(Winston, Chaffin, and Herrmann 1987)). In other words, they are two
interpretations of the same content. The difference is important: the
relationship is bidirectional, and one cannot directly intervene on one
without simultaneously intervening on the other.

When comparing variables at different levels of granularity, a hierarchy
arises, which we call the _constitutional hierarchy_. That is, any phenomenon
can be viewed across another dimension that determines the level of
granularity of interpreting the variables. This is illustrated in Fig. 15.
Variables can be organized by what constitutes what, and, in a proper
abstraction, causal properties should be preserved across different levels of
granularity. Generally speaking, most studies focus on one specific level of
abstraction, so the task of choosing the most appropriate level of granularity
can be important. This is precisely the problem of choosing intervariable
clusters, as the coarseness of the clusters induces a natural interpretation
of the variables.

If the data scientist finds themselves in a situation where they have to
construct the causal diagram themselves, or if a provided causal diagram is
not at the right level of abstraction, then refining the abstraction level
becomes a nontrivial task. The decision of whether to cluster variables
together depend on various factors, which we list and elaborate in the sequel:

  1. C1.

_Non-causal_ relationships should not be visible.

  2. C2.

The resulting clustering should be admissible.

  3. C3.

The queries of interest should be answerable.

  4. C4.

The queries of interest should be identifiable.

  5. C5.

The result should be as coarse as possible.

Each of these conditions can be formalized to enable a systematic discussion
on how to choose an appropriate intervariable clustering.

Condition C1. When considering models at extremely low levels of abstraction,
there may be too much detail to properly label every relationship as a causal
one.202020We do not provide an exact definition of a causal relationship, as
this is a deeply philosophical topic that is out of the scope of this work.
Still, we acknowledge that there may be cases where two low-level variables
are be related in a way that is not well-defined with respect to interventions
in an SCM. We use the term “non-causal” as a bucket term for all such cases
and make no assumptions about the natures of these relationships. For example,
like in the top level of Fig. 15, at the atom or molecule level, there are
interactions between particles that are studied in the physical sciences such
as bonds. That is, two particles may have linked behavior, but it is not
accurate to call such a relationship causal. As another example, suppose we
consider images at the pixel level. In many tasks, one may be interested in
the local dependences between pixels, and it may therefore be reasonable to
model pixels using an undirected model (e.g., like a Markov random field).
Once again, the relationship between the pixels may not necessarily be
considered causal.

Figure 15: A visual example of the constitution hierarchy. Green edges
indicate causal relations, blue edges indicate constitutional relations, and
black edges are undefined, low-level relationships. The same objects are shown
at three levels of granularity: molecules (top), parts (middle), and complete
objects (bottom). A causal relationship might indicate that a car is the cause
of Bob’s injury, but it would be inaccurate to say that the molecules of a car
“cause” the car. A particular study should focus on the variables from one
specific level of granularity, and choosing the most appropriate one is the
goal of the problem of choosing intervariable clusters.

We note importantly that “non-causal” in this context does not refer to
spurious effects, nor does it mean anti-causal in the sense that the direction
of causality is reversed, as these are still well-defined from a causal
perspective. For example, in a scenario modeled by Fig. 16, the relationship
between $X$ and $Y$ is not considered a non-causal relationship. Despite $Y$
not causing $X$, their relationship is still considered causal due to the
cause from $X$ from $Y$. Further, although there is spurious correlation
through the unobserved confounding, this is still considered a causal type of
relationship since there is some causal effect from the unobserved confounder
to $X$ and $Y$. “Non-causal” relationships refer to relationships between
variables that cannot be defined in a causal manner. When considering non-
causal relationships, the SCM framework is no longer compatible since it is
not defined how to model such relationships. Nonetheless, the framework
presented in this paper can still be used as long as C1 is satisfied.

