Draft:Causal equality notation

In nature and human societies, many phenomena have causal relationships where one phenomenon A (a cause) impacts another phenomenon B (an effect). Establishing causal relationships is the aim of many scientific studies across fields ranging from physics and biology to social sciences and economics. It can also be considered a prerequisite for effective policy making.

Mathematical expressions are often used to model relationships between phenomena. Causal relationships can be encoded within mathematical expressions using causal equalities. Examples below illustrate the use of causal equality notation. Additional notation is provided in the table of causal equalities below.

Example

Physics example

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

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

where $I$ represents intensity of sunlight (J$\cdot$s$^{-1}$$\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$\cdot$s$^{-2}$), and $h$ represents the height the rock is lifted (m). To keep track of the causal nature of this expression, an arrow may be placed over the equals sign pointing from the cause to the effect:

$I\times t~{\overset {\rightarrow }{=}}~m\times g\times h~$.

In this example, the sunlight causes the stone to rise, not the other way around; lifting the stone will not result in turning on the sun to illuminate the solar panel. The arrow graphically represents the fact that the sunlight intensity $I$ causes the rock's height $h$ to change, but changing the rock's height $h$ does not affect the intensity $I$ of the sunlight incident on the solar panel.

Table of causal equality notation

Other forms of causal relationships also exist. For instance, two quantities $f(y)$ and $g(y)$ can both be caused by a confounding variable $y$, but not by each other. Imagine a garbage strike in a large city, $y$, causes an increase in the smell of garbage, $f(y)$ and an increase in the rat population $g(y)$. Even though $g(y)$ does not cause $f(y)$ and vice-versa, one can write an equation relating $g(y)$ and $f(y)$. The following table contains notation representing a variety of ways that $y$, $f(y)$ and $g(y)$ may be related to each other..

Causal equality notation

Symbolic expression	Defined relationships between $y$, $f(y)$ and $g(y)$
$y~{\overset {\rightarrow }{=}}~f\left(y\right)$	$f(y)$ is caused by $y$. The dependent variable is $f(y)$. The independent variable is $y$.
$y~{\overset {\leftarrow }{=}}~f\left(y\right)$	$y$ is caused by $f(y)$. The independent variable is $f(y)$. The dependent variable is $y$.
$y~{\overset {\leftrightarrow }{=}}~f\left(y\right)$	$f(y)$ and $y$ are mutually dependent, or bi-directionally causal.
$y~{\overset {\rightarrow }{=}}~f\left(y\right)$ $y~{\overset {\rightarrow }{=}}~g\left(y\right)$	Correlation: $f(y)$ and $g(y)$ are both caused by $y$: $f(y)~{\overset {\curvearrowleft \curvearrowright }{=}}~g(y)$. If a bi-directional causal relationship may exist, but this is not yet established, the notation $f\left(y\right)~{\overset {\curvearrowleft ?\curvearrowright }{=}}~g\left(y\right)$ can be used.
$y~{\overset {\rightarrow }{=}}~f\left(y\right)$ $y~{\overset {\leftarrow }{=}}~g\left(y\right)$	$g(y)$ causes $y~$ which in turn causes $f(y)$: $g(y)~{\overset {\rightarrow }{=}}~f(y)$
$y~{\overset {\leftarrow }{=}}~f\left(y\right)$ $y~{\overset {\rightarrow }{=}}~g\left(y\right)$	$f(y)$ causes $y~$ which in turn causes $g(y)$: $g(y)~{\overset {\leftarrow }{=}}~f(y)$.
$y~{\overset {\leftarrow }{=}}~f\left(y\right)$ $y~{\overset {\leftarrow }{=}}~g\left(y\right)$	Uncertainty/bicausal: $y~$ can be caused by $f(y)$ or $g(y)$: $f\left(y\right)~{\overset {\curvearrowright \curvearrowleft }{=}}~g\left(y\right)$, or $g(y)~{\overset {>-<}{=}}~f(y)$
$y~{\overset {\leftrightarrow }{=}}~f\left(y\right)$ $y~{\overset {\rightarrow }{=}}~g\left(y\right)$	$f(y)$ and $y$ are bi-directionally causal. $g(y)$ is caused by $f(y)$
$y~{\overset {\rightarrow }{=}}~f\left(y\right)$ $y~{\overset {\leftrightarrow }{=}}~g\left(y\right)$	$g(y)$ and $y$ are bi-directionally causal. $f(y)$ is caused by $g(y)$
$y~{\overset {\leftrightarrow }{=}}~f\left(y\right)$ $y~{\overset {\leftrightarrow }{=}}~g\left(y\right)$	$g(y)$ causes $f(y)$ and $f(y)~$ causes $g(y)$: $f(y)~{\overset {\leftrightarrow }{=}}~g(y)~$. $g(y)$ and $f(y)$ are bi-directionally causal.
$y_{1}~{\overset {arb.}{=}}~f\left(y_{1}\right)$ $y_{2}~{\overset {arb.}{=}}~g\left(y_{2}\right)$	Mismatched indices indicate that for any arbitrary causal relation between $y_{1}$ and $f(y_{1})$ or $y_{2}$ and $g(y_{2})$, $f(y_{1})$ and $g(y_{2})$ cannot be related.

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

Other similar conventions

Do-calculus, and specifically the do operator, uses alternative notation to deal with similar concepts to those discussed above, but in the language of probability. A notation used in do-calculus is, for instance:

$P(Y|do(X))=P(Y)~$,

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 $Y$.

References

Category:Causality Category:Notation