Network Structure Learning Under Uncertain Interventions

Abstract

Gaussian Directed Acyclic Graphs (DAGs) represent a powerful tool for learning the network of dependencies among variables, a task which is of primary interest in many fields and specifically in biology. Different DAGs may encode equivalent conditional independence structures, implying limited ability, with observational data, to identify causal relations. In many contexts however, measurements are collected under heterogeneous settings where variables are subject to exogenous interventions. Interventional data can improve the structure learning process whenever the targets of an intervention are known. However, these are often uncertain or completely unknown, as in the context of drug target discovery. We propose a Bayesian method for learning dependence structures and intervention targets from data subject to interventions on unknown variables of the system. Selected features of our approach include a DAG-Wishart prior on the DAG parameters, and the use of variable selection priors to express uncertainty on the targets. We provide theoretical results on the correct asymptotic identification of intervention targets and derive sufficient conditions for Bayes factor and posterior ratio consistency of the graph structure. Our method is applied in simulations and real-data world settings, to analyze perturbed protein data and assess antiepileptic drug therapies. Details of the MCMC algorithm and proofs of propositions are provided in the supplementary materials, together with more extensive results on simulations and applied studies. Supplementary materials for this article are available online.

Keywords:

• DAG-Wishart prior
• Directed acyclic graph
• Interventional data
• Target discovery

Supplementary Materials

The file supplementary_material.pdf provides supplemental information to our paper, and is organized as follows. Sections 1 and 2 contain proofs and discussions of Propositions 3.1 and 3.2. In Section 3 we show and discuss results on graph consistency. Sections 4 and 5 provide details about the proposed MCMC scheme and additional simulated results. Finally, Sections 6 and 7 contain sensitivity analyses to hyperparameter choices and further results for the two real data applications.

Acknowledgments

The authors gratefully acknowledge helpful discussions with Guido Consonni (UCSC) during the drafting of the paper.

Additional information

Funding

Research of F.C. was partially supported by UCSC (D1 and 2019-D.3.2 research grants). S.P. acknowledges partial support from UNIMIB research project 2020-ATE-0519 and Swiss National Foundation project CRSK-2_190656.