Hitting the nail on its arrowhead, a.k.a. when does $X$ cause $Y$?

PoC Post Series

• PoC #8: Inferred Causation, $IC$, and ${IC}^*$
• ➡️ PoC #9: Potential, Genuine, Temporal Causes and Spurious Association
• PoC #10: Interventions and Identifiability

Time-agnostic Relationships

Causal inference in practice is often ambiguous: depending on the available data, we might only be able to identify the graph up to its Markov equivalence class. The presence of latent variables increases the space of possible structures; thus, it is worth defining a set of concepts to express the different variants of cause-effect relationships.

For this, we will start from $IC^*$ and will quantify the edges from a causal perspective. The algorithm results in a PDAG with four edge types, as discussed in PoC #8. We can think about those as expressing our “certainty” whether a relationship is causal:

1. $a\rightarrow^{*} b$: $a$ causes $b$
2. $a\rightarrow b$: $a$ potentially causes $b$
3. $a \leftrightarrow b$: $a$ and $b$ are spuriously associated
4. $a-b:$ we have no clue whether the relationship is causal or spurious (the direction is also unknown)

In the following, adjacency is used to emphasize that there is an edge between $a$ and $b$ (including $\leftrightarrow$, thus, stretching the traditional edge concept as this is a shorthand for a latent common cause $a\leftarrow L \rightarrow b$); whereas a context $S$ means a set of variable assignments (only w.r.t. the observed variables - latent values cannot be assigned as they are unobservable; $S$ can also be the empty set). Context-specific independence $(\perp_c)$ is a shorthand for $X \perp Y | Z, C=c$.

Time-agnosticity in the section title refers to the fact that the below concepts are general, they do not exploit temporal information.

That will be considered in the next section and will make our life easier.

From now on, we will only use edges as in DAGs (edge types 1 and 3, but without a marking).

Potential Cause

The first level of causally useful relationships fall into the category of potential causes - this can be thought as a necessary condition of $X$ causing $Y$. If $X$ is not a potential cause of $Y$, then $X$ cannot cause $Y$.

$X$ is a potential cause of $Y$ (that is inferable from $P$) if:

1. $X$ and $Y$ are dependent in every context ($X \not\perp_c Y$).
2. There exists a variable $Z$ and a context $S:$
   1. $X \perp Z | S$
   2. $Y \not\perp Z | S$

The definition requires adjacency (including latent common causes) to ensure that $X$ and $Y$ are dependent irrespective what we condition on.

That is, the definition excludes indirect causes - i.e., when $X$ influences $Y$ via observable mediator nodes such as $X\rightarrow Z \rightarrow Y$ or $X \leftrightarrow Z \leftrightarrow Y$.

The reason why the pair $X$ is considered a potential cause is the exclusion of $X\leftarrow Y$; thus leaving only the options:

• $X \rightarrow Y$
• $X \leftrightarrow Y$

Namely, if $X\leftarrow Y$ would be possible, then that would mean that $X \not\perp Z | S$ - as there is an edge, and so an active path, between $Y$ and $X$.

If $X$ is a potential cause of $Y$, then $X$ either causes $Y$ or they have the same parent. That is, potential causes are a superset of genuine causes and spurious associations.

Genuine Cause

Potential causes are useful to restrict the space of causes, but genuine causal relationships are the most important for us. For real causal relationships are more restrictive, we expect to have more conditions.

$X$ is the genuine cause of $Y$ if there exists a variable $Z$ such that either:

1. $X$ and $Y$ are dependent in any context ($X \not\perp_c Y $) and $\exists$ context $S$:
   1. $Z$ is a potential cause of $X$
   2. $Y \not\perp Z | S$
   3. $Y \perp Z | S \cup X$
2. $X$ and $Y$ are in a transitive closure of criterion 1.

Our ultimate goal is to identify genuine causes and we use the potential cause definition as a stepping stone. The trick here is that when reasoning about the causal relationship of $X$ and $Y$, a third variable $Z$ will be used as a potential cause.

By requiring $Z$ to be the potential cause of $X$, we imply either

• $Z \rightarrow X$ or
• $Z \leftrightarrow X$

Additionally, conditions 2 and 3 rule out $X \leftarrow Y$ and $X \leftrightarrow Y$ (context-specific dependence means there is an $X-Y$ edge), as they require that the effect of $Z$ on $Y$ is mediated by $X$ (otherwise, conditioning on $X$ could not block the path). Thus, the latent common cause scenario $X \leftrightarrow Y$ is infeasible.

