In order to model such continuous underlying bivariate generative process $\mathcal{B}_{\mathcal{G}, f, P_N}$ which is assumed to have generated the data, the author introduce the notion of candidate model $\hat{\mathcal{B}}_{\hat{\mathcal{G}}, \hat{f}, Q_N}$ described by: 
 - a structure defined by a causal orientation X → Y or Y → X 
 - its estimated mechanisms modeled by $\hat{f}$
 -  an estimated distribution of noise $Q_N$ 

Candidate Bivariate Models

In order to evaluate the quality of a candidate generative model, we introduce a fit score $S_{\hat{\mathcal{B}}} (\theta)$ of a candidate model $\hat{\mathcal{B}}_{\hat{\mathcal{G}}, \hat{f}, Q_N}(\theta)$. This score has been implemented in different ways in the literature. It has always the property to be minimal when P = Q (perfect fit in the large sample limit).

Here are some example of scores:
 - Log-Likelihood Parametric Scores
 - Implicit Fit Score Computed as an Independence Score Between Cause and Noise Variable
 - Non-parametric Scores

Model Fitness Score

The authors introduce a term of complexity of a candidate bivariate model $\hat{\mathcal{B}}_{\hat{\mathcal{G}}, \hat{f}, Q_N}(\theta)$ that we note $C_{\hat{B}}(\theta)$. This complexity notion refers to a simplicity prior on the underlying data generative process. It has been handled in different ways in the literature from fix a class of admissible mechanisms to more flexible criteria.

Th ecomplexity term can be defined through different approaches, including:
 - Explicit Class of Admissible Mechanisms
 - Kolmogorov Complexity

Complexity of the Model

Since we have defined the complexity $C_{\hat{\mathcal{B}}}(\theta)$ and the fit $S_{\hat{\mathcal{B}}}(\theta)$ of a candidate bivariate model parameterized by the vector of parameters θ . From a general point of view, we can now frame the model selection as a bi-objective trade-off with a Pareto front of optimal models.

This general bi-objective trade-off between fit and complexity is very general in science. It has been seen from two different angles. Some of the modeling approaches favor models of lower complexity, while others favor models with better explanatory power:
 - KISS Approach (Keep It Simple, Stupid)
 - KIDS Approach (Keep It Descriptive, Stupid)

Bi-objective Trade-Off between Fit and Complexity

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 .

University of Michigan - Ann Arbor

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.

Generative Models for Cause-Effect Problem

 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

$X$ 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

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

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