  One of the hallmark achievements of the theory of graphical models and
Bayesian model selection is the celebrated greedy equivalence search (GES)
algorithm due to Chickering and Meek. GES is known to consistently estimate the
structure of directed acyclic graph (DAG) models in various special cases
including Gaussian and discrete models, which are in particular curved
exponential families. A general theory that covers general nonparametric DAG
models, however, is missing. Here, we establish the consistency of greedy
equivalence search for general families of DAG models that satisfy smoothness
conditions on the Markov factorization, and hence may not be curved exponential
families, or even parametric. The proof leverages recent advances in
nonparametric Bayes to construct a test for comparing misspecified DAG models
that avoids arguments based on the Laplace approximation. Nonetheless, when the
Laplace approximation is valid and a consistent scoring function exists, we
recover the classical result. As a result, we obtain a general consistency
theorem for GES applied to general DAG models.
