An Algorithmic and a Geometric Characterization of Coarsening at Random
Richard Gill and Peter Grünwald
Annals of Statistics 2007.

## Abstract

We show that the class of conditional distributions satisfying the Coarsening at Random (CAR) property has a simple algorithmic description based on randomized uniform multicovers, which are combinatorial objects generalizing the notion of partition of a set. The maximum needed {\em height\/} of the multicovers is exponential in the number of points in the sample space. This algorithmic characterization stems 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 can be useful in parsimonious statistical modelling of CAR mechanisms.