Graphical models such as Markov random fields (MRFs) that are associated with undirected graphs, and Bayesian networks (BNs) that are associated with directed acyclic graphs, have proven to be a very popular approach for reasoning under uncertainty, prediction problems and causal inference. Parametric MRF likelihoods are well-studied for Gaussian and categorical data. However, in more complicated parametric and semi-parametric settings, likelihoods specified via clique potential functions are generally not known to be congenial {(jointly well-specified)} or non-redundant. Congenial and non-redundant DAG likelihoods are far simpler to specify in both parametric and semi-parametric settings by modeling Markov factors in the DAG factorization. However, DAG likelihoods specified in this way are not guaranteed to coincide in distinct DAGs within the same Markov equivalence class. This complicates likelihoods based model selection procedures for DAGs by ``sneaking in'' potentially unwarranted assumptions about edge orientations. In this paper we link a density function decomposition due to Chen with the clique factorization of MRFs described by Lauritzen to provide a general likelihood for MRF models. The proposed likelihood is composed of variationally independent, and non-redundant closed form functionals of the observed data distribution, and is sufficiently general to apply to arbitrary parametric and semi-parametric models. We use an extension of our developments to give a general likelihood for DAG models that is guaranteed to coincide for all members of a Markov equivalence class. Our results have direct applications for model selection and semi-parametric inference.

翻译：与无向图相关的马尔可夫随机场（MRF）以及与有向无环图相关的贝叶斯网络（BN）等图形模型，已被证明是不确定性推理、预测问题和因果推断中非常流行的工具。参数化MRF似然对于高斯数据和分类数据已有充分研究。然而，在更复杂的参数化和半参数化设置中，通过团势函数指定的似然通常已知不具有一致性（联合正确指定）或非冗余性。在参数化和半参数化设置中，通过建模DAG分解中的马尔可夫因子，一致且非冗余的DAG似然要简单得多。然而，以这种方式指定的DAG似然并不保证在同一马尔可夫等价类中的不同DAG上一致。这通过"暗中引入"关于边方向可能无根据的假设，使基于似然的DAG模型选择程序复杂化。在本文中，我们将Chen的密度函数分解与Lauritzen描述的MRF团分解联系起来，为MRF模型提供了一种通用似然。所提出的似然由变分独立且非冗余的观测数据分布闭式泛函组成，并且足够通用，可适用于任意参数化和半参数化模型。我们通过扩展我们的研究，为DAG模型提供了一种通用似然，该似然保证在马尔可夫等价类的所有成员中一致。我们的结果对模型选择和半参数推断具有直接应用价值。