A Categorical Approach to DIBI Models
Tao Gu, Jialu Bao, Justin Hsu, Alexandra Silva, Fabio Zanasi
TL;DR
The paper develops a uniform, category-theoretic framework for constructing DIBI models using string diagrams in Markov categories, unifying probabilistic and relational notions of conditional independence (CI) under a single formalism. It defines DIBI frames from a pair of monoids with units and provides a semantics for CI via formula satisfaction, implemented in the probabilistic setting by input-preserving kernels as Kleisli morphisms and the frame $\mathbf{PrFr}[\mathrm{Val}]$ with the natural valuation $\mathcal{V}_{\mathrm{nat}}$. The core contribution is the construction of $\mathbf{Fr}(\mathbb{C}[\theta])$, a general DIBI frame that yields a DIBI model for any Markov category $\mathbb{C}$, recovering standard probabilistic and relational models as instances and enabling continuous probabilistic models like $\mathbb{Stoch}$ and other measure-theoretic Markov categories. The work also introduces a syntactic DIBI model and discusses comparisons with existing categorical CI notions, highlighting potential for completeness results and broader applicability to CI reasoning in abstract probabilistic and relational settings.
Abstract
The logic of Dependence and Independence Bunched Implications (DIBI) is a logic to reason about conditional independence (CI); for instance, DIBI formulas can characterise CI in probability distributions and relational databases, using the probabilistic and relational DIBI models, respectively. Despite the similarity of the probabilistic and relational models, a uniform, more abstract account remains unsolved. The laborious case-by-case verification of the frame conditions required for constructing new models also calls for such a treatment. In this paper, we develop an abstract framework for systematically constructing DIBI models, using category theory as the unifying mathematical language. In particular, we use string diagrams -- a graphical presentation of monoidal categories -- to give a uniform definition of the parallel composition and subkernel relation in DIBI models. Our approach not only generalises known models, but also yields new models of interest and reduces properties of DIBI models to structures in the underlying categories. Furthermore, our categorical framework enables a logical notion of CI, in terms of the satisfaction of specific DIBI formulas. We compare it with string diagrammatic approaches to CI and show that it is an extension of string diagrammatic CI under reasonable conditions.