Lower Bounds on the Size of Markov Equivalence Classes
This work addresses a fundamental limitation in causal inference for researchers, revealing that without strict assumptions, observational data may provide much less information about causal structures than previously thought.
The paper tackles the problem of understanding the size of Markov equivalence classes in causal discovery, showing that relaxing assumptions like acyclicity or causal sufficiency leads to exponentially large expected class sizes in various random graph settings.
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.