# NTS-NOTEARS: Learning Nonparametric DBNs With Prior Knowledge

arXiv: https://arxiv.org/abs/2109.04286 · HTML: https://mlsanity.com/paper/2109.04286/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 47 references that are in the map.
- First posted on arXiv 9 Sep 2021.

## Builds on (most cited first, 20 of 20)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Detecting causal associations in large nonlinear time series datasets](https://mlsanity.com/paper/1702.07007/) (2017)
- [Gradient-Based Neural DAG Learning](https://mlsanity.com/paper/1906.02226/) (2019)
- [Learning Sparse Nonparametric DAGs](https://mlsanity.com/paper/1909.13189/) (2019)
- [DYNOTEARS: Structure Learning from Time-Series Data](https://mlsanity.com/paper/2002.00498/) (2020)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game](https://mlsanity.com/paper/2102.13647/) (2021)
- [Discovering contemporaneous and lagged causal relations in autocorrelated nonlinear time series datasets](https://mlsanity.com/paper/2003.03685/) (2020)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [High-recall causal discovery for autocorrelated time series with latent confounders](https://mlsanity.com/paper/2007.01884/) (2020)
- [Conditional independence testing based on a nearest-neighbor estimator of conditional mutual information](https://mlsanity.com/paper/1709.01447/) (2017)
- [Causal Discovery with General Non-Linear Relationships Using Non-Linear ICA](https://mlsanity.com/paper/1904.09096/) (2019)
- [Economy Statistical Recurrent Units For Inferring Nonlinear Granger Causality](https://mlsanity.com/paper/1911.09879/) (2019)
- [Earth System Modeling 2.0: A Blueprint for Models That Learn From Observations and Targeted High-Resolution Simulations](https://mlsanity.com/paper/1709.00037/) (2017)
- [Interpretable Models for Granger Causality Using Self-explaining Neural Networks](https://mlsanity.com/paper/2101.07600/) (2021)
- [End-to-End Environmental Sound Classification using a 1D Convolutional Neural Network](https://mlsanity.com/paper/1904.08990/) (2019)
- [Network structure from rich but noisy data](https://mlsanity.com/paper/1703.07376/) (2017)
- [Cracking the Black Box: Distilling Deep Sports Analytics](https://mlsanity.com/paper/2006.04551/) (2020)
- [High-Dimensional Poisson DAG Model Learning Using ℓ1-Regularized Regression](https://mlsanity.com/paper/1810.02501/) (2018)
- [The Grow-Shrink strategy for learning Markov network structures constrained by context-specific independences](https://mlsanity.com/paper/1407.8088/) (2014)

