Social and economic studies are often implemented as complex survey designs. For example, multistage, unequal probability sampling designs utilised by federal statistical agencies are typically constructed to maximise the efficiency of the target domain level estimator (e.g. indexed by geographic area) within cost constraints for survey administration. Such designs may induce dependence between the sampled units; for example, with employment of a sampling step that selects geographically indexed clusters of units. A sampling‐weighted pseudo‐posterior distribution may be used to estimate the population model on the observed sample. The dependence induced between coclustered units inflates the scale of the resulting pseudo‐posterior covariance matrix that has been shown to induce under coverage of the credibility sets. By bridging results across Bayesian model misspecification and survey sampling, we demonstrate that the scale and shape of the asymptotic distributions are different between each of the pseudo‐maximum likelihood estimate (MLE), the pseudo‐posterior and the MLE under simple random sampling. Through insights from survey‐sampling variance estimation and recent advances in computational methods, we devise a correction applied as a simple and fast postprocessing step to Markov chain Monte Carlo draws of the pseudo‐posterior distribution. This adjustment projects the pseudo‐posterior covariance matrix such that the nominal coverage is approximately achieved. We make an application to the National Survey on Drug Use and Health as a motivating example and we demonstrate the efficacy of our scale and shape projection procedure on synthetic data on several common archetypes of survey designs.

中文翻译：

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复杂采样下伪贝叶斯推理的不确定性估计

社会和经济研究通常作为复杂的调查设计来实施。例如，联邦统计机构使用的多阶段，不等概率抽样设计通常被构造为在调查管理的成本约束内最大化目标域级别估计器（例如，按地理区域索引）的效率。这样的设计可能会引起采样单位之间的依赖性。例如，采用一个抽样步骤，该步骤选择按地理索引编制的单位簇。抽样加权伪后验分布可用于估计观察样本的总体模型。共聚单元之间诱导的依赖性扩大了所产生的伪后协方差矩阵的规模，该伪后后协方差矩阵已显示出可信度覆盖范围不足。通过跨越贝叶斯模型错误指定和调查抽样的结果，我们证明了在简单随机抽样下，每个伪最大似然估计（MLE），伪后验和MLE之间的渐近分布的规模和形状是不同的。通过从调查抽样方差估计和计算方法的最新进展中得出的见解，我们设计出一种校正方法，将其作为伪后分布的马尔可夫链蒙特卡罗图的简单而快速的后处理步骤应用。此调整投影伪后协方差矩阵，以便大致实现标称覆盖率。