An Algorithmic and a geometric characterization of coarsening at random

TL;DR

This paper provides a geometric and algorithmic framework for understanding coarsening at random (CAR) mechanisms in discrete data, revealing their complexity and potential limitations in statistical modeling.

Contribution

It introduces a simple, robust algorithmic description of CAR distributions using randomized multicovers and characterizes their geometric structure as a convex polytope.

Findings

CAR mechanisms can be described by randomized multicovers

The complexity of CAR mechanisms can be exponential in sample size

The set of CAR distributions forms a convex polytope

Abstract

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.