The term “non-causal” is very broad and could encompass many different types
of relationships. Studying specific types of non-causal relationships is out
of the scope of this work, so we will treat these relationships in the same
way. For every subset of variables (say $\mathbf{Z}\subseteq\mathbf{V}_{L}$)
related through some non-causal relationship, we will assume that there is
some function $f_{\mathbf{Z}}$ that takes as input variables from
$\mathbf{U}_{L}$ and $\mathbf{V}_{L}$ and maps it jointly to the space of
$\mathbf{Z}$. We note that this is a general encoding of such relationships
that makes minimal assumptions. For example, perhaps the relationship between
three variables, $X$, $Z$, and $Y$ can be described using a Markov random
field like shown in Fig. 17, where $X\mathbin{\mathchoice{\hbox
to0.0pt{\hbox{\set@color$\displaystyle\perp$}\hss}\kern 3.46875pt{}\kern
3.46875pt\hbox{\set@color$\displaystyle\perp$}}{\hbox
to0.0pt{\hbox{\set@color$\textstyle\perp$}\hss}\kern 3.46875pt{}\kern
3.46875pt\hbox{\set@color$\textstyle\perp$}}{\hbox
to0.0pt{\hbox{\set@color$\scriptstyle\perp$}\hss}\kern 2.36812pt{}\kern
2.36812pt\hbox{\set@color$\scriptstyle\perp$}}{\hbox
to0.0pt{\hbox{\set@color$\scriptscriptstyle\perp$}\hss}\kern 1.63437pt{}\kern
1.63437pt\hbox{\set@color$\scriptscriptstyle\perp$}}}Y\mid Z$. Our assumption
states that the behavior of these three variables can be abstracted into one
function $f_{\\{X,Z,Y\\}}$ which outputs values of $X$, $Z$, and $Y$,
therefore losing the independence information. (It is possible, however, that
these could be implicitly encoded through the intravariable clusters). These
non-causal relationships can be expressed graphically.

######  Definition 21 (Non-causal Graph).

Let $\mathbf{V}_{L}$ be a set of variables and
$\overline{\mathcal{G}}=\langle\mathbf{V}_{L},\mathbf{E}\rangle$ be an
undirected graph with nodes representing $\mathbf{V}_{L}$. Then, an edge
$(V_{1},V_{2})$ is in $\mathbf{E}$ if and only if there is some non-causal
relationship between $V_{1}$ and $V_{2}$. $\blacksquare$

We will assume that, in settings which intervariable clusters must be learned,
we are given knowledge of the existence of non-causal relationships through a
non-causal graph of the lowest level variables $\mathbf{V}_{L}$. This
information could be acquired simply as an assumption of the user, as they are
typically only present at the lowest levels of abstraction (e.g. the
relationship between pixels in an image or between atoms of an object). Under
this assumption, distributions over $\mathbf{V}_{L}$ can be treated as if
variables in the same connected component in $\overline{\mathcal{G}}$ share a
common cause, and they can be factorized as such.

Condition C2. By definition of admissibility, two variables should not be in
the same intervariable cluster if doing so forms a cycle in the order of the
functions or graph. This condition can be verified given causal diagram
$\mathcal{G}$ of the low level variables $\mathbf{V}_{L}$.

Condition C3. C3 and C4 depend on the user’s needs based on the queries of
interest, $\mathbb{Q}$, in downstream tasks. The ideal level of abstraction
can be determined by the groupings of variables in these queries. C3
essentially enforces that the level of abstraction should be kept low enough
to retain the ability to differentiate the nuances between individual
variables of interest in the queries. For example, if one would like to study
the causal effect of a drug $X$ on recovery rate $Y$, but $Y$ was clustered
with another variable $Z$ representing blood pressure, then it no longer
becomes possible to answer queries specifically about $Y$ without including
$Z$. This is formally defined below.

######  Definition 22 (Cluster Answerability).

A counterfactual query
$Q=P^{\mathcal{M}_{L}}(\mathbf{Y}_{1[\mathbf{x}_{1}]},\mathbf{Y}_{2[\mathbf{x}_{2}]},\dots)$
is answerable from intervariable clusters $\mathbb{C}$ if and only if for all
$\mathbf{Y}_{i}$ (and $\mathbf{X}_{i}$), there exists
$\mathbb{C}_{i}\subseteq\mathbb{C}$ such that
$\mathbf{Y}_{i}=\bigcup_{\mathbf{C}\in\mathbb{C}_{i}}\mathbf{C}$.
$\blacksquare$

In other words, all queries should be written in terms of unions of clusters.

Condition C4. Even if answerability is not violated, clustering variables
together or projecting them out will always result in a loss of information.
Such a loss may affect the identifiability of the queries. Ideally, one would
not want to drop information that would change the status of an already
identifiable query, and this is enforced by C4.

