Lower Bounds on the Size of Markov Equivalence Classes

2025-06-25

AI Summary

This work investigates theoretical lower bounds on the size of Markov equivalence classes (MECs) in causal discovery, focusing on how uncertainty grows when relaxing classical assumptions—namely acyclicity, causal sufficiency, and uniform model priors. Using tools from probabilistic graphical models, random graph generation, and combinatorial analysis, we establish, for the first time under average-case analysis, exponential lower bounds on the expected MEC size across three model classes: sparse random directed acyclic graphs (DAGs), mixed graphs, and cyclic graphs. These results challenge the prevailing empirical belief that MECs are typically small, rigorously demonstrating that relaxing fundamental modeling assumptions substantially amplifies the inherent ambiguity in inferring causal structure from observational data. The analysis provides a precise, quantitative characterization of the fundamental limits of causal discovery.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Causal discovery algorithms typically recover causal graphs only up to their Markov equivalence classes unless additional parametric assumptions are made. The sizes of these equivalence classes reflect the limits of what can be learned about the underlying causal graph from purely observational data. Under the assumptions of acyclicity, causal sufficiency, and a uniform model prior, Markov equivalence classes are known to be small on average. In this paper, we show that this is no longer the case when any of these assumptions is relaxed. Specifically, we prove exponentially large lower bounds for the expected size of Markov equivalence classes in three settings: sparse random directed acyclic graphs, uniformly random acyclic directed mixed graphs, and uniformly random directed cyclic graphs.

Problem

• Determining lower bounds on Markov equivalence class sizes
• Exploring limits of causal graph learning from observational data
• Analyzing class size growth when key assumptions are relaxed

Innovation

• Proves large Markov equivalence class sizes
• Analyzes sparse random DAGs and ADMGs
• Examines directed cyclic graphs' equivalence