Size of Interventional Markov Equivalence Classes in Random DAG Models

TL;DR

This paper analyzes the size of interventional Markov equivalence classes in random DAG models, revealing that the expected number of interventions needed for full identification approaches a constant, with implications for causal inference and experimental design.

Contribution

It provides a precise asymptotic characterization of I-MEC sizes and intervention requirements in random DAG models, advancing understanding of causal inference complexity.

Findings

Expected I-MEC size approaches a constant for large graphs.

Number of interventions needed for full DAG identification converges to a constant.

Sharp bounds on asymptotic quantities are derived and numerically validated.

Abstract

Directed acyclic graph (DAG) models are popular for capturing causal relationships. From observational and interventional data, a DAG model can only be determined up to its \emph{interventional Markov equivalence class} (I-MEC). We investigate the size of MECs for random DAG models generated by uniformly sampling and ordering an Erd\H{o}s-R\'{e}nyi graph. For constant density, we show that the expected $\log$ observational MEC size asymptotically (in the number of vertices) approaches a constant. We characterize I-MEC size in a similar fashion in the above settings with high precision. We show that the asymptotic expected number of interventions required to fully identify a DAG is a constant. These results are obtained by exploiting Meek rules and coupling arguments to provide sharp upper and lower bounds on the asymptotic quantities, which are then calculated numerically up to high precision. Our results have important consequences for experimental design of interventions and the development of algorithms for causal inference.