
In this work, we develop methodology and procedures for nonlinear causal discovery for grouped data. Building upon the RESIT algorithm introduced by \citet{Peters2014}, we propose \textit{MURGS}, a pruning procedure to perform model selection after a causal order has been learned. We derive a closed-form backfitting update within a block coordinate descent framework. Extensive experiments on synthetic data demonstrate that \textit{GRESIT} combined with \textit{MURGS} achieves superior performance, clearly outperforming other state-of-the-art causal discovery algorithms. Moreover, evaluation on real-world manufacturing data with partially known causal structure further substantiates the practical applicability of our method. We have implemented all methods presented in this paper in a Python library, which is publicly available at \url{https://github.com/boschresearch/gresit}.

The current limitations of our work include the assumptions on nonlinearity and differentiability imposed on the function class \(\mathcal{F}\). Next to rather standard assumptions (acyclicity and i.i.d.~data) we assume the absence of unobserved confounding. Addressing these limitations presents promising directions for future work.

Overall, our contributions offer a flexible framework for causal discovery in grouped data under the GANM, laying the groundwork for both theoretical developments and practical applications in complex real-world systems.
