# Causal Modeling with Stationary Diffusions

arXiv: https://arxiv.org/abs/2310.17405 · HTML: https://mlsanity.com/paper/2310.17405/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 90 references that are in the map.
- First posted on arXiv 26 Oct 2023.

## Builds on (most cited first, 29 of 29)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Causality for Machine Learning](https://mlsanity.com/paper/1911.10500/) (2019)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [DYNOTEARS: Structure Learning from Time-Series Data](https://mlsanity.com/paper/2002.00498/) (2020)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Causal Consistency of Structural Equation Models](https://mlsanity.com/paper/1707.00819/) (2017)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Measuring Sample Quality with Stein's Method](https://mlsanity.com/paper/1506.03039/) (2015)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems](https://mlsanity.com/paper/1808.00924/) (2018)
- [Permutation-based Causal Inference Algorithms with Interventions](https://mlsanity.com/paper/1705.10220/) (2017)
- [Learning Gaussian Networks](https://mlsanity.com/paper/1302.6808/) (2013)
- [On the Identifiability and Estimation of Causal Location-Scale Noise Models](https://mlsanity.com/paper/2210.09054/) (2022)
- [Neural graphical modelling in continuous-time: consistency guarantees and algorithms](https://mlsanity.com/paper/2105.02522/) (2021)
- [From Ordinary Differential Equations to Structural Causal Models: the deterministic case](https://mlsanity.com/paper/1408.2063/) (2014)
- [backShift: Learning causal cyclic graphs from unknown shift interventions](https://mlsanity.com/paper/1506.02494/) (2015)
- [Causal models for dynamical systems](https://mlsanity.com/paper/2001.06208/) (2020)
- [Active Learning for Optimal Intervention Design in Causal Models](https://mlsanity.com/paper/2209.04744/) (2022)
- [Causal Modeling of Dynamical Systems](https://mlsanity.com/paper/1803.08784/) (2018)
- [Causal machine learning for single-cell genomics](https://mlsanity.com/paper/2310.14935/) (2023)
- [Cyclic Causal Discovery from Continuous Equilibrium Data](https://mlsanity.com/paper/1309.6849/) (2013)
- [NODAGS-Flow: Nonlinear Cyclic Causal Structure Learning](https://mlsanity.com/paper/2301.01849/) (2023)
- [Matching a Desired Causal State via Shift Interventions](https://mlsanity.com/paper/2107.01850/) (2021)
- [Graphical continuous Lyapunov models](https://mlsanity.com/paper/2005.10483/) (2020)
- [A Polynomial-Time Algorithm for Deciding Markov Equivalence of Directed Cyclic Graphical Models](https://mlsanity.com/paper/1302.3600/) (2013)

## Built on by (most cited first, 19 of 19)

- [Identifying Drift, Diffusion, and Causal Structure from Temporal Snapshots](https://mlsanity.com/paper/2410.22729/) (2024)
- [Scalable and Flexible Causal Discovery with an Efficient Test for Adjacency](https://mlsanity.com/paper/2406.09177/) (2024)
- [Counterfactual reasoning: an analysis of in-context emergence](https://mlsanity.com/paper/2506.05188/) (2025)
- [Towards Continuous-time Causal Foundation Models](https://mlsanity.com/paper/2605.28880/) (2026)
- [No Foundations without Foundations -- Why semi-mechanistic models are essential for regulatory biology](https://mlsanity.com/paper/2501.19178/) (2025)
- [Latent Causal Diffusions for Single-Cell Perturbation Modeling](https://mlsanity.com/paper/2601.15341/) (2026)
- [Efficient Learning of Stationary Diffusions with Stein-type Discrepancies](https://mlsanity.com/paper/2601.16597/) (2026)
- [Causal Representation Meets Stochastic Modeling under Generic Geometry](https://mlsanity.com/paper/2602.05033/) (2026)
- [ExDAG: an MIQP Algorithm for Learning DAGs](https://mlsanity.com/paper/2406.15229/) (2024)
- [Identifying biological perturbation targets through causal differential networks](https://mlsanity.com/paper/2410.03380/) (2024)
- [Towards Identifiability of Interventional Stochastic Differential Equations](https://mlsanity.com/paper/2505.15987/) (2025)
- [A Generative Framework for Causal Estimation via Importance-Weighted Diffusion Distillation](https://mlsanity.com/paper/2505.11444/) (2025)
- [Learning Joint Interventional Effects from Single-Variable Interventions in Additive Models](https://mlsanity.com/paper/2506.04945/) (2025)
- [Cyclic Counterfactuals under Shift-Scale Interventions](https://mlsanity.com/paper/2510.25005/) (2025)
- [Sign Identifiability of Causal Effects in Stationary Stochastic Dynamical Systems](https://mlsanity.com/paper/2603.08311/) (2026)
- [Generative Modeling with Flux Matching](https://mlsanity.com/paper/2605.07319/) (2026)
- [Reconciling Causality and Non-Equilibrium Thermodynamics with Hamiltonian Causal Models](https://mlsanity.com/paper/2606.04822/) (2026)
- [Non-parametric recovery of causal diffusion mechanisms from steady-state observations](https://mlsanity.com/paper/2606.30467/) (2026)
- [One Intervention per Component is Enough: Towards Identifiability in Linear Stochastic Dynamics from Steady State](https://mlsanity.com/paper/2609.19955/) (2026)

## Abstract (arXiv)

We develop a novel approach towards causal inference. Rather than structural equations over a causal graph, we learn stochastic differential equations (SDEs) whose stationary densities model a system's behavior under interventions. These stationary diffusion models do not require the formalism of causal graphs, let alone the common assumption of acyclicity. We show that in several cases, they generalize to unseen interventions on their variables, often better than classical approaches. Our inference method is based on a new theoretical result that expresses a stationarity condition on the diffusion's generator in a reproducing kernel Hilbert space. The resulting kernel deviation from stationarity (KDS) is an objective function of independent interest.