Reconstruction of a directed acyclic graph with intervention.

Si Peng,Xiaotong Shen,Wei Pan

doi:10.1214/20-ejs1767

Abstract

Identification of causal relations among variables is central to many scientific investigations, as in regulatory network analysis of gene interactions and brain network analysis of effective connectivity of causal relations between regions of interest. Statistically, causal relations are often modeled by a directed acyclic graph (DAG), and hence that reconstruction of a DAG's structure leads to the discovery of causal relations. Yet, reconstruction of a DAG's structure from observational data is impossible because a DAG Gaussian model is usually not identifiable with unequal error variances. In this article, we reconstruct a DAG's structure with the help of interventional data. Particularly, we construct a constrained likelihood to regularize intervention in addition to adjacency matrices to identify a DAG's structure, subject to an error variance constraint to further reinforce the model identifiability. Theoretically, we show that the proposed constrained likelihood leads to identifiable models, thus correct reconstruction of a DAG's structure through parameter estimation even with unequal error variances. Computationally, we design efficient algorithms for the proposed method. In simulations, we show that the proposed method enables to produce a higher accuracy of reconstruction with the help of interventional observations.

Highlights

Directed acyclic graph (DAG) models are useful to describe pairwise causal relations between random variables, defined by a certain Markov property [5], with each node representing one variable and each directed edge representing the corresponding pairwise causal relation
It is generally believed that interventions may help the reconstruction of a DAG’s structure, when a DAG model is not identifiable from data, that is, DAGs in a Markov equivalence class are not distinguishable based on observational data alone [16]
Example 3 (Sparse neighborhood with a causal model satisfying (3)). This example is modified from Example 1, in which the intervention matrix is diagonal, and the squared diagonals are generated from a sequence from 0.5 to 1 with spaced points so that the variance constraint is satisfied

Summary

Introduction

Directed acyclic graph (DAG) models are useful to describe pairwise causal relations between random variables, defined by a certain Markov property [5], with each node representing one variable and each directed edge representing the corresponding pairwise causal relation. It is generally believed that interventions may help the reconstruction of a DAG’s structure, when a DAG model is not identifiable from data, that is, DAGs in a Markov equivalence class are not distinguishable based on observational data alone [16]. For instance, intervention occurs in a form of randomized treatments in a clinical trial or a form of gene knockdown or knockout experiments in systems biology. In such a situation, some or all system variables are controlled, permitting direction estimation of ambiguous edges connecting to these controlled variables. It is practically important to design a reconstruction method for interventional data, permitting the identification of a DAG’s structure. One exception is a Bayesian method of [4], which is designed for a low-dimensional problem without theoretical guarantee, due to the super-exponential complexity in the number of nodes

Objectives

Methods

Conclusion