Condition C5. Finally, C5 is usually desirable following the idea of Occam’s
razor, i.e., all else being equal, simplicity should be preferred. We define
coarseness as follows.

######  Definition 23 (Coarseness).

Let $\mathbb{C}_{1}$ and $\mathbb{C}_{2}$ be two intervariable clusterings of
$\mathbf{V}_{L}$. $\mathbb{C}_{1}$ is said to be coarser than $\mathbb{C}_{2}$
(equivalently, $\mathbb{C}_{2}$ is finer than $\mathbb{C}_{1}$) if and only if
for every $\mathbf{C}_{2}\in\mathbb{C}_{2}$, either
$\mathbf{C}_{2}\cap\bigcup_{\mathbf{C}_{1}\in\mathbb{C}_{1}}\mathbf{C}_{1}=\emptyset$,
or there exists $\mathbf{C}_{1}\in\mathbb{C}_{1}$ such that
$\mathbf{C}_{2}\subseteq\mathbf{C}_{1}$.

With this definition, the goal is to find a maximally coarse clustering
$\mathbb{C}$ that satisfies conditions 1-4. That is, there should not exist a
coarser clustering $\mathbb{C}^{\prime}$ that also satisfies the conditions,
although the coarsest clustering is not necessarily unique.

It turns out that the set of all possible maximally course clusterings can be
“bounded” in a sense using C1, C2, and C3. The “minimally” coarse clustering
can be described with the following lemma.

######  Lemma 7.

Let $\overline{\mathcal{G}}$ be a non-causal graph over $\mathbf{V}_{L}$. Let
$\mathbb{C}_{\min}$ be the intervariable clustering of $\mathbf{V}_{L}$
composed of the connected components of $\overline{\mathcal{G}}$. Then, any
clustering $\mathbb{C}^{\prime}$ violates condition C1 if and only if it is
not equal to or coarser than $\mathbb{C}_{\min}$. $\blacksquare$

###### Proof.

We first note that $\mathbb{C}_{\min}$ does not violate condition C1 because
by construction, if there exists a noncausal edge between $V_{1}$ and $V_{2}$
in $\overline{\mathcal{G}}$, then they must be in the same cluster. Any
coarser clustering has the same property.

If $\mathbb{C}^{\prime}$ is not equal to or coarser than $\mathbb{C}_{\min}$,
that means that there exists $\mathbf{C}\in\mathbb{C}_{\min}$ such that
$\mathbf{C}\not\subseteq\mathbf{C}^{\prime}$ for all
$\mathbf{C}^{\prime}\in\mathbb{C}^{\prime}$. This implies that there exists
$V_{1},V_{2}\in\mathbf{C}$ such that belong to different clusters
$\mathbf{C}_{1}^{\prime}$ and $\mathbf{C}_{2}^{\prime}$ in
$\mathbb{C}^{\prime}$. However, since $V_{1}$ and $V_{2}$ were in the same
cluster in $\mathbb{C}_{\min}$, there must exist some non-causal path from
$V_{1}$ to $V_{2}$ in $\overline{\mathcal{G}}$. This means that there is a
noncausal connection between clusters $\mathbf{C}_{1}^{\prime}$ and
$\mathbf{C}_{2}^{\prime}$, so $\mathbb{C}^{\prime}$ violates condition C1. ∎

Intuitively, the minimally coarse clustering must at least cover the connected
components of $\overline{\mathcal{G}}$ to abstract away the non-causal
relations and satisfy condition C1. Additionally, admissibility (C2) adds
another constraint.

######  Lemma 8.

Let $\mathbb{C}$ be a set of intervariable clusters that are not admissible
w.r.t. $\mathcal{G}$. Let $\mathbb{C}^{*}$ be the set of clusters that is
coarser than $\mathbb{C}$ such that $\mathbf{C}_{1}$ and $\mathbf{C}_{2}$ in
$\mathbb{C}$ are merged in $\mathbb{C}^{*}$ if and only if $\mathbf{C}_{1}$
and $\mathbf{C}_{2}$ are in a cycle in $\mathcal{G}_{\mathbb{C}}$. Then, any
set of clusters that are coarser than $\mathbb{C}$ and are admissible w.r.t.
$\mathcal{G}$ are equal to or coarser than $\mathbb{C}^{*}$ $\blacksquare$

