# Distinguishing Cause from Effect with Causal Velocity Models

arXiv: https://arxiv.org/abs/2502.05122 · HTML: https://mlsanity.com/paper/2502.05122/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 7 Feb 2025.

## Builds on (most cited first, 21 of 21)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Score matching enables causal discovery of nonlinear additive noise models](https://mlsanity.com/paper/2203.04413/) (2022)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [Learning Differential Equations that are Easy to Solve](https://mlsanity.com/paper/2007.04504/) (2020)
- [Optimal score estimation via empirical Bayes smoothing](https://mlsanity.com/paper/2402.07747/) (2024)
- [Gradient Estimators for Implicit Models](https://mlsanity.com/paper/1705.07107/) (2017)
- [On the Identifiability and Estimation of Causal Location-Scale Noise Models](https://mlsanity.com/paper/2210.09054/) (2022)
- [Causal normalizing flows: from theory to practice](https://mlsanity.com/paper/2306.05415/) (2023)
- [Scalable Causal Discovery with Score Matching](https://mlsanity.com/paper/2304.03382/) (2023)
- [Counterfactual Identifiability of Bijective Causal Models](https://mlsanity.com/paper/2302.02228/) (2023)
- [Causal Discovery with Score Matching on Additive Models with Arbitrary Noise](https://mlsanity.com/paper/2304.03265/) (2023)
- [Identifying Patient-Specific Root Causes with the Heteroscedastic Noise Model](https://mlsanity.com/paper/2205.13085/) (2022)
- [Nonparametric Score Estimators](https://mlsanity.com/paper/2005.10099/) (2020)
- [Optimal transport for causal discovery](https://mlsanity.com/paper/2201.09366/) (2022)
- [Conditional distribution variability measures for causality detection](https://mlsanity.com/paper/1601.06680/) (2016)
- [On the pitfalls of Gaussian likelihood scoring for causal discovery](https://mlsanity.com/paper/2210.11104/) (2022)
- [Rank-Based Causal Discovery for Post-Nonlinear Models](https://mlsanity.com/paper/2302.12341/) (2023)
- [Counterfactual Cocycles: A Framework for Robust and Coherent Counterfactual Transports](https://mlsanity.com/paper/2405.13844/) (2024)

## Built on by (most cited first, 6 of 6)

- [AutoSOTA: An End-to-End Automated Research System for State-of-the-Art AI Model Discovery](https://mlsanity.com/paper/2604.05550/) (2026)
- [Causal Discovery via Quantile Partial Effect](https://mlsanity.com/paper/2509.12981/) (2025)
- [DAG-FM: A Foundation Model for Causal Discovery under Heterogeneous Causal Mechanisms](https://mlsanity.com/paper/2607.11510/) (2026)
- [On the Identifiability of Causal Graphs with the Invariance Principle](https://mlsanity.com/paper/2510.13583/) (2025)
- [Causal Discovery via Transformed Low-Rank Quantile Surfaces](https://mlsanity.com/paper/2609.16931/) (2026)
- [Decoupled Causal Discovery](https://mlsanity.com/paper/2609.23535/) (2026)

## Abstract (arXiv)

Bivariate structural causal models (SCM) are often used to infer causal direction by examining their goodness-of-fit under restricted model classes. In this paper, we describe a parametrization of bivariate SCMs in terms of a causal velocity by viewing the cause variable as time in a dynamical system. The velocity implicitly defines counterfactual curves via the solution of initial value problems where the observation specifies the initial condition. Using tools from measure transport, we obtain a unique correspondence between SCMs and the score function of the generated distribution via its causal velocity. Based on this, we derive an objective function that directly regresses the velocity against the score function, the latter of which can be estimated non-parametrically from observational data. We use this to develop a method for bivariate causal discovery that extends beyond known model classes such as additive or location scale noise, and that requires no assumptions on the noise distributions. When the score is estimated well, the objective is also useful for detecting model non-identifiability and misspecification. We present positive results in simulation and benchmark experiments where many existing methods fail, and perform ablation studies to examine the method's sensitivity to accurate score estimation.