用范畴论统一解释贝叶斯网络与马尔可夫网络的转换机制
Bayesian Networks, Markov Networks, Moralisation, Triangulation: a Categorical Perspective
- 将两类网络视为语法到语义的函子，转换通过函子预复合定义
- 道德化纯语法，三角化依赖语义，分离了形式与实质操作
- 揭示变量消除算法本质是独立函子，适合形式化推理研究者
道德化和三角化是概率分布图形化表示间转换的两种方法：前者将有向的贝叶斯网络转化为无向的马尔可夫网络，后者实现反向转换。本文提出一个范畴论框架，将这两种变换建模为从贝叶斯网络范畴到马尔可夫网络范畴的函子。两类网络本身被表示为从‘语法’域到‘语义’余域的函子。值得注意的是，道德化可基于语法结构归纳定义，而三角化则依赖语义信息。这促使我们重新审视变量消除算法——它在此框架中也被视为一个独立函子，将三角化过程分为纯粹语法与纯粹语义两部分。该方法为概率图模型理论引入了函子视角，凸显了语法与语义修改的本质区别。
原文摘要 · Abstract (English)
Moralisation and Triangulation are transformations allowing to switch between different ways of factoring a probability distribution into a graphical model. Moralisation allows to view a Bayesian network (a directed model) as a Markov network (an undirected model), whereas triangulation addresses the opposite direction. We present a categorical framework where these transformations are modelled as functors between a category of Bayesian networks and one of Markov networks. The two kinds of network (the objects of these categories) are themselves represented as functors from a `syntax' domain to a `semantics' codomain. Notably, moralisation and triangulation can be defined inductively on such syntax via functor pre-composition. Moreover, while moralisation is fully syntactic, triangulation relies on semantics. This leads to a discussion of the variable elimination algorithm, reinterpreted here as a functor in its own right, that splits the triangulation procedure in two: one purely syntactic, the other purely semantic. This approach introduces a functorial perspective into the theory of probabilistic graphical models, which highlights the distinctions between syntactic and semantic modifications.
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