预测变量冗余下基于密度比后验压缩的贝叶斯模型平均
Bayesian Model Averaging under Predictor Redundancy via Density-Ratio Posterior Compression
AI总结：
针对预测变量冗余导致后验质量分散的问题，提出通过密度比后验压缩生成硬或软支持区域报告，提供可计算的失真度量、诊断指标和误差界，实现用少量区域替代大量单个支持并保持主要后验信息。
摘要：
支持索引回归中的贝叶斯模型平均会在活跃预测变量支持上诱导后验分布。在预测变量冗余下，后验质量可能分散在许多几乎可互换的支持上，使得精确支持总结不稳定或难以解释，即使预测是稳定的。我们研究如何报告已经拟合的贝叶斯模型平均后验而不改变贝叶斯目标。报告使用支持空间的硬或软区域，其压缩报告法则通过显式密度比与参考后验进行比较。该比率给出可计算的总变差和Kullback-Leibler失真、有界预测总结的界、保留质量诊断和后备权重诊断。该框架涵盖固定硬区域、度量球区域、后验聚类区域和池化剪枝区域字典。我们证明了这些区域报告的精确误差公式和验证界，并给出了少数区域可以替代一长串单个支持的条件。在模拟中，我们的区域报告通常给出更短更清晰的总结，同时保留主要后验信息，密度比诊断显示何时信息丢失过多。
英文摘要：
Bayesian model averaging in support-indexed regression induces a posterior distribution over active predictor supports. Under predictor redundancy, posterior mass can spread across many nearly interchangeable supports, making exact-support summaries unstable or hard to interpret even when prediction is stable. We study how to report an already fitted Bayesian model averaging posterior without changing the Bayesian target. A report uses hard or soft regions of support space, and its compressed reporting law is compared with the reference posterior through an explicit density ratio. This ratio gives computable total-variation and Kullback--Leibler distortion, bounds for bounded predictive summaries, retained-mass diagnostics, and fallback-weight diagnostics. The framework covers fixed hard regions, metric-ball regions, posterior-cluster regions, and pooled-pruned region dictionaries. We prove exact error formulas and validation bounds for these region reports, and give conditions under which a few regions can replace a long list of individual supports. In simulations, our region reports often give shorter and clearer summaries while preserving the main posterior information, and the density-ratio diagnostics show when too much information has been lost.
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