Causal Theories: A Categorical Perspective on Bayesian Networks
Causal Theories: A Categorical Perspective on Bayesian Networks
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发表时间：
2013-01
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通讯作者：
Brendan Fong
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作者：
Brendan Fong
在这篇论文中，我们开发了一个新的因果推理的形式化图形框架。首先回顾monoidal范畴及其相关的图形语言，然后从范畴的角度重新审视概率论，并介绍贝叶斯网络，一种描述因果关系的现有结构。出于这些动机，我们提出了一个新的代数结构，我们称之为因果理论。这些都采取了一个对称monoidal范畴的形式，对象表示变量和态射的方式推导信息的一个变量从另一个。使用这些结构进行推理的一个主要优点是，由此产生的态射的图形表示与这些变量之间的信息流的直觉很好地匹配。这些类别可以在其他类别中建模，为变量和态射提供具体的解释。特别是，我们将看到，可测空间和随机映射范畴中的模型提供了贝叶斯网络的轻微概括，并且自然地形成了一个类别。最后，我们讨论这个范畴，分类态射和讨论一些基本的通用结构。错误：（i）第41-42页：因果理论的对象是$V$中的词，而不是集合，我们包括交换作为生成态射，服从定义对称monoidal范畴的恒等式。(ii)因果模型是一个强对称monoidal函子。
In this dissertation we develop a new formal graphical framework for causal reasoning. Starting with a review of monoidal categories and their associated graphical languages, we then revisit probability theory from a categorical perspective and introduce Bayesian networks, an existing structure for describing causal relationships. Motivated by these, we propose a new algebraic structure, which we term a causal theory. These take the form of a symmetric monoidal category, with the objects representing variables and morphisms ways of deducing information about one variable from another. A major advantage of reasoning with these structures is that the resulting graphical representations of morphisms match well with intuitions for flows of information between these variables. These categories can then be modelled in other categories, providing concrete interpretations for the variables and morphisms. In particular, we shall see that models in the category of measurable spaces and stochastic maps provide a slight generalisation of Bayesian networks, and naturally form a category themselves. We conclude with a discussion of this category, classifying the morphisms and discussing some basic universal constructions. ERRATA: (i) Pages 41-42: Objects of a causal theory are words, not collections, in $V$, and we include swaps as generating morphisms, subject to the identities defining a symmetric monoidal category. (ii) Page 46: A causal model is a strong symmetric monoidal functor.