A selection bias corresponds to an unobserved variable on which the two variables $$X$$ and $$Y$$ were implicitly conditioned: $$X\rightarrow W \leftarrow Y$$. In the generative cause-effect problem setting, it is assumed that the sample $$\{(x_i,y_i)\}_{i=1}^n$$, corresponding to the variables $$X$$ and $$Y$$, was collected without selection bias.

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Let $$X$$ and $$Y$$ be two one-dimensional random variables in $$R$$ with joint distribution $$P_{X,Y}$$. The main assumptions usually involved in this problem setting include:

- **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 in Generative Cause-Effect Models

 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

In the probelm setting, the time for which the observed data have been collected is not available, so time is not explicitly modeled even though it is assumed that causes precede their effects.

Assumption 2: Time

Under this assumption it is assumed that X and Y are not common consequences of the same hidden variables ( X ← Z → Y ). 

This causal sufficiency assumption is made by many methods of the cause-effect pair literature as it allows to considerably simplify the cause-effect pair problem. However there are many realistic cases where there are potential confounding effects that can affect both variables (the most typical examples of confounding factors are the age or the gender in epidemiological studies).

Assumption 5: Causal Sufficiency

The orders of magnitude of the variable as well as their physical units are not considered in the problem. In most methods, the variables X and Y are re-scaled with zero mean and unit variance.

Assumption 9: Variable Units

In the context of cause-effect pairs in machine learning, it is generally assumed that there is no measurement noise. Measurement noise may occur if, for example, altitude is not measured precisely, with its noisy version denoted as $$	ilde{Z}$$. However, the variable temperature $$T$$ is still a function of the original variable $$Z$$ that is not corrupted by measurement noise. This is related to a cause-effect pair problem between $$T$$ and $$	ilde{Z}$$ in the presence of a latent hidden variable, which is the original $$Z$$.

Assumption 8: Measurement Noise

A feedback loop occurs when both $$X$$ causes $$Y$$ and $$Y$$ causes $$X$$ ($$X \Leftrightarrow Y$$). Even if the notion of time is not explicitly modeled, this situation can arise in cross-sectional studies where data are collected over a period of time (e.g., temperature and the amount of sea ice). In the generative cause-effect problem setting, it is assumed that feedback loops are excluded. Therefore, the causal graph between $$X$$ and $$Y$$ conforms to the literature on Directed Acyclic Graphs (DAGs) restricted to two variables.

Assumption 6: Exclusion of Feedback Loops

In generative cause-effect models, it is assumed that the given observations $$\mathcal{D} = \{(x_i,y_i)\}_{i=1}^n$$ of the random variables $$X$$ and $$Y$$ are independent and identically distributed, drawn from the joint distribution $$P_{X,Y}$$.

Independent and Identically Distributed Samples Assumption in Generative Cause-Effect Models

In generative cause-effect models, the constraint relation assumption excludes cases where a dependency with an undirected relation $$X - Y$$ arises because two variables are linked by a logical or mathematical formula, rather than a causal relation. When variables are linked by an equivalent mathematical relation, one can be deduced from the other without any notion of an underlying causal system generating one variable from the other. Such deterministic, non-causal dependencies are not considered in the standard generative cause-effect problem setting.

Constraint Relation Assumption in Generative Cause-Effect Models

Selection Bias Assumption in Generative Cause-Effect Models

The classical faithfulness assumption used in graphical causal inference, transposed for two variables, states that if there is a causal relation between $$X$$ and $$Y$$, the two variables are not independent.

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