Bayesian Inference for Risk Minimization via Exponentially Tilted Empirical Likelihood
提出一种用指数倾斜经验似然替代传统似然的贝叶斯推断方法，解决模型误设或部分指定时的稳健性问题，使贝叶斯可信区间在频率意义下自动校准，适用于分位数回归、风险最小化学习等场景。
Abstract The celebrated Bernstein von-Mises theorem ensures credible regions from a Bayesian posterior to be well-calibrated when the model is correctly-specified, in the frequentist sense that their coverage probabilities tend to the nominal values as data accrue. However, this conventional Bayesian framework is known to lack robustness when the model is misspecified or partly specified, for example, in quantile regression, risk minimization based supervised/unsupervised learning and robust estimation. To alleviate this limitation, we propose a new Bayesian inferential approach that substitutes the (misspecified or partly specified) likelihoods with proper exponentially tilted empirical likelihoods plus a regularization term. Our surrogate empirical likelihood is carefully constructed by using the first-order optimality condition of empirical risk minimization as the moment condition. We show that the Bayesian posterior obtained by combining this surrogate empirical likelihood and a prior is asymptotically close to a normal distribution centering at the empirical risk minimizer with an appropriate sandwich-form covariance matrix. Consequently, the resulting Bayesian credible regions are automatically calibrated to deliver valid uncertainty quantification. Computationally, the proposed method can be easily implemented by Markov Chain Monte Carlo sampling algorithms. Our numerical results show that the proposed method tends to be more accurate than existing state-of-the-art competitors.