Pharmaceutical companies are interested in determining the influence of genes on each other. A dataset may consist of jointly recorded gene activities for $N$ pairs of genes $( X_k , Y_k ), k = 1, ⋯ N$ and for $n$ patients $S_k = \{( x_{k1} , y_{k 1 }), ⋯ , ( x_{ kn} , y_{kn} \} $. 

Determining which gene influences which other gene is a costly experimental process, so only a subset of pairs of genes are generally labeled with ground truth of causal relationship $G = g_k$ . 

The labeled dataset $\{( S_1 , g_1 ), ⋯ , ( S_k , g_k ), ⋯\}, k = 1⋯ N$ , is an empirical sample of the “mother distribution” over pairs of genes and their causal labels, which may be used as training data to obtain a classifier, such that predictions of unknown causal labels can be made on new pairs of genes.

University of Michigan - Ann Arbor

The meaning of “mother distribution” may seem mysterious. Here are few examples of causal discovery problems, which could clarify this notion. 
 - Example 1: Genomics
 - Example 2: Ecology
 - Example 3: Social Sciences

For notational simplicity we sometimes reformulate $P_{Π = π} ( X , Y )$ as $P ( X_π , Y_π )$, the distribution over two random variables indexed by $π$ . In the examples, $π$ is just a discrete index $k$ .

Examples of Mother Distribution

 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

Example 1: Genomics

A country conducts a census survey, including socio-demographic questions (age, gender, education, profession, number of children, salary, etc.). The question is to determine which factor influences which other factor. 

The dataset consists of joint answers from $n$ citizens for factor pairs $( X_k , Y_k ), k = 1, ⋯ N : S_k = \{( x_{k 1} , y_{k 1} ), ⋯ , ( x_{kn} , y_{kn} \} $. Prior knowledge allows us to determine the ground truth for a few pairs $( X_k , Y_k )$ (for example age can influence wages and not vice-versa). 

This constitutes an empirical “mother distribution” of pairs $\{( S_1 , g_1 ), ⋯ , ( S_k , g_k ), ⋯ , ( S_N , g_N )\} $ from which we might train a classifier to label other similar pairs with their causal direction.

Example 3: Social Science

To illustrate a mother distribution in causal discovery, consider an ecological study conducted to understand a forest ecosystem and preserve biodiversity. The objective is to determine which factor influences which other factor. The dataset consists of $$n$$ jointly recorded values of many variables $$(X_k, Y_k)$$, where $$k = 1, \dots, N$$, in different locations (measuring factors such as soil, humidity, lighting, or the presence of certain plants or animals). This results in samples $$S_k = \{ (x_{k1}, y_{k1}), \dots, (x_{kn}, y_{kn}) \}$$. Determining which factor influences another is a complicated process; however, prior knowledge of physics and biology may allow researchers to label some pairs with ground truth $$G = g_k$$ (for example, the aspect of a slope can influence hill shade, but not vice versa). The labeled dataset thus obtained, $$\{ (S_1, g_1), \dots, (S_N, g_N) \}$$, is an empirical sample of the "mother distribution". It is hoped that more pairs can be automatically labeled using a classifier trained on such data, provided the mechanisms of the other pairs bear some similarity.

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