  We show that the class of conditional distributions satisfying the coarsening
at random (CAR) property for discrete data has a simple and robust algorithmic
description based on randomized uniform multicovers: combinatorial objects
generalizing the notion of partition of a set. However, the complexity of a
given CAR mechanism can be large: the maximal "height" of the needed
multicovers can be exponential in the number of points in the sample space. The
results stem from a geometric interpretation of the set of CAR distributions as
a convex polytope and a characterization of its extreme points. The hierarchy
of CAR models defined in this way could be useful in parsimonious statistical
modeling of CAR mechanisms, though the results also raise doubts in applied
work as to the meaningfulness of the CAR assumption in its full generality.
