ExMAG: Learning of Maximally Ancestral Graphs

TL;DR

ExMAG addresses causal structure learning under latent confounding by learning maximal ancestral graphs (MAGs) through a score-based branch-and-cut approach. It formulates MAG discovery as a compact mixed-integer quadratic program using edge matrices for directed and bidirected relations and a lazy constraint separation routine to enforce MAG properties, including cycles and inducing paths. Empirically, ExMAG delivers superior structural accuracy and competitive runtimes compared with state-of-the-art baselines on synthetic and real-world financial datasets, demonstrating resilience to hidden confounding. This method advances practical causal discovery in settings with unobserved confounders and complex mixed-graph representations, enabling more reliable causal inference in domains where latent factors are prevalent.

Abstract

In mixed graphs, there are both directed and undirected edges. An extension of acyclicity to this mixed-graph setting is known as maximally ancestral graphs. This extension is of considerable interest in causal learning in the presence of confounders. There, directed edges represent a clear direction of causality, while undirected edges represent confounding. We propose a score-based branch-and-cut algorithm for learning maximally ancestral graphs. The algorithm produces more accurate results than state-of-the-art methods, while being faster to run on small and medium-sized synthetic instances.

Figures (8)

• Figure 1: Ground truth with the confounder of Department on the Berkeley graduate admission example (left, \ref{['fig:1a']}), a dynamic Bayesian network trained on the data (center, \ref{['fig:1b']}), and ExMAG output (right, \ref{['fig:1c']}). While the dynamic Bayesian network suggests a causal relationship between gender and admission, ExMAG correctly identifies the confounding. See the supplementary material for details.
• Figure 2: SHD values (in the vertical axis) for different settings of $d$ (in the horizontal axis) and $n$ (horizontal choice of the graph). The plots in the vertical dimension differ according to the dataset used. Standard deviations are depicted as the blurred regions, and dashed lines are the maximum values. See supplementary materials for results on more datasets and error information.
• Figure 3: SHD values on 3BF datasets.
• Figure 4: Run times of the compared algorithms in seconds.
• Figure 5: Heatmap of weight matrix $W$ (left) and bidirectional weight matrix $W$ (right) on the financial dataset.