Authors: Bruno Petrungaro, Anthony C. Constantinou
Organizations: Bayesian Artificial Intelligence research lab, MInDS research group, Queen Mary University of London, Mile End Road, London, E1 4NS, UK.
Causal Bayesian Networks (CBNs) are a powerful tool for reasoning under uncertainty about complex real-world problems. Such problems evolve over time, responding to external shocks as they occur. To support decision-making, CBNs require a cause-and-effect map of the variables under consideration, known as the network's structure. Learning the graphical structure of a causal model from data remains challenging; learning it from time-series data is even harder because dependencies may arise at different time lags. Existing time-series causal discovery methods often assume a fixed lag window and do not explicitly optimise edge-specific lags. We propose a Tabu-based structure learning algorithm that searches for a time-ordered directed structure (i.e., where every edge respects time) while allowing edge-specific lags up to a specified maximum lag. The approach uses a decomposable BIC-based score with node-specific effective sample sizes and an explicit lag-length penalty encouraging parsimonious delay assignments while preserving efficient local score updates. We provide theoretical guarantees of validity and local optimality, and we also describe a parallel implementation for improved scalability. In simulations, the method recovered graph structure competitively and estimated lags accurately when true adjacencies were recovered. On a real-world UK COVID-19 policy dataset, the learnt structure was dominated by short delays while retaining a substantial minority of longer-lag dependencies, consistent with delayed behavioural and epidemiological effects.
Figures & tables
N(G)={G′G′ is obtained by one edge modification from G and is acyclic}.
Algorithm 1 Tabu Search
Figure 1: Graph with edge-specific lags. Solid arrows are selected; dashed is a candidate long-lag edge.
Figure 2: Lag distribution of directed edges in the learnt time-series causal graph (maximum lag of 6 ). Most dependencies are assigned to lag 1, with a long tail of edges at lags 2 to 6.
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: Sweep-level F1 results across the 10 one factor at a time simulation sweeps. Each panel varies one factor while holding the others at their baseline values. Points show the mean performance over 5 trials for each setting.
Figure 4: Sweep-level SHD results across the 10 one factor at a time simulation sweeps. Lower values indicate better structural recovery. Points show the mean performance over 5 trials for each setting.
Figure 5: Sweep-level BSF results across the 10 one factor at a time simulation sweeps. Higher values indicate better balanced recovery of edges and non-edges. Points show the mean performance over 5 trials for each setting.
Figure 6: Sweep-level lag-MAE results across the 10 one factor at a time simulation sweeps. Lower values indicate more accurate lag recovery on correctly recovered adjacencies. Points show the mean performance over 5 trials for each setting.
Leveraging deep learning for causal discovery in time series remains challenging because existing neural methods predominantly rely on component-wise architectures that fail to capture shared system dynamics or employ decoupled post-hoc graph extraction that risks overfitting to spurious correlations. We propose Mask2Cause, an end-to-end framework that recovers the underlying causal graph directly during the forecasting forward pass. Our approach introduces an Inverted Variable Embedding and an Adjacency-Constrained Masked Attention mechanism, trained with homoscedastic or heteroscedastic objectives to capture causal influences in both mean and variance. Empirical results on diverse benchmarks, from synthetic chaotic dynamics to realistic biological simulations, demonstrate state-of-the-art causal discovery with significantly reduced parameter complexity compared to standard baselines. We further show that inferred causal structures can be used to reduce parameter count of forecasting models by more than 70% on average while maintaining predictive accuracy.
Omar Muhammad, Pasupuleti Dhruv Shivkant, Deepak N. Subramani
Causal discovery with instantaneous effects in multivariate time series is challenging, as the instantaneous structure must be acyclic. Prior methods enforce this by either separating instantaneous and lagged estimation into multi-stage pipelines or imposing algebraic acyclicity constraints via complex augmented Lagrangian optimization, both of which incur high computational cost. In this work, we propose a different approach: we learn a differentiable permutation of variables using the Gumbel--Sinkhorn operator and triangularize the instantaneous coefficient matrix of a Structural Vector Autoregressive (SVAR) model in the learned order. This converts acyclicity from a hard constraint into a parameterization and keeps it valid throughout optimization. In doing so, our method enables unified, continuous optimization with gradient-based learning, leading to improved efficiency in time--series causal discovery. Across three real-world benchmarks, our method achieves the best overall performance compared with 12 baselines in both discovery accuracy and efficiency. On the large-scale benchmark, it further demonstrates strong scalability, achieving more than a 6x speedup over competing methods.
Understanding causal dependencies in observational data is critical for informing decision-making. These relationships are often modeled as Bayesian Networks (BNs) and Directed Acyclic Graphs (DAGs). Existing methods, such as NOTEARS and DAG-GNN, often face issues with scalability and stability in high-dimensional data, especially when there is a feature-sample imbalance. Here, we show that the denoising score matching objective of diffusion models could smooth the gradients for faster, more stable convergence. We also propose an adaptive k-hop acyclicity constraint that improves runtime over existing solutions that require matrix inversion. We name this framework Denoising Diffusion Causal Discovery (DDCD). Unlike generative diffusion models, DDCD utilizes the reverse denoising process to infer a parameterized causal structure rather than to generate data. We demonstrate the competitive performance of DDCDs on synthetic benchmarking data. We also show that our methods are practically useful by conducting qualitative analyses on two real-world examples. Code is available at this url: https://github.com/haozhu233/ddcd.
Hao Zhu, Di Zhou, Donna Slonim
Beth Israel Deaconess Medical Center Harvard Medical School Boston, MA, USA · Department of Computer Science Tufts University Medford, MA, USA