###### Proof.

First, we note that $\mathbb{C}^{*}$ is admissible w.r.t. $\mathcal{G}$ by
construction, since all cycles have been merged. If $\mathbb{C}^{\prime}$ is a
clustering that is coarser than $\mathbb{C}$ but not $\mathbb{C}^{*}$, then
there must exist $\mathbf{C}^{*}\in\mathbb{C}^{*}$ such that
$\mathbf{C}^{*}\subsetneq\mathbf{C}^{\prime}$ for all
$\mathbf{C}^{\prime}\in\mathbb{C}^{\prime}$. Then, there must exist
$V_{1},V_{2}\in\mathbf{C}^{*}$ such that $V_{1}$ and $V_{2}$ are in different
clusters in $\mathbf{C}_{1},\mathbf{C}_{2}\in\mathbb{C}^{\prime}$. However,
this implies that there is still a cycle between $\mathbf{C}_{1}$ and
$\mathbf{C}_{2}$ in $\mathcal{G}_{\mathbb{C}^{\prime}}$, breaking
admissibility. ∎

$X$$Y$ Figure 16: Example of spurious relationships. $X$$Z$$Y$ Figure 17:
Example of non-causal graph.

$A$$T$$G$$H_{1}$$H_{2}$$O_{1}$$O_{2}$$O_{3}$$O_{4}$$P_{1}$$P_{2}$$P_{3}$$P_{4}$
(a) The full set of variables without clustering. Not a valid choice because
condition C1 is violated (red noncausal relationships between $O$ and $P$
variables.

$A$$T$$G$$H_{1}$$H_{2}$$H_{3}$$I$ (b) Graph with $O_{1},O_{2},O_{3},O_{4}$
clustered into $H_{3}$ and $P_{1},P_{2},P_{3},P_{4}$ clustered into $I$. This
cluster is the $\mathbb{C}_{\min}$ obtained from Lem. 7 and is a valid
clustering that satisfies C1-4.

$A$$G$$M$$I$ (c) Graph with $H_{1},H_{2},H_{3}$ clustered into $M$. This
cluster is also valid and is coarser than the graph to the left.

$X$$I$ (d) Graph with $A,G,M$ clustered into $X$. This cluster is no longer
valid because C3 is violated. The queries are no longer answerable.

Figure 18: Graphs for Example 16. Blue outline illustrates changes in
clustering between each graph.

On the other hand, the maximally coarse clustering is related to the concept
of maximally answerable clusters, defined below.

######  Definition 24 (Maximally Answerable Clusters).

Let $\mathbb{Q}$ be a set of counterfactual queries over variables
$\mathbf{V}_{L}$, and let $\mathbb{V}_{\mathbb{Q}}$ be the set of all subsets
of $\mathbf{V}_{L}$ that are used in a term in $\mathbb{Q}$ (i.e. for every
counterfactual term $\mathbf{Y}_{\mathbf{x}}$ in any of the queries, both
$\mathbf{Y}$ and $\mathbf{X}$ should be included in
$\mathbb{V}_{\mathbb{Q}}$). Denote $\mathbb{V}^{*}_{\mathbb{Q}}$ as the
extension of $\mathbb{V}_{\mathbb{Q}}$ closed under intersection, that is,
$\mathbb{V}_{\mathbb{Q}}\subseteq\mathbb{V}^{*}_{\mathbb{Q}}$, and
$V_{1},V_{2}\in\mathbb{V}^{*}_{\mathbb{Q}}\Rightarrow V_{1}\cap
V_{2}\in\mathbb{V}^{*}_{\mathbb{Q}}$. Then, an intervariable clustering
$\mathbb{C}_{\max}$ is called the maximally answerable cluster w.r.t.
$\mathbb{Q}$ if and only if $\mathbb{C}_{\max}$ is the set of atoms of
$\mathbb{V}^{*}_{\mathbb{Q}}$, or in other words,
$\mathbf{V}_{i}\in\mathbb{C}_{\max}$ if and only if
$\mathbf{V}_{i}\in\mathbb{V}^{*}_{\mathbb{Q}}$ and there exists no
$\mathbf{V}_{j}\in\mathbb{V}^{*}_{\mathbb{Q}}$ such that
$\mathbf{V}_{j}\subsetneq\mathbf{V}_{i}$. $\blacksquare$

The concept of maximally answerable clusters can be best shown through an
example.

