On the optimality of conditional expectation as a Bregman predictor Export

@article{citeulike:1611930, abstract = {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.}, author = {Banerjee, A. and Guo, Xin and Wang, Hui}, booktitle = {Information Theory, IEEE Transactions on}, citeulike-article-id = {1611930}, citeulike-linkout-0 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1459065}, journal = {Information Theory, IEEE Transactions on}, keywords = {bregman}, number = {7}, pages = {2664--2669}, posted-at = {2007-09-01 08:03:21}, priority = {2}, title = {On the optimality of conditional expectation as a Bregman predictor}, url = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1459065}, volume = {51}, year = {2005} }

TY - JOUR ID - citeulike:1611930 L3 - citeulike-article-id:1611930 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 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. IS - 7 JF - Information Theory, IEEE Transactions on EP - 2669 TI - On the optimality of conditional expectation as a Bregman predictor VL - 51 T2 - Information Theory, IEEE Transactions on SP - 2664 KW - bregman AU - Banerjee, A AU - Guo, Xin AU - Wang, Hui PY - 2005/// UR - http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1459065 ER -