$X$ and $Y$ are two one-dimensional random variables in $R$ with joint distribution $P_{X,Y}$. 

The main assumptions usually involved in the problem setting includes:
 - Assumption 1:  Identically and Distributed Samples
 - Assumption 2: Time
 - Assumption 3: Faithfulness
 - Assumption 4: Selection Bias
 - Assumption 5: Causal Sufficiency
 - Assumption 6: Feedback Loops
 - Assumption 7: Constraint Relation
 - Assumption 8: Measurement Noise
 - Assumption 9: Variable Units

Notations and Assumptions

A bivariate structural model noted is a triplet $\mathcal{B}_{\mathcal{G}, f, P_N}$, where $\mathcal{G}$ is the causal graph X → Y or Y → X , $f = ( f_X , f_Y )$ is a couple of possibly nonlinear functions and $( N_X , N_Y )$ are two independent random variables drawn according to continuous distribution $P_N = (P_{N_X}, P_{N_Y})$, such that: 
 - $X := f_X ( N_X )$ and $Y := f_Y ( X , N_Y )$ if X → Y 
 - $Y := f_Y ( N_Y )$ and $X := f_X ( Y , N_X )$ if Y → X

Bivariate Structural Model

The determination of the best explainable model is based in the literature on these main lines: 
 - Different candidate bivariate models (hypotheses) are evaluated in both directions.
 - For each candidate model one evaluates a score monitoring the trade-off between the fit of the model (meaning its adequacy to the observational data) and the complexity of the mechanisms involved. 
 - The model with the best score is returned, with its corresponding causal arrow X → Y or Y → X .

Determination of the Best Explainable Model

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

The cause-effect inference problem is relatively new in the Machine Learning literature and there are still a lot of open problems to be addressed in order to build a robust tool that can be used by the practitioner to confirm for example if a given treatment has an impact or not on a given disease or to discover if a gene has a regulatory power on an other one. Here are the open problems:
 - Relax the Causal Sufficiency Assumption
 - Need for Real Datasets of a Big Size
 - Biased Assessment Due to Artifacts in Data
 - Extension of the Generative Approach for Categorical Variables
 - Extension of the Pairwise Settnng for Complete Graph Inference
 - Computational Complexity Limitations
 - Relax Restrictive Assumptions on Causal Mechanisms

Open Problems and Extensions

Generative algorithms solve the cause-effect pair problem from a reverse engineering perspective: given the available measurements $\{ (x_i,y_i) \}_{i=1}^n $ of the two variables $X$ and $Y$, the task is to discover the underlying causal process that led the variables to take the values they have.

Such methods allow gaining not only a clue about the causal direction but also information about the mechanism itself, making causal discovery less of a black box decision process.

University of Michigan - Ann Arbor

Cause-effect methods can be divided into two types:
 - Generative Models 
 - Discriminative Models

Cause-Effect 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

Cause Effect Pairs in Machine Learning

Discriiminative algorithms do not attempt to reverse engineer the data generating process, they merely classify the empirical joint distribution of two variables $X$ and $Y$ (a scatter plot) as being a $X$ cause $Y$ or a $Y$ cause $X$ (or neither).

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