Identifying and computing the exact core-determining class - LUO, Ye (Institutional Knowledge at Singapore Management University, 2018)
The indeterministic relations between unobservable events andobserved outcomes in partially identified models can be characterized bya bipartite graph. Given a probability measure on observed outcomes, theset of feasible probability measures on unobservable events can be definedby a set of linear inequality constraints, according to Artstein’s Theorem.This set of inequalities is called the “core-determining class”. However, thenumber of inequalities defined by Artstein’s Theorem is exponentially increasing with the number of unobservable events, and many inequalitiesmay in fact be redundant. In this paper, we show that the “exact coredetermining class”, i.e., the smallest possible core-determining class, canbe characterized by a set of combinatorial rules of the bipartite graph. Weprove that if the bipartite graph and the measure on observed outcomesare non-degenerate, the exact core-determining class is unique and it onlydepends on the structure of the bipartite graph. We then propose an algorithm that explores the structure of the bipartite graph to construct theexact core-determining class. We design and implement the model and algorithm in a set of examples to show that our methodology could efficientlydiscard the redundant inequalities that are not useful to identify the parameter of interest. We also demonstrate that, by using the inequalitiescorresponding to the exact core-determining class to perform set inference,the power of test statistics against local alternatives can be improved.