When dealing with incomplete data in statistical learning, or incomplete observations in probabilistic inference, one needs to distinguish the fact that a certain event is observed from the fact that the observed event has happened. Since the modeling and computational complexities entailed by maintaining this proper distinction are often prohibitive, one asks for conditions under which it can be safely ignored. Such conditions are given by the missing at random (mar) and coarsened at random (car) assumptions. In this paper we provide an in-depth analysis of several questions relating to mar/car assumptions. Main purpose of our study is to provide criteria by which one may evaluate whether a car assumption is reasonable for a particular data collecting or observational process. This question is complicated by the fact that several distinct versions of mar/car assumptions exist. We therefore first provide an overview over these different versions, in which we highlight the distinction between distributional and coarsening variable induced versions. We show that distributional versions are less restrictive and sufficient for most applications. We then address from two different perspectives the question of when the mar/car assumption is warranted. First we provide a static analysis that characterizes the admissibility of the car assumption in terms of the support structure of the joint probability distribution of complete data and incomplete observations. Here we obtain an equivalence characterization that improves and extends a recent result by Grunwald and Halpern. We then turn to a procedural analysis that characterizes the admissibility of the car assumption in terms of procedural models for the actual data (or observation) generating process. The main result of this analysis is that the stronger coarsened completely at random (ccar) condition is arguably the most reasonable assumption, as it alone corresponds to data coarsening procedures that satisfy a natural robustness property.

本文详细分析了与missing at random假设和coarsened at random假设相关的几个问题并提供了一些评估数据收集或观察过程中ccar假设是否合理的标准。同时研究表明ccar条件是最合理的假设，因为它代表了满足一种自然鲁棒性属性的数据粗化过程。

统计和概率推断中的可忽略性