基于指数倾斜经验似然的贝叶斯风险最小化统计推断
Statistical Inference for Bayesian Risk Minimization via Exponentially Tilted Empirical Likelihood
摘要：
著名的Bernstein von-Mises定理确保了当模型被正确指定时，贝叶斯后验的可信区域被很好地校准，在频率论意义上，随着数据的积累，它们的覆盖概率趋于标称值。然而，众所周知，当模型被错误指定或仅部分指定时，这种传统的贝叶斯框架缺乏鲁棒性，例如在分位数回归、基于风险最小化的监督/无监督学习和稳健估计中。为了克服这个困难，我们提出了一个新的贝叶斯推理方法，取代（错误指定或部分指定）的似然与适当的指数倾斜的经验似然加上正则化项。我们的代理经验似然精心构造使用的一阶最优性条件的经验风险最小化的矩条件。我们表明，贝叶斯后验组合这种替代经验似然和先验是渐近接近的正态分布集中在经验风险最小化的协方差阵采取适当的三明治形式。因此，所得到的贝叶斯可信区域被自动校准，以提供有效的不确定性量化。在计算上，所提出的方法可以很容易地实现马尔可夫链蒙特卡罗采样算法。我们的数值结果表明，所提出的方法往往比现有的国家的最先进的竞争对手更准确。
英文摘要：
The celebrated Bernstein von-Mises theorem ensures that credible regions from Bayesian posterior are 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 only partly specified, such as in quantile regression, risk minimization based supervised/unsupervised learning and robust estimation. To overcome this difficulty, 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 the empirical risk minimization as the moment condition. We show that the Bayesian posterior obtained by combining this surrogate empirical likelihood and the prior is asymptotically close to a normal distribution centering at the empirical risk minimizer with covariance matrix taking an appropriate sandwiched form. 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.
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