Causal notation#Notations

What follows does not necessarily assume the convention whereby $y$ denotes an independent variable, and

$f(y)$ denotes a function of the independent variable $y$. Instead, $y$ and $f(y)$ denote two quantities with an a priori unknown causal relationship, which can be related by a mathematical expression.

Imagine the number of days of weather below one degrees Celsius, $y$, causes ice to form on a lake, $f(y)$, and it causes bears to go into hibernation $g(y)$. Even though $g(y)$ does not cause $f(y)$ and vice versa, one can write an equation relating $g(y)$ and $f(y)$. This equation may be used to successfully calculate the number of hibernating bears $g(y)$, given the surface area of the lake covered by ice. However, melting the ice in a region of the lake by pouring salt onto it, will not cause bears to come out of hibernation. Nor will waking the bears by physically disturbing them cause the ice to melt. In this case the two quantities $f(y)$ and $g(y)$ are both caused by a confounding variable $y$ (the outdoor temperature), but not by each other. $f(y)$ and $g(y)$ are related by correlation without causation.

Suppose an ideal solar-powered system is built such that if it is sunny and the sun provides an intensity $I$ of $100$ watts incident on a $1$m$^2$ solar panel for $10~$ seconds, an electric motor raises a $2$kg stone by $50$ meters, $h(I)$. More generally, we assume the system is described by the following expression:

$I\; \backslash times\; A\; \backslash times\; t\; =\; m\; \backslash times\; g\; \backslash times\; h\; ~$,

where $I$ represents intensity of sunlight (J$\backslash cdot$s$^\{-1\}$$\backslash cdot$m$^\{-2\}$), $A$ is the surface area of the solar panel (m$^\{2\}$), $t$ represents time (s), $m$ represents mass (kg), $g$ represents the acceleration due to Earth's gravity ($9.8$ m$\backslash cdot$s$^\{-2\}$), and $h$ represents the height the rock is lifted (m).

In this example, the fact that it is sunny and there is a light intensity $I$, causes the stone to rise $h(I)$, not the other way around; lifting the stone (increasing $h(I)$) will not result in turning on the sun to illuminate the solar panel (an increase in $I$). The causal relationship between $I$ and $h(I)$ is unidirectional.

Smoking, $f(y)$, and exposure to asbestos, $g(y)$, are both known causes of cancer, $y$. One can write an equation $f(y)\; =\; g(y)$ to describe an equivalent carcinogenicity between how many cigarettes a person smokes, $f(y)$, and how many grams of asbestos a person inhales, $g(y)$. Here, neither $f(y)$ causes $g(y)$ nor $g(y)$ causes $f(y)$, but they both have a common outcome.

Consider a barter-based economy where the number of cows $C$ one owns has value measured in a standard currency of chickens, $y$. Additionally, the number of barrels of oil $B$ one owns has value which can be measured in chickens, $y$. If a marketplace exists where cows can be traded for chickens which can in turn be traded for barrels of oil, one can write an equation $C(y)\; =\; B(y)$ to describe the value relationship between cows $C$ and barrels of oil $B$. Suppose an individual in this economy always keeps half of their value in the form of cows and the other half in the form of barrels of oil. Then, increasing their number of cows $C(y)$ by offering them 4 cows, will eventually lead to an increase in their number of barrels of oil $B(y)$, or vice versa. In this case, the mathematical equality $C(y)\; =\; B(y)$ describes a bidirectional causal relationship.

In chemistry, many chemical reactions are reversible and described using equations which tend towards a dynamic chemical equilibrium. In these reactions, adding a reactant or a product causes the reaction to occur producing more product, or more reactant, respectively. It is standard to draw “harpoon-type” arrows in place of an equals sign, {{eqm}}, to denote the reversible nature of the reaction and the dynamic causal relationship between reactants and products.

Do-calculus, and specifically the do operator, is used to describe causal relationships in the language of probability. A notation used in do-calculus is, for instance:{{Hitchcock, Christopher, "Causal Models", The Stanford Encyclopedia of Philosophy (Spring 2023 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL = }}

:$P(Y|do(X))\; =\; P(Y)~$,

which can be read as: “the probability of $Y$ given that you do $X$”. The expression above describes the case where $Y$ is independent of anything done to $X$. It specifies that there is no unidirectional causal relationship where $X$ causes $Y$.

A causal diagram consists of a set of nodes which may or may not be interlinked by arrows. Arrows between nodes denote causal relationships with the arrow pointing from the cause to the effect. There exist several forms of causal diagrams including Ishikawa diagrams, directed acyclic graphs, causal loop diagrams, and why-because graphs (WBGs). The image below shows a partial why-because graph used to analyze the capsizing of the Herald of Free Enterprise. Image:Herald of Free Enterprise WBG.png ]]

Junction patterns can be used to describe the graph structure of Bayesian networks. Three possible patterns allowed in a 3-node directed acyclic graph (DAG) include:

{{COI|reason=The editor adding this section appears to be the author of the cited paper, raising concerns about self-promotion and undue weight.|date=February 2025}}

Various forms of causal relationships exist. For instance, two quantities $a(s)$ and $b(s)$ can both be caused by a confounding variable $s$, but not by each other. Imagine a garbage strike in a large city, $s$, causes an increase in the smell of garbage, $a(s)$ and an increase in the rat population $b(s)$. Even though $b(s)$ does not cause $a(s)$ and vice versa, one can write an equation relating $b(s)$ and $a(s)$. The following table contains notation representing a variety of ways that $s$, $a(s)$ and $b(s)$ may be related to each other. Van Horne N. and Mukherjee M. Improved description of trapped ions as a modular electromechanical system, J. Appl. Phys. 135, 154401 (2024)

It should be assumed that a relationship between two equations with identical senses of causality (such as $s\; ~\backslash overset\{\backslash rightarrow\}\{=\}~\; a\backslash left(s\backslash right)$, and $s\; ~\backslash overset\{\backslash rightarrow\}\{=\}~\; b\backslash left(s\backslash right)$) is one of pure correlation unless both expressions are proven to be bi-directional causal equalities. In that case, the overall causal relationship between $b(s)$ and $a(s)$ is bi-directionally causal.

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Category:Notation