全贝叶斯频谱聚类与不确定性量化在小区域估计中的应用
Fully Bayesian Spectral Clustering and Benchmarking with Uncertainty Quantification for Small Area Estimation
AI总结：
本文提出了一种基于频谱聚类的贝叶斯模型，用于小区域估计，通过引入条件后验均方误差来量化不确定性，并通过模拟和案例研究验证了其有效性。
摘要：
在本文中，受机器学习技术的启发，我们提出了一种新的贝叶斯模型用于小区域估计（SAE），即带有频谱聚类的Fay-Herriot模型（FH-SC）。与传统方法不同，FH-SC中的聚类基于利用外部协变量的频谱聚类算法，而非地理或行政标准。FH-SC模型的主要优势在于其在整合现有SAE方法（带或不带聚类随机效应）方面的灵活性。为了实现基准测试，我们利用后验投影的理论框架进行约束贝叶斯推断，并推导出在FH-SC模型下后验均值的新的Rao-Blackwell（RB）估计量的闭合表达式。此外，我们引入了一种新的不确定性度量方法，即条件后验均方误差（CPMSE），该方法可推广到其他贝叶斯SAE估计器。我们通过基于模型和数据的模拟研究来评估CPMSE的频率特性。所提出的方法受哥伦比亚市镇中互联网接入家庭比例估计的现实案例研究的启发。最后，我们通过我们的案例研究展示了FH-SC相对于现有贝叶斯和频率方法的优势。
英文摘要：
In this work, inspired by machine learning techniques, we propose a new Bayesian model for Small Area Estimation (SAE), the Fay-Herriot model with Spectral Clustering (FH-SC). Unlike traditional approaches, clustering in FH-SC is based on spectral clustering algorithms that utilize external covariates, rather than geographical or administrative criteria. A major advantage of the FH-SC model is its flexibility in integrating existing SAE approaches, with or without clustering random effects. To enable benchmarking, we leverage the theoretical framework of posterior projections for constrained Bayesian inference and derive closed form expressions for the new Rao-Blackwell (RB) estimators of the posterior mean under the FH-SC model. Additionally, we introduce a novel measure of uncertainty for the benchmarked estimator, the Conditional Posterior Mean Square Error (CPMSE), which is generalizable to other Bayesian SAE estimators. We conduct model-based and data-based simulation studies to evaluate the frequentist properties of the CPMSE. The proposed methodology is motivated by a real case study involving the estimation of the proportion of households with internet access in the municipalities of Colombia. Finally, we also illustrate the advantages of FH-SC over existing Bayesian and frequentist approaches through our case study.
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