Causal Theories: A Categorical Perspective on Bayesian Networks (1301.6201v1)

Abstract: 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.

• The paper introduces "causal theories," a categorical framework extending Bayesian networks with a graphical calculus to provide a structured way to reason about causal mechanisms.
• This framework associates a category with a causal structure (DAG), where objects represent variables and morphisms represent information flow, generalizing Bayesian network factorization.
• Unlike traditional Bayesian networks focusing on factorization existence, this framework explicitly chooses factorization, offering a more robust structure for causal reasoning.

Causal Theories: A Categorical Perspective on Bayesian Networks

The paper "Causal Theories: A Categorical Perspective on Bayesian Networks" by Brendan Fong explores the development of a formal graphical framework for causal reasoning using category theory. This research stands at the intersection of probability theory, graphical models, and monoidal categories, aiming to provide an enriched structure for understanding causal relationships between random variables.

The central concept introduced is the "causal theory," a symmetric monoidal category $\mathcal{C}_G$ associated with a causal structure $G$ (represented as a directed acyclic graph). In this setting, objects represent variables, and morphisms symbolize ways of deriving information about one variable from another. The notion of causal theories generalizes Bayesian networks by making explicit the factorization of joint probability distributions in terms of graphical causal mechanisms.

Key Contributions and Results

1. Extension of Bayesian Networks: The research extends Bayesian networks into a categorical framework. Bayesian networks are known for encoding causal relationships through joint distributions and conditional independence properties. Fong's work enriches this understanding by associating these networks with categories that emphasize causal mechanisms and the flows of information.
2. Graphical Calculi and Causal Reasoning: The causal theories are equipped with graphical calculi, which allow intuitive visual reasoning about the propagation of causality. The graphical language aids in visualizing the results of compositions of morphisms, providing clarity in understanding complex relations.
3. Models in Different Categories: The paper investigates causal models within various categories. For instance, in the category $Stoch$, which is rich with stochastic maps, the causal models recover and slightly generalize Bayesian networks. The research shows that by choosing stochastic maps as morphisms for causal mechanisms, one retrieves the factorization of joint distributions that comply with the structure of a Bayesian network.
4. Morphism Factorization: A key theoretical result is that morphisms of stochastic causal models in the category $CGStoch_{\text{SSM}}$ factor into a coarse graining followed by an embedding. This factorization aligns with the properties of deterministic maps but adds a richer semantic meaning in the stochastic setting.
5. Lack of Certain Universal Properties: The paper demonstrates that categories of causal models generally lack certain universal constructions, such as initial objects and products. This absence signals an inherent limitation in how these models can be combined or initiated, contrasting with the often-behaved category $Stoch$.
6. Comparison with Bayesian Networks: The paper highlights an important distinction—while Bayesian networks focus on the existence of a suitable factorization upholding the causal map, causal theories explicitly choose a factorization, thus providing a more robust framework for reasoning about causality.

Implications and Future Directions

• Broader Applicability: By framing causal reasoning within categorical topology, causal theories could find applicability in various fields beyond artificial intelligence, such as systems biology and complex systems analysis.
• Further Integration with Quantum Mechanics: The structural parallels between causal theories and quantum processes (e.g., through categories such as $\mathbf{Hilb}$—the category of Hilbert spaces) might allow for extensions into quantum causal reasoning.
• Enhanced Algorithms: Algorithms for causal inference, such as those used in Gibbs sampling or Markov Chain Monte Carlo methods, could be reformulated within this categorical framework to leverage graphical calculi for improved computational effectiveness.
• Exploration of Combinatorial Properties: Considering the limited combinations of conditional independence relations expressible by existing causal structures, there is an open investigation into more general causal frameworks that can account for a broader set of independencies.

In conclusion, this paper introduces a novel algebraic structure that enhances our ability to reason about causality. By leveraging the formalism of category theory, it provides a foundation for future exploration of complex causal relationships with a mathematically rigorous and visually comprehensible approach. This work represents a meaningful step toward a more comprehensive mathematical language for causality in complex systems.