  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.
