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.

中文翻译：

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贝叶斯指数倾斜经验似然不等概率抽样下的推断

在存在不等概率抽样的情况下，完全贝叶斯推理需要比频率论半参数方法更强的数据生成分布结构假设，但提供了改进小样本推理和方便证据合成的潜力。我们证明了贝叶斯指数倾斜的经验似然可用于将贝叶斯推理的实际优势与频率论方法的鲁棒性和有吸引力的大样本特性相结合。估计量定义为无偏估计方程的解，可用于通过一组相应的矩约束来定义半参数模型。我们证明了 Bernstein-von Mises 定理，该定理表明由所得的指数倾斜经验似然构造的后验近似正态，以具有匹配渐近方差的所选估计量为中心；因此，后验具有类似于估计量的属性，例如双重鲁棒性，并且任何可信集合的频率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属性的现有估计器的修改版本，例如保证估计器位于参数空间内。与现有的贝叶斯建议不同，我们的方法没有规定特定的先验选择或需要后验方差校正，并且模拟表明它在频率学标准方面提供了卓越的性能。并且任何可信集合的频率论覆盖率大约等于它的可信度。所提出的方法可用于获得具有改进属性的现有估计器的修改版本，例如保证估计器位于参数空间内。与现有的贝叶斯建议不同，我们的方法没有规定特定的先验选择或需要后验方差校正，并且模拟表明它在频率学标准方面提供了卓越的性能。并且任何可信集合的频率论覆盖率大约等于它的可信度。所提出的方法可用于获得具有改进属性的现有估计器的修改版本，例如保证估计器位于参数空间内。与现有的贝叶斯建议不同，我们的方法没有规定特定的先验选择或需要后验方差校正，并且模拟表明它在频率学标准方面提供了卓越的性能。