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.

Causal relationships manifest as statistical dependence, but statistical models are not able to distinguish between when a variable causes the other, and when a variable is caused by the other. Causal discovery refers to the problem of determining the causal direction with only passively observed data from the system, without performing experimental intervention. This paper follows up on recent work that selects the causal direction to be the one that best fits some presupposed hypothesis, formulated as a restricted statistical model. Here, this is done by introducing a new way of formulating such a hypothesis via the incremental effects of the cause on the effect, which we call causal velocity models. This formulation greatly enlarges the types of hypotheses that can be made about the causal relationship. Although simple models for the causal effect are used here, they perform very well for causal relationships that are not well-explained by existing methods.