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.

翻译：图模型与贝叶斯模型选择理论的一项标志性成就是由Chickering和Meek提出的著名贪心等价搜索（GES）算法。已知GES能在多种特殊情况下（包括高斯模型和离散模型——这些模型特别属于曲指数族）一致地估计有向无环图（DAG）模型的结构。然而，一个涵盖一般非参数DAG模型的通用理论尚属空白。本文中，我们针对满足马尔可夫分解光滑性条件的一般DAG模型族建立了贪心等价搜索的一致性，因此这些模型可能不属于曲指数族，甚至可能不是参数模型。该证明利用非参数贝叶斯的最新进展，构建了一种用于比较误设DAG模型的检验方法，从而避免了基于拉普拉斯近似的论证。尽管如此，当拉普拉斯近似有效且存在一致评分函数时，我们仍能恢复经典结果。由此，我们得到了将GES应用于一般DAG模型的通用一致性定理。