TY - JOUR

T1 - On the optimality of conditional expectation as a Bregman predictor

AU - Banerjee, Arindam

AU - Guo, Xin

AU - Wang, Hui

PY - 2005/7

Y1 - 2005/7

N2 - 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 L2 predictor (also known as "the least-mean-square error" predictor) of X, among all (Borel measurable) functions of Z. In this correspondence, 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 L2 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 ali constants y is attained by the expectation E[X], then F is a BLF.

AB - 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 L2 predictor (also known as "the least-mean-square error" predictor) of X, among all (Borel measurable) functions of Z. In this correspondence, 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 L2 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 ali constants y is attained by the expectation E[X], then F is a BLF.

KW - Bregman loss functions (BLFs)

KW - Conditional expectation

KW - Prediction

UR - http://www.scopus.com/inward/record.url?scp=23744473964&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23744473964&partnerID=8YFLogxK

U2 - 10.1109/TIT.2005.850145

DO - 10.1109/TIT.2005.850145

M3 - Article

AN - SCOPUS:23744473964

SN - 0018-9448

VL - 51

SP - 2664

EP - 2669

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 7

ER -