# Time series causal discovery with variable lags

arXiv: https://arxiv.org/abs/2605.04081 · HTML: https://mlsanity.com/paper/2605.04081/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 37 references that are in the map.
- First posted on arXiv 15 Apr 2026.

## Builds on (most cited first, 9 of 9)

- [Evaluation: from precision, recall and F-measure to ROC, informedness, markedness and correlation](https://mlsanity.com/paper/2010.16061/) (2020)
- [Detecting causal associations in large nonlinear time series datasets](https://mlsanity.com/paper/1702.07007/) (2017)
- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [High-recall causal discovery for autocorrelated time series with latent confounders](https://mlsanity.com/paper/2007.01884/) (2020)
- [Large-scale empirical validation of Bayesian Network structure learning algorithms with noisy data](https://mlsanity.com/paper/2005.09020/) (2020)
- [Evaluating structure learning algorithms with a balanced scoring function](https://mlsanity.com/paper/1905.12666/) (2019)
- [Online Causal Structure Learning in the Presence of Latent Variables](https://mlsanity.com/paper/1904.13247/) (2019)
- [Transforming Causality: Transformer-Based Temporal Causal Discovery with Prior Knowledge Integration](https://mlsanity.com/paper/2508.15928/) (2025)
- [Econometric vs. Causal Structure-Learning for Time-Series Policy Decisions: Evidence from the UK COVID-19 Policies](https://mlsanity.com/paper/2603.00041/) (2026)

## Abstract (arXiv)

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.