Equivalence in Non-Recursive Structural Equation Models

Introduction In the last decade, there has been considerable progress in understanding a certain class of statistical models, known as directed acyclic graph (DAG) models, which encode independence, and conditional independence constraints. (See Pearl, 1988). This research has had fruitful results in many areas: there is now a relatively clear causal interpretation of these models, there are efficient procedures for determining the statistical indistinguishability of DAG’s, reliable algorithms for generating a class of DAG models from sample data and background knowledge, etc. Two important elements in these investigations were: First, a purely graphical condition for calculating the conditional independence relations entailed by a DAG. Second, a ‘local’ characterization of equivalence between two graphs, in the sense that all of the same conditional independencies are entailed by each graph. Such a local characterization was essential in allowing the construction of efficient algorithms which could search the whole class of DAG models and to find those which fitted the given data.