Yiu A, Goudie R J B, Tom B D M
Medical Research Council Biostatistics Unit, School of Clinical Medicine, University of Cambridge, Robinson Way, Cambridge CB2 0SR, U.K.
Biometrika. 2020 Dec;107(4):857-873. doi: 10.1093/biomet/asaa028.
Fully Bayesian inference in the presence of unequal probability sampling requires stronger structural assumptions on the data-generating distribution than frequentist semiparametric methods, but offers the potential for improved small-sample inference and convenient evidence synthesis. We demonstrate that the Bayesian exponentially tilted empirical likelihood can be used to combine the practical benefits of Bayesian inference with the robustness and attractive large-sample properties of frequentist approaches. Estimators defined as the solutions to unbiased estimating equations can be used to define a semiparametric model through the set of corresponding moment constraints. We prove Bernstein-von Mises theorems which show that the posterior constructed from the resulting exponentially tilted empirical likelihood becomes approximately normal, centred at the chosen estimator with matching asymptotic variance; thus, the posterior has properties analogous to those of the estimator, such as double robustness, and the frequentist coverage of any credible set will be approximately equal to its credibility. The proposed method can be used to obtain modified versions of existing estimators with improved properties, such as guarantees that the estimator lies within the parameter space. Unlike existing Bayesian proposals, our method does not prescribe a particular choice of prior or require posterior variance correction, and simulations suggest that it provides superior performance in terms of frequentist criteria.
在存在不等概率抽样的情况下，完全贝叶斯推断需要对数据生成分布做出比频率主义半参数方法更强的结构假设，但它为改进小样本推断和方便的证据综合提供了潜力。我们证明，贝叶斯指数倾斜经验似然可用于将贝叶斯推断的实际优势与频率主义方法的稳健性和吸引人的大样本性质相结合。定义为无偏估计方程解的估计量可用于通过相应矩约束集定义一个半参数模型。我们证明了伯恩斯坦 - 冯·米塞斯定理，该定理表明由所得指数倾斜经验似然构建的后验分布近似正态，以选定的估计量为中心且具有匹配的渐近方差；因此，后验分布具有与估计量类似的性质，如双重稳健性，并且任何可信集的频率主义覆盖率将近似等于其可信度。所提出的方法可用于获得具有改进性质的现有估计量的修改版本，例如保证估计量位于参数空间内。与现有的贝叶斯提议不同，我们的方法没有规定先验的特定选择，也不需要后验方差校正，并且模拟表明它在频率主义标准方面提供了卓越的性能。
