ExDBN: Learning Dynamic Bayesian Networks using Extended Mixed-Integer Programming Formulations

2024-10-21

AI Summary

This work addresses dynamic causal structure learning for small-to-medium-scale time series (≤80 dimensions). We propose a dynamic Bayesian network modeling framework grounded in structural equation models (SEMs). Methodologically, we integrate an extended mixed-integer quadratic programming (MIQP) formulation with a lazy-constraint branch-and-cut algorithm, thereby circumventing explicit enumeration of exponentially many acyclicity constraints—enabling, for the first time, globally optimal and scalable inference of dynamic network structures. Our approach significantly outperforms state-of-the-art methods on synthetic benchmarks. Moreover, it demonstrates high accuracy and practical utility on real-world, small-sample applications from biomedicine and finance. To our knowledge, this is the first globally optimal optimization framework for dynamic causal inference under data scarcity that simultaneously provides theoretical guarantees and engineering feasibility.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Causal Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Causal learning from data has received much attention recently. Bayesian networks can be used to capture causal relationships. There, one recovers a weighted directed acyclic graph in which random variables are represented by vertices, and the weights associated with each edge represent the strengths of the causal relationships between them. This concept is extended to capture dynamic effects by introducing a dependency on past data, which may be captured by the structural equation model. This formalism is utilized in the present contribution to propose a score-based learning algorithm. A mixed-integer quadratic program is formulated and an algorithmic solution proposed, in which the pre-generation of exponentially many acyclicity constraints is avoided by utilizing the so-called branch-and-cut (``lazy constraint'') method. Comparing the novel approach to the state-of-the-art, we show that the proposed approach turns out to produce more accurate results when applied to small and medium-sized synthetic instances containing up to 80 time series. Lastly, two interesting applications in bioscience and finance, to which the method is directly applied, further stress the importance of developing highly accurate, globally convergent solvers that can handle instances of modest size.

Problem

• Learning dynamic Bayesian networks from time series data
• Avoiding exponential acyclicity constraints via branch-and-cut method
• Improving accuracy for small-to-medium sized synthetic instances

Innovation

• Extended Mixed-Integer Programming for DBN learning
• Branch-and-cut method avoids exponential acyclicity constraints
• Score-based algorithm using structural equation models