Causal computations in Semi Markovian Structural Causal Models using divide and conquer

TL;DR

This paper extends a divide-and-conquer algorithm for bounding counterfactuals from Markovian to semi-Markovian structural causal models, addressing confounding effects and evaluating new solution strategies.

Contribution

It introduces methods to adapt existing algorithms to semi-Markovian models, enabling handling of confounding relationships in causal inference.

Findings

Proposed alternative strategies for semi-Markovian models

Theoretical analysis of solution strategies

Computational evaluation demonstrating effectiveness

Abstract

Recently, Bj{\o}ru 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.