Causal computations in Semi Markovian Structural Causal Models using divide and conquer
Anna Rodum Bjøru, Rafael Cabañas, Helge Langseth, Antonio Salmerón
TL;DR
The paper tackles bounding counterfactuals in semi-Markovian SCMs when exogenous marginals are incompletely known. It extends the Divide and Conquer for Causal Computation (DCCC) framework to handle exogenous confounding, deriving exact extensions for specific semi-Markovian topologies and two Markovian approximations (endogenous merge and exogenous split). By incorporating observational and interventional data, the authors show tighter bounds and, in some cases, identifiability of interventional queries, while demonstrating substantial computational advantages over existing methods like EMCC and Gibbs sampling. The results indicate that combining observational and experimental data yields the most informative bounds, and the proposed approximations offer a favorable trade-off between accuracy and efficiency. The work lays groundwork for generalizing DCCC to broader semi-Markovian structures and provides practical guidelines for translating semi-Markovian problems into solvable Markovian equivalents.
Abstract
Recently, Bjøru et al. proposed a novel divide-and-conquer algorithm for bounding counterfactual probabilities in structural causal models (SCMs). They assumed that the SCMs were learned from purely observational data, leading to an imprecise characterization of the marginal distributions of exogenous variables. Their method leveraged the canonical representation of structural equations to decompose a general SCM with high-cardinality exogenous variables into a set of sub-models with low-cardinality exogenous variables. These sub-models had precise marginals over the exogenous variables and therefore admitted efficient exact inference. The aggregated results were used to bound counterfactual probabilities in the original model. The approach was developed for Markovian models, where each exogenous variable affects only a single endogenous variable. In this paper, we investigate extending the methodology to \textit{semi-Markovian} SCMs, where exogenous variables may influence multiple endogenous variables. Such models are capable of representing confounding relationships that Markovian models cannot. We illustrate the challenges of this extension using a minimal example, which motivates a set of alternative solution strategies. These strategies are evaluated both theoretically and through a computational study.