Non-parametric recovery of causal diffusion mechanisms from steady-state observations

2026-06-29

AI Summary

This study addresses the problem of recovering the causal diffusion mechanism of a continuous-time sparse multivariate stochastic system from steady-state cross-sectional observations alone—a setting relevant to domains like gene expression where repeated temporal measurements are infeasible. Assuming the system follows a stationary diffusion process with a known causal graph structure, the authors propose a nonparametric kernel method to identify and consistently estimate its drift functions, thereby reconstructing the system’s infinitesimal time-evolution dynamics. This work establishes, for the first time, nonparametric identifiability of causal diffusion drift functions using only static data, without requiring time-series observations. By integrating cross-validation with low-frequency sampling theory, the method enables effective estimation. Theoretical analysis confirms the consistency of the estimator, and simulations demonstrate its empirical validity, opening a new avenue for inferring dynamic causal mechanisms from static snapshots.

Abstract

We consider sparse multivariate stochastic systems that evolve in continuous time according to a causal mechanism and present methodology to recover the system's time-infinitesimal transition mechanism from mere cross-sectional data. This observational paradigm is motivated by applications such as gene expression analysis, where destructive experimental techniques may only allow recording data once over a cell's lifetime. Precisely, we assume the system follows a time-homogeneous diffusion process that has reached an equilibrium distribution at observation time. Further, we assume the causal mechanism is fully described by the diffusion drift, is acyclic, and its causal structure graph is known. In this setting, we prove that the full causal mechanism, i.e., the drift function, can be non-parametrically identified under a weak non-explosion criterion. We derive a non-parametric kernel estimator for this challenging inverse problem and prove its consistency. Moreover, we propose a cross-validation scheme for hyperparameter tuning, illustrate the behavior of our estimator in simulations, and we discuss connections with irreversible generative diffusion models and low-frequency sampled data.

Problem

• causal diffusion
• steady-state observations
• non-parametric identification
• drift function
• inverse problem

Innovation

• non-parametric identification
• causal diffusion
• steady-state data
• drift estimation
• inverse problem