非参数图模型的贪婪等价搜索
Greedy equivalence search for nonparametric graphical models
AI总结：
针对一般非参数 DAG 模型缺乏一致性理论的问题，本文利用非参数贝叶斯进展构建模型检验方法，证明了贪婪等价搜索（GES）在满足马尔可夫分解平滑条件的一般 DAG 模型族上的一致性。
摘要：
图模型理论和贝叶斯模型选择领域的标志性成就之一，是由 Chickering 和 Meek 提出的著名的贪婪等价搜索（GES）算法。已知 GES 在多种特殊情况下能够一致地估计有向无环图（DAG）模型的结构，包括高斯模型和离散模型，它们属于特定的弯曲指数族。然而，涵盖一般非参数 DAG 模型的一般性理论目前尚属缺失。在此，我们针对满足马尔可夫分解平滑条件的一般 DAG 模型族，建立了贪婪等价搜索的一致性，这些模型族因此可能不属于弯曲指数族，甚至不属于参数模型。该证明利用了非参数贝叶斯的最新进展，构建了一种用于比较错误设定 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，我们恢复了经典结果。由此，我们得到了一个关于将 GES 应用于一般 DAG 模型的一般性一致性定理。
英文摘要：
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.
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