Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes

Inferring the causal structure underlying stochastic dynamical systems from observational data holds great promise in domains ranging from science and health to finance. Such processes can often be accurately modeled via stochastic differential equations (SDEs), which naturally imply causal relationships via 'which variables enter the differential of which other variables'. In this paper, we develop a kernel-based test of conditional independence (CI) on 'path-space'-e.g., solutions to SDEs, but applicable beyond that-by leveraging recent advances in signature kernels. We demonstrate strictly superior performance of our proposed CI test compared to existing approaches on path-space and provide theoretical consistency results. Then, we develop constraint-based causal discovery algorithms for acyclic stochastic dynamical systems (allowing for self-loops) that leverage temporal information to recover the entire directed acyclic graph. Assuming faithfulness and a CI oracle, we show that our algorithms are sound and complete. We empirically verify that our developed CI test in conjunction with the causal discovery algorithms outperform baselines across a range of settings. is the solution of the following path-dependent integral equation: This 'kernel trick' allows us to evaluate the signature kernel without explicit computation of the signature transform by solving the partial differential equation (PDE) in eq. ( 3 ). We refer to Salvi et al. (2021a) for a numerical approximation scheme to solve this hyperbolic PDE and its error rates and to Appendix A.2 for more mathematical details on the applicability in our setting. In our experiments, we use the JAX library sigkerax to efficiently solve eq. ( 3 ). Conditional independence tests. Conditional independence (CI) forms the foundation of graphical models. CI testing involves assessing whether two random variables, X and Y , are independent