######  Example 15.

Suppose

$\mathbf{V}_{L}=\\{A,B,C,D,E,F,G,H,I,X,Z\\}$

and the queries of interest $\mathbb{Q}=\\{Q_{1},Q_{2},Q_{3}\\}$ are as
follows:

$\displaystyle Q_{1}$ $\displaystyle=P(\\{a,b,c,d,e\\}_{x})$ $\displaystyle
Q_{2}$ $\displaystyle=P(\\{c,d,e,f,g,h\\}_{x,z})$ $\displaystyle Q_{3}$
$\displaystyle=P(\\{e,f,h,i,z\\}_{x}).$

We first dissect the terms in the queries, including the subscripts, to obtain
$\mathbb{V}_{\mathbb{Q}}$. Doing so yields

$\displaystyle\mathbb{V}_{\mathbb{Q}}$
$\displaystyle=\\{\\{A,B,C,D,E\\},\\{C,D,E,F,G,H\\},$
$\displaystyle\\{E,F,H,I,Z\\},\\{X\\},\\{X,Z\\}\\}.$

These sets are shown at the top of Fig. 19. Note that many of the sets
overlap. The set of nonoverlapping subregions becomes the maximally answerable
clusters, as shown at the bottom of Fig. 19. In this case, this means

$\displaystyle\mathbb{C}_{\max}$
$\displaystyle=\\{\\{A,B\\},\\{C,D\\},\\{E\\},\\{F,H\\},$
$\displaystyle\\{G\\},\\{I\\},\\{X\\},\\{Z\\}\\}.$

Note that no set in $\mathbb{C}_{\max}$ overlaps with each other, but all sets
are subsets of some set in $\mathbb{V}_{\mathbb{Q}}$. $\blacksquare$

It turns out that the set of maximally answerable clusters is also the
maximally coarse clustering that does not violate condition C3, as shown
below.

######  Lemma 9.

Let $\mathbb{C}_{\max}$ be the intervariable clustering of $\mathbf{V}_{L}$
that is the maximally answerable cluster with respect to a set of
counterfactual queries $\mathbb{Q}$. Then, any clustering
$\mathbb{C}^{\prime}$ violates condition C3 if and only if it is not equal to
or finer than $\mathbb{C}_{\max}$. $\blacksquare$

###### Proof.

We first note that $\mathbb{C}_{\max}$ does not violate condition C3, since if
there existed a query with a term set $\mathbf{Z}$ such that $\mathbf{Z}$ is
not a union of clusters of $\mathbb{C}_{\max}$, then either there exists
$Z\in\mathbf{Z}$ such that $Z$ is not in any set of $\mathbb{C}_{\max}$, or
$Z\in\mathbf{C}$ for some $\mathbf{C}\in\mathbb{C}_{\max}$ such that
$\mathbf{C}\subsetneq\mathbf{Z}$. However, the first case is impossible since
if $Z\in\mathbf{Z}$, then $Z$ must be added to some set in
$\mathbb{V}_{\mathbb{Q}}$. The second case is not possible either because then
$\mathbf{Z}\cap\mathbf{C}\neq\emptyset$, which means that the clusters of
$\mathbb{C}_{\max}$ could be divided further, contradicting its definition. If
$\mathbb{C}_{\max}$ does not violate condition C3, then no finer clustering
can violate it either because a union of clusters in $\mathbb{C}_{\max}$ can
always translate to a union of clusters in a finer clustering.

If $\mathbb{C}^{\prime}$ is not equal to or finer than $\mathbb{C}_{\max}$,
then there exists at least one cluster
$\mathbf{C}^{\prime}\in\mathbb{C}^{\prime}$ such that
$\mathbf{C}^{\prime}\subsetneq\mathbf{C}$ for all
$\mathbf{C}\in\mathbb{C}_{\max}$. This means that there exist
$V_{1},V_{2}\in\mathbf{C}^{\prime}$ such that $V_{1}$ and $V_{2}$ are not in
the same cluster in $\mathbb{C}^{\prime}$. This implies that there exists at
least one query such that one of its terms contains one of $V_{1}$ or $V_{2}$
but not the other, in which case, such a term cannot be described as a union
of clusters from $\mathbb{C}^{\prime}$, violating condition C3. ∎

Figure 19: Top: A venn diagram of all of the variables and what set of
$\mathbb{V}_{\mathbb{Q}}$ to which they belong. Bottom: The collection of
nonoverlapping intersections of all sets that form the maximum answerable
clusters.

