for all Bregman loss functions (BLFs), of which the L/sup 2/ loss function is a special case. Moreover, under mild conditions, we show that the BLFs are exhaustive, i.e., if for every random variable X, the infimum of E[F(X,y)] over all constants y is attained by the expectation E[X], then F is a BLF.

This paper provides necessary and sufficient conditions for general loss functions under which the conditional expectation is the unique optimal predictor of a random variable. Further, using such loss functions, we give an exact characterization of the difference between the two sides of Jensen's inequality.