On the optimality of conditional expectation as a Bregman predictor

Given a probability space (Ω,F , P ), a F -measurable random variable X , and a sub-σ-algebra G ⊂ F , it is well known that the conditional expectation E[X|G] is the optimal L-predictor (also known as “the least mean square error” predictor) of X among all the G-measurable random variables [8, 11]. In this paper, we provide necessary and sufficient conditions under which the conditional expectation is the unique optimal predictor. We show that E[X|G] is the optimal predictor for all Bregman Loss Functions (BLFs), of which L loss function is a special case. Moreover, under mild conditions, we show that BLFs are exhaustive. Namely, if the inÞmum of E[F (X,Y )] over all the G-measurable random variables Y and for any variable X is attained at the conditional expectation E[X|G], then F is a BLF.