The Lauritzen-Chen Likelihood For Graphical Models
The Lauritzen-Chen Likelihood For Graphical Models
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发表时间：
2022-07
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通讯作者：
I. Shpitser
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作者：
I. Shpitser
图形模型，例如与无向图相关联的马尔可夫随机场（MRF）和与有向无环图相关联的贝叶斯网络（BN），已被证明是一种非常流行的不确定性推理，预测问题和因果推理的方法。参数MRF似然是高斯和分类数据的研究。然而，在更复杂的参数和半参数设置中，通过团势函数指定的似然性通常不知道是合意的{（联合良好指定的）}或非冗余的。通过在DAG分解中对马尔可夫因子进行建模，在参数和半参数设置中指定合适的和非冗余的DAG似然要简单得多。然而，以这种方式指定的DAG似然不能保证在相同的马尔可夫等价类内的不同DAG中一致。这使得基于似然性的DAG模型选择过程复杂化，因为“潜入”了关于边缘方向的潜在不必要的假设。在本文中，我们链接的密度函数分解由于陈与集团分解的MRF描述的Lauritzen提供一个一般的可能性MRF模型。建议的可能性是由变分独立的，和非冗余的封闭形式泛函的观察到的数据分布，是足够的一般适用于任意参数和半参数模型。我们使用我们的发展的扩展，给出了一般的可能性DAG模型，保证符合马尔可夫等价类的所有成员。我们的结果有直接的应用模型选择和半参数推断。
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.