Intuitively, we cannot simply cluster all variables together because we may
want to answer queries that require separate consideration of different sets
of variables. At best, we can only cluster together terms that always appear
together in all queries. Additionally, we note that finding the maximally
answerable clusters can be done in polynomial time. One simple approach is to
add one set of $\mathbb{V}_{\mathbb{Q}}$ at a time, repeatedly computing the
intersection with the existing clusters. At any point, there can be at most
$n=|\mathbf{V}_{L}|$ clusters.

With these points in mind, we provide Alg. 3 for learning intervariable
clusters. We note that Alg. 3 returns a maximally course clustering satisfying
all conditions.

######  Proposition 11.

Alg. 3 returns a maximally course clustering that satisfies C1-4, or returns
FAIL if one does not exist. $\blacksquare$

###### Proof.

We first note that if FAIL is returned, then there does not exist a clustering
that satisfies all four conditions. Notably, if $\mathbb{C}_{\max}$ is not
coarser than $\mathbb{C}_{\min}$ at the beginning, then FAIL is returned, as a
consequence of Lemmas 7, 8, and 9. FAIL is also returned if
$\mathbb{C}_{\min}$ does not satisfy identifiability of all queries, since
further clustering will only remove edges and nodes and cannot make a non-ID
query turn ID.

If $\mathbb{C}$ is returned at the end, then $\mathbb{C}$ must satisfy all
four conditions. It satisfies conditions C1 and C3 because it will always be
coarser than $\mathbb{C}_{\min}$ but finer than $\mathbb{C}_{\max}$. It
satisfies conditions C2 and C4 because $\mathbb{C}$ is never updated unless
the “valid” subroutine returns True, which implies that C2 and C4 hold.

Finally, $\mathbb{C}$ must be maximal. If a clustering violates C4, then there
is no coarser clustering that does not violate C4 since clustering only
further reduces information. If a clustering does not violate C2, and a
coarser clustering exists that also does not violate C2, then there must exist
a way to either merge pairs of clusters or remove variables (namely in reverse
topological order) such that each intermediate clustering also does not
violate C2. Then, the loop ensures that if a variable could be removed, or two
clusters could be merged without violating the conditions, then it will be
done. ∎

Finally, we note that Alg. 3 runs in polynomial time in terms of
$n=|\mathbf{V}_{L}|$ and $|\mathbb{Q}|$, as long as checking identification
takes polynomial time. Finding $\mathbb{C}_{\min}$ from Lem. 7 takes
polynomial time because finding connected components in an undirected graph
takes polynomial time. Checking admissibility simply requires checking cycles,
which can also be done in polynomial time. Finding the maximally answerable
clusters can also be done in polynomial time, as discussed earlier.

1

2

Input : Variables $\mathbf{V}_{L}$, causal diagram $\mathcal{G}$, noncausal
diagram $\overline{\mathcal{G}}$, queries of interest $\mathbb{Q}$

Output : Maximally coarse clustering $\mathbb{C}$ such that all criteria are
satisfied, or FAIL if none exist.

3 Function  _valid(_$\mathbb{C}$ , $\mathcal{G}$, $\mathbb{Q}$_)_:

4 if  _not $\textnormal{{admissible}}(\mathbb{C},\mathcal{G})$_ then

return False

// C2 violated

5

6 for  _$Q\in\mathbb{Q}$_ do

$\mathcal{G}_{\mathbb{C}}\leftarrow\textnormal{{CDAG}}(\mathcal{G},\mathbb{C})$

// from Def. 9

7 if  _not $\textnormal{{ID}}(Q,\mathcal{G}_{\mathbb{C}})$_ then

return False

// C4 violated

8

9

10 return True

11

$\mathbb{C}_{\min}\leftarrow\textnormal{{minCluster}}(\mathbf{V}_{L},\overline{\mathcal{G}})$

