Here are some examples of non-parametric methods:
 - Probabilistic Latent Variable Models
 - Conditional GAN for Causal Discovery
 - Causal Generative Neural Network

University of Michigan - Ann Arbor

Methods with a restricted class of mechanisms always assumed that the causal mechanisms belong to a restricted class of functions $\mathcal{F}$. However, this priori restriction poses serious practical limitations when the task is to infer the causal direction on real data. Indeed in reality the mechanisms are often far from linearity and the interaction between noise and cause may be more complex than additive or even post-nonlinear noise. 

This is why more general methods have been proposed following pioneer works by Stegle et al. These methods in general offer better overall results on real data as they are more flexible, but they come with a loss of theoretical identifiability results, as no explicit restriction on the class of function is imposed. The causal direction is often recovered by setting a smooth prior on the complexity of the mechanisms.

Non-Parametric Methods

 Isabelle Guyon, Alexander Statnikov, Berna Bakir Batu (2019)
Cause Effect Pairs in Machine Learning (The Springer Series on Challenges in Machine Learning)
https://www.amazon.com/Effect-Machine-Learning-Springer-Challenges/dp/3030218090

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