Tam [2026] shows that combining Bethel multivariate allocation with Hierarchical Bayes (HB) small area models can substantially reduce survey sample sizes while maintaining domain-level precision and near-nominal coverage of posterior credible intervals (CrIs). This paper extends that framework to cross-classified statistics derived from HBcalibrated unit record data. Its central contribution is a Post-Hoc Inference Engine (PHIE) that propagates uncertainty from HB domain posterior draws to arbitrary cross-tabulations. PHIE transforms each MCMC draw via chi-square calibration to produce replicate survey weights, from which CrIs are obtained. Three tiers of statistics are identified. Tier 1-E cells reproduce calibration totals and yield exact posterior CrIs. Tier 2 cells involve filtered sums of calibration variables; PHIE alone undercovers, but a Calibrated Bayes interval (CBI), augmenting PHIE with design-based compositional variance, restores near-nominal coverage. Tier 3-NCV cells involve non-calibration variables; a ratio-based CBI linked to a correlated calibration variable achieves reliable coverage even under weak correlation. A key empirical finding is that uncertainty in cross-tabulations is driven primarily by compositional sampling variability rather than HB model uncertainty. Resulting CBI-based coefficients of variation remain within standard publication thresholds.

翻译：Tam [2026]证明了将Bethel多元分配与分层贝叶斯(HB)小区域模型相结合，可在维持领域精度和后验可信区间(CrIs)近似名义覆盖率的同时，显著降低调查样本量。本文将该框架扩展至基于HB校准个体记录数据的交叉分类统计量。其核心贡献在于提出后验推断引擎(PHIE)，用于将HB领域后验抽取的不确定性传播至任意交叉制表。PHIE通过卡方校准对每次MCMC抽取进行变换，生成复制调查权重，并据此获得CrIs。研究识别出三类统计量：第一类(E类)单元格可复现校准总量并产生精确后验CrIs；第二类单元格涉及校准变量的过滤和，仅使用PHIE会覆盖不足，但通过校准贝叶斯区间(CBI)——即在PHIE基础上附加基于设计的构成方差——可恢复近名义覆盖率；第三类(NCV类)单元格涉及非校准变量，基于与相关校准变量关联的比率型CBI，即使在弱相关条件下也能实现可靠覆盖。关键实证发现是：交叉制表的不确定性主要由构成抽样变异驱动，而非HB模型不确定性。由此得到的CBI基变异系数仍保持在标准出版阈值之内。