Identifying and Computing the Exact Core-determining Class

Abstract

The indeterministic relations between unobservable events and observed outcomes in partially identified models can be characterized by a bipartite graph. Given a probability measure on observed outcomes, the set of feasible probability measures on unobservable events can be defined by a set of linear inequality constraints, according to Artstein's Theorem. This set of inequalities is called the “core-determining class”. However, the number of inequalities defined by Artstein's Theorem is exponentially increasing with the number of unobservable events, and many inequalities may in fact be redundant. In this paper, we show that the exact core-determining class, i.e., the smallest possible core-determining class, can be characterized by a set of combinatorial rules of the bipartite graph. We prove that if the bipartite graph and the measure on observed outcomes are non-degenerate, the exact core-determining class is unique and it only depends on the structure of the bipartite graph. We then propose an algorithm that explores the structure of the bipartite graph to construct the exact core-determining class. We design and implement the model and algorithm in a set of examples to show that our methodology could efficiently discard the redundant inequalities that are not useful to identify the parameter of interest. We also demonstrate that, by using the inequalities corresponding to the exact core-determining class to perform set inference, the power of test statistics against local alternatives can be improved.