Here are some examples of the methods that exploit the independence between cause and mechanism:
 - NonLinear Deterministic Mechanism
 - Unsupervised Inverse Regression
 - Causal Inference via Kernel Deviance Measures
 - Cause-Effect Inference by Comparing Regression Errors

Examples of Methods Exploiting the Independence between Cause and Mechanism

Information Geometric Causal Inference (IGCI) is an approach used to infer causal direction in a pairwise setting. It is based on the independence between the cause and the causal mechanism: under the strong assumption that variables $$X$$ and $$Y$$ are related by a bijective relation, the distribution of the cause $$P_X$$ is independent from the mechanism $$P_{Y | X}$$, while this independence does not hold in the opposite direction. This approach is also related to complexity-based methods on causal mechanisms.

Information Geometric Causal Inference (IGCI)

Methods exploiting the independence between cause and mechanism infer causal direction based on the principle that when $$X \to Y$$, the distribution of the cause $$P_X$$ should not contain information that can be useful to derive the conditional model $$Q_{Y | X}$$ on the data. This approach leverages the notion of complexity of causal mechanisms.

University of Michigan - Ann Arbor

Google

According to the type of functional involved, their fit scores, their complexity scores and their way to deal with the trade-off fit/complexity, three main families of methods emerge: 
 - Methods with a restricted class of mechanisms. 
 - Non-parametric methods. 
 - Methods exploiting the independence between cause and mechanism.

Main Algorithms for Cause-Effect Inference

In causal inference, the basic postulate that "the factorization of the joint density function $$P_{cause,effect}$$ into $$P_{cause} P_{effect|cause}$$ should lead to a simpler model than $$P_{effect} P_{cause|effect}$$" can be expressed using the Kolmogorov complexity framework: $$K(P_{cause}) + K(P_{effect|cause}) \overset{+}\leq K(P_{effect}) + K(P_{cause|effect})$$ This inequality derives from the postulate of algorithmic independence between the distribution of the cause $$P_{cause}$$ and the distribution of the causal mechanism $$P_{effect|cause}$$, stated by Janzing and Schölkopf as: $$I(P_{cause} : P_{effect|cause}) \overset{+}=0$$ where $$I(P_{cause} : P_{effect|cause})$$ denotes algorithmic mutual information. Since Kolmogorov complexity and algorithmic mutual information are not computable in practice, they have inspired practical implementations such as model selection with the Minimum Message Length (MML) principle and methods exploiting the independence between cause and mechanism.

Kolmogorov Complexity in Causal Inference

 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

Cause Effect Pairs in Machine Learning

Methods Exploiting the Independence between Cause and Mechanism

In causal inference, non-parametric methods do not restrict causal mechanisms to a specific class of functions $$\mathcal{F}$$. This flexibility allows them to better model complex, real-world data where the interaction between cause and noise may be non-linear or non-additive. While these methods often yield better empirical results than restricted-class approaches, the lack of explicit functional restrictions leads to a loss of theoretical identifiability guarantees. To recover causal direction, non-parametric methods typically rely on setting a smooth prior on the complexity of the causal mechanisms.

Non-Parametric Methods in Causal Inference

This class of causal inference methods relies on a restrictive class of admissible mechanisms $$\mathcal{F}$$ and focuses on identifiability results. The main idea is to show that in some cases there exist $$f\in \mathcal{F}$$ and $$Q_N$$ such that the hypothesis $$\mathcal{B}_{\mathcal{G}, f, Q_N}$$ holds in the causal direction $$\mathcal{G}$$ with respect to the observed data while there do not exist any $$f' \in \mathcal{F}$$ nor such $$Q'_N$$ that $$\mathcal{B}_{\mathcal{G'}, f', Q'_N}$$ holds in the opposite causal direction $$\mathcal{G'}$$.

Learn Before

Related

Learn After