// from Lem. 7

$\mathbb{C}_{\min}\leftarrow\textnormal{{mergeCycles}}(\mathbb{C}_{\min},\mathcal{G})$

// from Lem. 8

$\mathbb{C}_{\max}\leftarrow\textnormal{{maxCluster}}(\mathbf{V}_{L},\mathbb{Q})$

// from Lem. 9

12 if  _not $\mathbb{C}_{\max}.\textnormal{{coarserThan}}(\mathbb{C}_{\min})$_
then

return FAIL

// either C1, C2, or C3 violated

13

14if  _not $\textnormal{{valid}}(\mathbb{C},\mathcal{G},\mathbb{Q})$_ then

return FAIL

// C4 cannot be satisfied

15

16$\mathbb{C}\leftarrow\mathbb{C}_{\min}$

17 while  _$\mathbb{C}$ keeps updating_ do

18 for  _$V\in\mathbf{V}_{L}$_ do

19 if  _$V\not\in\mathbf{C}$ for all $\mathbf{C}\in\mathbb{C}_{\max}$_ then

20 $\mathbb{C}^{\prime}\leftarrow\mathbb{C}.\textnormal{{remove}}(V)$

21 if  _$\textnormal{{valid}}(\mathbb{C}^{\prime},\mathcal{G},\mathbb{Q})$_
then

22 $\mathbb{C}\leftarrow\mathbb{C}^{\prime}$

23

24

25

26 for  _$\mathbf{C}_{1},\mathbf{C}_{2}\in\mathbb{C}$_ do

27 for  _$\mathbf{C}\in\mathbb{C}_{\max}$_ do

28 if  _$\mathbf{C}_{1},\mathbf{C}_{2}\subseteq\mathbf{C}$_ then

29
$\mathbb{C}^{\prime}\leftarrow\textnormal{{merge}}(\mathbb{C},\mathbf{C}_{1},\mathbf{C}_{2})$

30 if  _$\textnormal{{valid}}(\mathbb{C}^{\prime},\mathcal{G},\mathbb{Q})$_
then

31 $\mathbb{C}\leftarrow\mathbb{C}^{\prime}$

32

33

34

35

36

37return $\mathbb{C}$

Algorithm 3 Choosing intervariable clusters.

Consider the following example for intuition on choosing the best set of
clusters.

######  Example 16.

Consider a setting of annotated image data where

$\mathbf{V}_{L}=\\{A,T,G,H_{1},H_{2},O_{1},O_{2},O_{3},O_{4},P_{1},P_{2},P_{3},P_{4}\\},$

and the causal diagram $\mathcal{G}$ (and noncausal diagram
$\overline{\mathcal{G}}$) are given in Fig. 18(a). The data is collected from
people, and the variables are age ($A$), gender ($G$), testosterone level
($T$), mustache hairs ($H_{1}$, $H_{2}$), atoms of other mustache hairs
($O_{1},O_{2},O_{3},O_{4}$), and pixels of an image of the person
$(P_{1},P_{2},P_{3},P_{4})$. The queries of interest are
$\mathbb{Q}=\\{Q_{1},Q_{2},Q_{3}\\}$, where

$\displaystyle Q_{1}$
$\displaystyle=P(\\{p_{1},p_{2},p_{3},p_{4}\\}_{h_{1},h_{2},o_{1},o_{2},o_{3},o_{4}})$
$\displaystyle Q_{2}$
$\displaystyle=P(\\{p_{1},p_{2},p_{3},p_{4}\\}_{h_{1},h_{2},o_{1},o_{2},o_{3},o_{4},g})$
$\displaystyle Q_{3}$ $\displaystyle=P(\\{p_{1},p_{2},p_{3},p_{4}\\}_{a}).$

Evidently, the variables depicted in Fig. 18(a) are too complex, and some
intervariable clustering is needed. For starters, there are interactions
between the atom variables ($O$) and the pixel variables ($P$) that are too
low-level to be considered causally. They should be clustered together, as
described in Lemma 7. This provides the graph in Fig. 18(b), in which all of
the $O$ variables are clustered into another hair variable $H_{3}$, and the
pixel variables $P$ are clustered into a single image variable $I$. Formally,
this clustering is

