In Nonparametric and High-Dimensional Models, Bayesian Ignorability is an Informative Prior

In problems with large amounts of missing data one must model two distinct data generating processes: the outcome process, which generates the response, and the missing data mechanism, which determines the data we observe. Under the <i>ignorability</i> condition of Rubin (1976), however, likelihood-based inference for the outcome process does not depend on the missing data mechanism so that only the former needs to be estimated; partially because of this simplification, ignorability is often used as a baseline assumption. We study the implications of Bayesian ignorability in the presence of high-dimensional nuisance parameters and argue that ignorability is typically incompatible with sensible prior beliefs about the amount of confounding bias. We show that, for many problems, ignorability directly implies that the prior on the selection bias is tightly concentrated around zero. This is demonstrated on several models of practical interest, and the effect of ignorability on the posterior distribution is characterized for high-dimensional linear models with a ridge regression prior. We then show both how to build high-dimensional models that encode sensible beliefs about the confounding bias and also show that under certain narrow circumstances ignorability is less problematic.

在存在大量缺失数据的问题中，需对两类截然不同的数据生成过程进行建模：其一为生成响应变量的结果生成过程（outcome process），其二为决定观测数据的缺失数据机制（missing data mechanism）。然而，在鲁宾（Rubin, 1976）提出的可忽略性（ignorability）条件下，针对结果生成过程的基于似然的推断并不依赖于缺失数据机制，因此仅需对前者进行估计。部分由于这一简化，可忽略性常被用作基准假设。我们探究了高维多余参数（nuisance parameters）场景下贝叶斯可忽略性的理论内涵，并指出可忽略性通常与关于混杂偏倚（confounding bias）程度的合理先验信念不相兼容。我们证明，在诸多问题中，可忽略性直接推导出选择偏倚（selection bias）的先验分布紧密集中于零点附近。这一结论在多个具有实际研究价值的模型中得到了验证；同时针对带有岭回归先验（ridge regression prior）的高维线性模型，我们刻画了可忽略性对其后验分布（posterior distribution）的影响。随后我们既展示了如何构建能够嵌入关于混杂偏倚合理先验信念的高维模型，也阐明了在某些特定的狭窄场景下，可忽略性的问题性会有所减弱。