Bayesian Networks and Proof-Nets: the proof-theory of Bayesian Inference

We study the correspondence between Bayesian Networks and graphical representation of proofs in linear logic. The goal of this paper is threefold: to develop a proof-theoretical account of Bayesian inference (in the spirit of the Curry-Howard correspondence between proofs and programs), to provide compositional graphical methods, and to take into account computational efficiency. We exploit the fact that the decomposition of a graph is more flexible than that of a proof-tree, or of a type-derivation, even if compositionality becomes more challenging.

💡 Research Summary

The paper establishes a deep correspondence between Bayesian networks (BNs) and proof‑nets, the graphical representation of proofs in multiplicative linear logic (MLL). Its primary aim is to give a Curry‑Howard‑style, proof‑theoretic account of Bayesian inference, to provide compositional graphical methods for probabilistic reasoning, and to address computational efficiency.

After introducing the two domains, the authors extend the syntax of MLL by adding a new node type called a “box”. Each box encapsulates a conditional probability table (CPT) and is internally built from sampling nodes (which generate Bernoulli distributions) and additive “if‑then‑else” constructions (using ⊕ and &). Positive atoms X⁺ represent output information, negative atoms X⁻ represent input, and the flow of information follows the Geometry of Interaction: it moves downwards on positive edges and upwards on negative edges. The box therefore behaves as a probabilistic module that can be plugged into a larger proof‑net.

The central technical contribution is the definition of Bayesian proof‑nets (bpn). A bpn is an MLL proof‑net whose boxes satisfy two constraints: (1) the positive conclusions of different boxes are pairwise distinct, and (2) the overall graph respects the usual correctness criterion (every cycle must contain at least two premises of a cut or contraction node). This ensures that every bpn is the image of a sequent‑calculus proof (sequentialization theorem).

Cut‑elimination and its dual, cut‑expansion, are interpreted as the core operations of Bayesian inference. A cut corresponds to the combination of factors (multiplying CPTs) and marginalisation (summing out variables). Eliminating a cut therefore implements variable elimination or message passing, while expanding a cut corresponds to factorising a joint distribution into smaller components. Because these operations are expressed as graph‑rewriting rules, the whole inference process can be visualised and manipulated directly on the proof‑net.

A major insight is that graph decomposition is far more flexible than proof‑tree decomposition. The authors prove (Theorem 5.2) that a large Bayesian network can be split into a collection of smaller proof‑nets, each of which can be evaluated independently with a reduced semantic cost. The cost model distinguishes the usual tensor product ⊗ (which would blow up to size 2ⁿ for n binary variables) from a “factor product” that behaves like the multiplication of CPTs and therefore stays compact. This mirrors the efficiency of standard Bayesian inference algorithms while retaining the formal guarantees of proof theory.

The paper also demonstrates a diagrammatic proof of d‑separation, showing that conditional independence criteria in BNs can be derived purely from the structural properties of proof‑nets. This provides a unified logical foundation for a key graphical reasoning tool used in probabilistic modeling.

Related work is surveyed: earlier proof‑net formalisms, categorical approaches to probability, and recent cost‑aware linear logic semantics. The authors position their contribution as a synthesis that inherits the compositionality of categorical string diagrams, the resource‑sensitivity of linear logic, and the practical factor‑based semantics of Bayesian networks.

In conclusion, the authors argue that viewing Bayesian inference through the lens of proof‑theory yields both conceptual clarity and practical benefits: inference steps become explicit graph rewrites, compositional reasoning is supported by the modular nature of proof‑nets, and computational overhead is mitigated by the factor product semantics. Future directions include extending the framework to multi‑valued or continuous random variables, handling dynamic Bayesian networks, and exploring automated tooling for proof‑net based inference.