The mention of the transitive closure means that $X$ is also a genuine cause of $Y$ if $X$ is a genuine cause of $W$ and $W$ is a genuine cause of $Y$.

Spurious Association

When a potential cause fails to fulfill the conditions of genuine causes, then the relationship is spurious.

$X$ and $Y$ are spuriously associated if

1. They are dependent in some context ($X \not\perp_c Y$) and
2. $\exists$ other variables $ Z_1, Z_2$ and contexts $S_1, S_2$:
   1. $X \not\perp Z_1 | S_1$
   2. $Y \perp Z_1 | S_1$
   3. $X \perp Z_2 | S_2$
   4. $Y \not\perp Z_2 | S_2$

The first requirement is that $X$ and $Y$ should be dependent in some context, as otherwise we could not think of them having any association between them.

The existence of other variables $Z_1, Z_2$ and contexts $S_1, S_2$ refers to other than $X$ and $Y$ for the variables and other than the context where $X$ and $Y$ are dependent for the contexts.

As you can see in the figure above, we cannot use $S_1$ or $S_2$ to make $X$ and $Y$ dependent - but conditioning on $Z_1$ and/or $Z_2$ makes $X \not\perp Y$.

When parsing the definition of spurious associations further, we can think of applying condition 2 (with its two requirements) of potential causes twice. $X \not\perp Z_1 | S_1$ and $Y \perp Z_1 | S_1$ rule out $X\leftarrow Y$, while $Y \not\perp Z_2 | S_2$ and $X \perp Z_2 | S_2$ exclude $X\rightarrow Y$. So the only remaining option is $X \leftrightarrow Y$, meaning that there is no causal influence between $X$ and $Y$.

Time-dependent Relationships

When we have temporal information (such as sensor measurements), the above definitions simplify. Intuitively, $X$ can only be a cause of $Y$ is $X$ precedes $Y$.

Potential Causation with Temporal Information

$X$ is a potential cause of $Y$ if:

1. $X$ precedes $Y$
2. $X$ and $Y$ are dependent in every context ($X \not\perp_c Y$).

The temporal information is used to exclude $X\leftarrow Y$. This agrees with our everyday intuition and makes causal inference simpler as there is no need for conditional independence tests with a third variable $Z$. The second condition requires the adjacency of $X$ and $Y$ (a latent common cause is still possible).

An alternative option to context-specific dependence that we will use for temporal genuine causes is requiring that

Context $S$ precedes $X$,

meaning that $S$ is not in the path $X-\dots-Y$ (as $S$ precedes $X$ precedes $Y$); thus, it is not the reason why $Z$ is dependent on $Y$ but not on $X$ (cf. potential causes). Note that temporal precedence is still required between $X$ and $Y$.

Genuine Causation with Temporal Information

$X$ is a genuine cause of $Y$ if there exists a variable $Z$ and a context $S$, both occurring before $X$, such that:

1. $X$ precedes $Y$
2. $Y \not\perp Z | S$
3. $Y \perp Z | S\cup X$

In this definition, temporal precedence requires $Z$ and $S$ preceding $X$ preceding $Y$ - so the only possible causal direction is $Z$ causing $X$ causing $Y$-, the two remaining conditions are the same as in the non-temporal definition. Note that the context-specific dependence of $X$ and $Y$ is not required - equivalently, we do not postulate in advance the existence of an $X-Y$ edge.

The context-dependence of $X$ and $Y$ is implied by the second and third conditions. They exclude both $Z$ and $S$ as a cause of $Y$, as otherwise it would not be possible for $X$ to block the dependence of $Z$ on $Y$.

Spurious Association with Temporal Information

$X$ and $Y$ are spuriously associated if $\exists S : X \not\perp Y |S$ and $\exists Z$ such that:

1. $X$ precedes $Y$
2. $Y \perp Z | S$
3. $X \not\perp Z | S$

• no genuine cause as that would require dependent Z and Y

In this case, condition 1 excludes $Y$ as a cause of $X$, while conditions 2 and 3 exclude $X$ as a cause of $Y$ (these are analogous to the sub-conditions of condition two in the definition of potential causes). Namely, having a genuine causal relationship would require $Y \not\perp Z | S$.

Summary

This time we made a deep dive into the taxonomy of causality zoo. Being able to understand the differences between genuine, potential causes, and spurious associations is crucial to reason about the output of a causal inference algorithm.