Compositional Inference for Bayesian Networks and Causality

2025-11-28

AI Summary

Probabilistic updating—i.e., conditioning on evidence—in Bayesian networks and causal inference is typically non-compositional due to normalization, hindering modular modeling and graphical derivation. This work introduces a formal framework for probabilistic reasoning based on string diagrams, whose core innovation is the “shadow normalization box” and its deletion rule, enabling fully compositional implementation of conditioning via graphical rewriting. By decoupling normalization from structural decomposition, the approach preserves the natural embedding of Bayesian network topology and uniformly supports observational, interventional, and counterfactual queries. Experiments demonstrate that the framework substantially improves modularity and verifiability of inference workflows, providing the first compositional, rigorously founded, and graphically intuitive basis for causal modeling.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Action, Change, and Causality

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Inference is a fundamental reasoning technique in probability theory. When applied to a large joint distribution, it involves updating with evidence (conditioning) in one or more components (variables) and computing the outcome in other components. When the joint distribution is represented by a Bayesian network, the network structure may be exploited to proceed in a compositional manner -- with great benefits. However, the main challenge is that updating involves (re)normalisation, making it an operation that interacts badly with other operations. String diagrams are becoming popular as a graphical technique for probabilistic (and quantum) reasoning. Conditioning has appeared in string diagrams, in terms of a disintegration, using bent wires and shaded (or dashed) normalisation boxes. It has become clear that such normalisation boxes do satisfy certain compositional rules. This paper takes a decisive step in this development by adding a removal rule to the formalism, for the deletion of shaded boxes. Via this removal rule one can get rid of shaded boxes and terminate an inference argument. This paper illustrates via many (graphical) examples how the resulting compositional inference technique can be used for Bayesian networks, causal reasoning and counterfactuals.

Problem

• Develops compositional inference for Bayesian networks
• Addresses renormalization challenges in probabilistic reasoning
• Introduces removal rule for shaded boxes in string diagrams

Innovation

• String diagrams with removal rule for normalization boxes
• Compositional inference via graphical disintegration technique
• Deletion of shaded boxes enables Bayesian network reasoning