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

TL;DR

This work tackles learning dynamic Bayesian networks by formulating score-based DAG structure learning as a mixed-integer quadratic program (MIQP). By introducing binary edge indicators and lazy cycle-exclusion constraints within a branch-and-bound-and-cut framework, the method achieves near-global optima while mitigating the curse of dimensionality. The ExDBN approach demonstrates improved accuracy on synthetic benchmarks up to 80 time series and showcases practical applications in finance (systemic risk via CDS networks) and biomedical time-series (Krebs cycle), where globally convergent solvers and tunable MIP gaps provide robust, interpretable causal structures. Overall, ExDBN advances scalable, exact-like DBN learning with strong performance guarantees and real-world impact potential.

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.

Figures (7)

• Figure 1: SHD for a test cases using ER-3-1, SF-3-1, ER-2-2 random ensembles, where the first number is the edge-vertex ratio on intra graph, the second is the autoregressive order. The edge-vertex ratio of inter graph is always 1. For the problem denoted by var suffix, the variance of Gaussian noise is randomly sampled from uniform distribution on interval $(0.6,1.2)$ for each variable. For the other problems, the variance is set to 1. The mean, standard deviation, and maximum over 10 algorithm runs is depicted. Each run is performed on different randomly sampled dataset.
• Figure 2: F1 score for test cases using ER-3-1, SF-3-1, ER-2-2 random ensembles.
• Figure 3: Comparison of ExDBN solution quality (SHD) and running time for the ER-2-2 and SF-3-1 ensembles with 25 variables and 250 samples. The mean, and standard deviation over 10 algorithm runs is depicted. Each run is performed on different randomly sampled dataset.
• Figure 4: SHD and F1 score on the SF-3-1 benchmark for larger problem instances. The number of variables $d$ ranges from 30 to 80, and the number of samples $n$ is set to $5d$.
• Figure 5: Number of lazy constraints added by ExDBN using "all cycles" variant on test cases using ER-3-1, SF-3-1, ER-2-2 random ensembles.

Theorems & Definitions (1)

• Remark 1