Likelihood-Based Block-Coordinate Descent Algorithm For Linear Structural Equation Models With Interventional Data

Abstract

This paper studies the computation of maximum likelihood estimates in linear Gaussian structural equation models (SEMs) by extending the block-coordinate descent (BCD) method to settings with both observational and interventional data. While the likelihood equations in linear SEMs are typically nonlinear and may admit multiple solutions even in the purely observational case, the BCD algorithm provides an efficient alternative by decomposing the optimization problem into low-dimensional subproblems with closed-form solutions. We formulatea general BCD-type estimation framework for linear SEMs without latent variables that integrates data from multiple interventional environments. In the presence of interventions, both the likelihood equations and the block-coordinate update equations can increase substantially in algebraic degree, making closed-form updates infeasible in general. We identify classes of directed graphs and intervention designs under which each block-update problem reduces to an equation system with algebraic degree two, and derive explicit closed-form solutions in these cases. Through a simulation study, we show that the proposed method is efficient and accurate, outperforming a simple aggregation method. We further illustrate the practical utility of the method using a protein-signaling network dataset, where the log-likelihood values of a cyclic graph and an acyclic graph are evaluated and compared.