Lower Bounds on the Size of Markov Equivalence Classes

TL;DR

This paper demonstrates that relaxing common assumptions in causal discovery can lead to exponentially larger Markov equivalence classes, highlighting fundamental limits in observational causal inference.

Contribution

The paper establishes exponential lower bounds on the expected size of Markov equivalence classes under relaxed assumptions in various graph models.

Findings

Markov equivalence classes can be exponentially large when assumptions are relaxed.

Expected size bounds vary across different graph models.

Results highlight limitations of causal discovery from observational data.

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.