In this paper we focus on the problem of assigning uncertainties to single-point predictions. We introduce a cost function that encodes the trade-off between accuracy and reliability in probabilistic forecast. We derive analytic formula for the case of forecasts of continuous scalar variables expressed in terms of Gaussian distributions. The Accuracy-Reliability cost function can be used to empirically estimate the variance in heteroskedastic regression problems (input dependent noise), by solving a two-objective optimization problem. The simple philosophy behind this strategy is that predictions based on the estimated variances should be both accurate and reliable (i.e. statistical consistent with observations). We show several examples with synthetic data, where the underlying hidden noise function can be accurately recovered, both in one and multi-dimensional problems. The practical implementation of the method has been done using a Neural Network and, in the one-dimensional case, with a simple polynomial fit.
Confidence Interval Construction and Conditional Variance Estimation with Dense ReLU Networks
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；University of California, Santa Cruz（加利福尼亚大学圣克鲁兹分校）；Hong Kong University of Science and Technology（香港科技大学）；Washington University in St. Louis（圣路易斯华盛顿大学）
Gradient Estimation for Binary Latent Variables via Gradient Variance Clipping
Columbia University（哥伦比亚大学）；Memorial Sloan Kettering Cancer Center（纪念斯隆-凯特琳癌症中心）；Howard Hughes Medical Institute（霍华德·休斯医学研究所）；University of Florida（佛罗里达大学）
Variance reduced Shapley value estimation for trustworthy data valuation
Virginia Tech（弗吉尼亚理工大学）；China Aerospace Academy of Systems Science and Engineering（中国航天系统科学与工程研究院）；Southern University of Science and Technology（南方科技大学）
Conditional Mean and Variance Estimation via \textit{k}-NN Algorithm with Automated Variance Selection
Harvard T.H. Chan School of Public Health（哈佛大学T.H. Chan公共卫生学院）；University of Santiago de Compostela（圣西蒙大学）；University of California, Los Angeles（加州大学洛杉矶分校）
Multivariate Density Estimation via Variance-Reduced Sketching
Committee on Computational and Applied Mathematics, University of Chicago（芝加哥大学计算与应用数学委员会）；Department of Statistics, University of Chicago（芝加哥大学统计学系）；Department of Applied and Computational Mathematics and Statistics, University of Notre Dame（南达科他大学应用与计算数学与统计学系）
