Post-Hoc Inference of Cross-Classified Statistics from Hierarchical Bayes Survey Weights

2026-04-28

AI Summary

This study addresses the challenge of propagating uncertainty for cross-classified statistics under hierarchical Bayesian (HB) calibration by introducing a Posterior Inference Engine (PHIE). PHIE transforms Markov chain Monte Carlo (MCMC) posterior draws from an HB model into replicate survey weights via chi-square calibration and integrates them with design-based variance to construct Calibrated Bayesian Intervals (CBI). By innovatively combining HB posteriors with post-calibration weights, this approach enables reliable inference for arbitrary cross-tabulations—a capability not previously attainable. The proposed three-tier classification framework and CBI methodology maintain near-nominal coverage even under weak correlation conditions, revealing that dominant uncertainty stems from compositional sampling variation. Empirical results demonstrate that CBI substantially outperforms inference based solely on PHIE across diverse cross-tabulated cells, with coefficients of variation meeting standard publication criteria.

Abstract

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.

Problem

• cross-classified statistics
• Hierarchical Bayes
• survey weights
• posterior credible intervals
• uncertainty propagation

Innovation

• Post-Hoc Inference Engine
• Hierarchical Bayes
• Calibrated Bayes Interval
• Cross-classified Statistics
• Survey Weight Calibration