用代数方法统一描述概率图模型的道德化与三角化。
An Algebraic Approach to Moralisation and Triangulation of Probabilistic Graphical Models
- 将贝叶斯网与马尔可夫网建模为范畴间的函子，实现变换的代数化。
- 道德化与三角化可通过语法上的归纳定义实现，操作为函子预复合。
- 适合形式化推理和模型转换的研究者，提供模块化理论框架。
道德化与三角化是概率图模型中不同因子分解方式之间的转换操作：道德化可将有向模型（贝叶斯网络）转化为无向模型（马尔可夫网络），而三角化则反之。本文提出一种范畴论框架，将这两种变换表示为从贝叶斯网络范畴到马尔可夫网络范畴的函子。两类网络本身也被视为从‘语法’域到‘语义’余域的函子。特别地，道德化与三角化可在语法结构上进行归纳定义，并以函子预复合的形式运作。该方法为概率图模型理论引入了模块化、代数化的视角。
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 works in 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 are definable inductively on such syntax, and operate as a form of functor pre-composition. This approach introduces a modular, algebraic perspective in the theory of probabilistic graphical models.
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