## Built on by (most cited first, 30 of 33)

- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [CausalTime: Realistically Generated Time-series for Benchmarking of Causal Discovery](https://mlsanity.com/paper/2310.01753/) (2023)
- [MECD: Unlocking Multi-Event Causal Discovery in Video Reasoning](https://mlsanity.com/paper/2409.17647/) (2024)
- [Causal-Copilot: An Autonomous Causal Analysis Agent](https://mlsanity.com/paper/2504.13263/) (2025)
- [Deep Causal Learning: Representation, Discovery and Inference](https://mlsanity.com/paper/2211.03374/) (2022)
- [Jacobian Regularizer-based Neural Granger Causality](https://mlsanity.com/paper/2405.08779/) (2024)
- [Dagma-DCE: Interpretable, Non-Parametric Differentiable Causal Discovery](https://mlsanity.com/paper/2401.02930/) (2024)
- [MECD+: Unlocking Event-Level Causal Graph Discovery for Video Reasoning](https://mlsanity.com/paper/2501.07227/) (2025)
- [Causal Climate Emulation with Bayesian Filtering](https://mlsanity.com/paper/2506.09891/) (2025)
- [psiDAG: Projected Stochastic Approximation Iteration for DAG Structure Learning](https://mlsanity.com/paper/2410.23862/) (2024)
- [Differentiable Constraint-Based Causal Discovery](https://mlsanity.com/paper/2510.22031/) (2025)
- [Flow based approach for Dynamic Temporal Causal models with non-Gaussian or Heteroscedastic Noises](https://mlsanity.com/paper/2506.17065/) (2025)
- [LOCAL: Learning with Orientation Matrix to Infer Causal Structure from Time Series Data](https://mlsanity.com/paper/2410.19464/) (2024)
- [Causal Discovery for Irregularly Time Series with Consistency Guarantees](https://mlsanity.com/paper/2507.03310/) (2025)
- [Incorporating Expert Knowledge into Bayesian Causal Discovery of Mixtures of Directed Acyclic Graphs](https://mlsanity.com/paper/2510.06735/) (2025)
- [Robust Causal Discovery under Imperfect Structural Constraints](https://mlsanity.com/paper/2511.06790/) (2025)
- [Coarse-to-Fine Learning of Dynamic Causal Structures](https://mlsanity.com/paper/2602.22532/) (2026)
- [A Knowledge-Informed Pretrained Model for Causal Discovery](https://mlsanity.com/paper/2603.20842/) (2026)
- [A Review and Roadmap of Deep Causal Model from Different Causal Structures and Representations](https://mlsanity.com/paper/2311.00923/) (2023)
- [Why Online Reinforcement Learning is Causal](https://mlsanity.com/paper/2403.04221/) (2024)
- [TS-CausalNN: Learning Temporal Causal Relations from Non-linear Non-stationary Time Series Data](https://mlsanity.com/paper/2404.01466/) (2024)
- [Fast Causal Discovery by Approximate Kernel-based Generalized Score Functions with Linear Computational Complexity](https://mlsanity.com/paper/2412.17717/) (2024)
- [Transformer Is Inherently a Causal Learner](https://mlsanity.com/paper/2601.05647/) (2026)
- [Causality-driven Sequence Segmentation for Enhancing Multiphase Industrial Process Data Analysis and Soft Sensing](https://mlsanity.com/paper/2407.05954/) (2024)
- [SpinSVAR: Estimating Structural Vector Autoregression Assuming Sparse Input](https://mlsanity.com/paper/2501.03130/) (2025)
- [Linear Causal Discovery with Interventional Constraints](https://mlsanity.com/paper/2510.26342/) (2025)
- [SC3D: Dynamic and Differentiable Causal Discovery for Temporal and Instantaneous Graphs](https://mlsanity.com/paper/2602.02830/) (2026)
- [Smoothing the Landscape: Causal Structure Learning via Diffusion Denoising Objectives](https://mlsanity.com/paper/2604.02250/) (2026)
- [From Causal Graphs to Markovian States: Multi-Order State Exposure for Deep RL](https://mlsanity.com/paper/2605.07057/) (2026)
- [TTCD:Transformer Integrated Temporal Causal Discovery from Non-Stationary Time Series Data](https://mlsanity.com/paper/2605.08111/) (2026)

## Abstract (arXiv)

We describe NTS-NOTEARS, a score-based structure learning method for time-series data to learn dynamic Bayesian networks (DBNs) that captures nonlinear, lagged (inter-slice) and instantaneous (intra-slice) relations among variables. NTS-NOTEARS utilizes 1D convolutional neural networks (CNNs) to model the dependence of child variables on their parents; 1D CNN is a neural function approximation model well-suited for sequential data. DBN-CNN structure learning is formulated as a continuous optimization problem with an acyclicity constraint, following the NOTEARS DAG learning approach. We show how prior knowledge of dependencies (e.g., forbidden and required edges) can be included as additional optimization constraints. Empirical evaluation on simulated and benchmark data show that NTS-NOTEARS achieves state-of-the-art DAG structure quality compared to both parametric and nonparametric baseline methods, with improvement in the range of 10-20% on the F1-score. We also evaluate NTS-NOTEARS on complex real-world data acquired from professional ice hockey games that contain a mixture of continuous and discrete variables. The code is available online.