Several inequalities are presented which, in part, generalize inequalities by Weinstein and Weiss, giving rise to new lower bounds for the Bayes risk under squared error loss. Consider a Bayesian setting in which y denotes the observed data and θ denotes an unknown scalar parameter. Suppose that E [ (θ − E(θ|y)) |y ] and E [ (θ − E(θ|y)) ] exist. For any estimator δ(y) of θ, E [ (θ − δ(y))|y ] ...

Mixability of a loss characterizes fast rates in the online learning setting of prediction with expert advice. The determination of the mixability constant for binary losses is straightforward but opaque. In the binary case we make this transparent and simpler by characterising mixability in terms of the second derivative of the Bayes risk of proper losses. We then extend this result to multicl...

X1, . . . , Xn iid ∼ N (θ, σ), with σ known. Our goal is to estimate θ under squared-error loss. For our first guess, pick the natural estimator X. Note that it has constant risk σ 2 n , which suggests minimaxity because we know that Bayes estimators with constant risk are also minimax estimators. However, X is not Bayes for any prior, because under squared-error loss unbiased estimators are Ba...

OBJECTIVE Risk stratification systems are used in cardiac surgery to estimate mortality risk for individual patients and to compare surgical performance between institutions or surgeons. This study investigates the suitability of six existing risk stratification systems for these purposes. METHODS Data on 5471 patients who underwent isolated coronary artery bypass grafting at two UK cardiac c...

A 3 r m h estimating the parameter 0 from a parametrized signal problem (with 0 5 0 5 L) observed through Gaussian white noise, four useful and computable lower bounds for the Bayes risk were developed. For problems with different L and Merent signal to noise ratios, some bounds am superior to the others. The lower bound obtained from taking the maximum of the four, serves not only as a good lo...