Structure learning and causal effect estimation from unbalanced groups

Causal implications can be undermined by control and treatment groups that are unbalanced by external confounders. Furthermore, the interventions on the treatment group can target unknown nodes of a Directed Acyclic Graph (DAG), with unknown alterations on the structure of dependencies. We propose a Bayesian methodology on a graph-driven multivariate Gaussian potential outcome model that identifies the unknown target nodes, quantifies the related causal effects and learns the network of dependencies pre- and post-interventions, accounting for observed confounders. For the purpose, we extend the DAG-Wishart prior to a Normal-DAG-Wishart prior in presence of covariates, whose conjugacy and marginal likelihood are derived. We first study the asymptotic properties of the propensity score Bayesian estimator, used to balance control and treatment groups from confounding effects. We then show the posterior ratio consistency of treated and untreated graphs, and the limiting distribution of the Average Treatment Effect estimator, under different asymptotic scenarios in terms of treated and control sample sizes. The theoretical results are validated on simulated data by an appropriately developed MCMC posterior sampler, and then implemented on Acute Myeloid Leukemia malignancies, to evaluate network dependencies, targets and causal effects of Histone Deacetylases inhibitor treatments.