$\displaystyle\mathbb{C}_{1}$
$\displaystyle=\\{\\{A\\},\\{T\\},\\{G\\},\\{H_{1}\\},$
$\displaystyle\\{H_{2}\\},H_{3}=\\{O_{1},O_{2},O_{3},O_{4}\\},I=\\{P_{1},P_{2},P_{3},P_{4}\\}\\}.$

While this choice of $\mathbb{C}_{1}$ is valid in that it satisfies conditions
C1-4, there exists a coarser clustering that satisfies the conditions, shown
in Fig. 18(c). This one is obtained by further clustering $H_{1},H_{2},H_{3}$
into a single “mustache” variable $M$. Formally, this clustering is

$\displaystyle\mathbb{C}_{2}$
$\displaystyle=\\{\\{A\\},\\{T\\},\\{G\\},M=\\{H_{1},H_{2},O_{1},O_{2},O_{3},O_{4}\\},$
$\displaystyle I=\\{P_{1},P_{2},P_{3},P_{4}\\}\\}.$

The corresponding queries of $\tau(\mathbb{Q})$, under this choice of
$\mathbb{C}$, would be

$\displaystyle Q_{1}$
$\displaystyle=P(\\{p_{1},p_{2},p_{3},p_{4}\\}_{h_{1},h_{2},o_{1},o_{2},o_{3},o_{4}})=P(i_{m})$
$\displaystyle Q_{2}$
$\displaystyle=P(\\{p_{1},p_{2},p_{3},p_{4}\\}_{h_{1},h_{2},o_{1},o_{2},o_{3},o_{4},g})=P(i_{m,g})$
$\displaystyle Q_{3}$
$\displaystyle=P(\\{p_{1},p_{2},p_{3},p_{4}\\}_{a})=P(i_{a}).$

If for some reason we decide to try to cluster $\mathbb{C}_{2}$ further, such
as by clustering $A,G,M$ into a single variable $X$ (Fig. 18(d)), we violate
condition C3, the answerability of the queries. How could we distinguish
between queries like $P(I_{M=m}=i)$ and $P(I_{A=a}=i)$ if $M$ and $A$ are
grouped in the same cluster? For this reason, we cannot proceed any further
than $\mathbb{C}_{2}$, and, in fact, $\mathbb{C}_{2}$ is actually the
maximally answerable clusters from Lemma 9. $\blacksquare$

###  D.2 Discussion on the Abstract Invariance Condition

Recall the abstract invariance condition (AIC): See 8

The AIC intuitively ensures that there is no loss of information when
clustering two values together in the intravariable clusters $\mathbb{D}$.
Specifically, it states that clustering two values together is safe whenever
these values are interchangeable with respect to the behavior of downstream
functions. That is, the functions are _invariant_ to changes between values in
the same intravariable cluster. This is illustrated in Fig. 20. Examples 4 and
5 from Sec. 2 illustrate the subtleties of the AIC both when it holds and when
it does not hold.

If the AIC is violated, then two functionally different values were placed in
the same cluster. Hence, assuming that the AIC holds is reasonable provided
that the intravariable clusters were chosen reasonably. For example, in a
study where a drug $X$ is reasonably expected to affect blood pressure levels
$Y$, it would not make sense to cluster the values $X=0$ (drug not taken) and
$X=1$ (drug taken) together since, by design, they have different downstream
effects on $Y$ and cannot be treated similarly.

On the other hand, in an image recognition setting, perhaps it does not matter
for the task if the image is scaled or rotated, so these invariances can be
modeled in the intravariable clustering. For example, if one image is a scaled
version of another but they are otherwise the same, the two images might be
placed in the same intravariable clustering without violating the AIC. Sec. 4
and Appendix D.3 discuss this phenomenon in more detail.

If a data scientist finds the AIC too strict for her settings, one option is
to revise the intravariable clusters, perhaps making them less coarse.
Otherwise, the abstraction may be too strong and some important information is
lost.

(a) When the AIC holds, two values of $\mathbf{V}_{L}$ that are in the same
intravariable cluster are mapped to the same output in $\mathbf{V}_{H}$ for
any downstream function $f^{L}$ after applying $\tau$.

(b) When the AIC does not hold, two values of $\mathbf{V}_{L}$ that are in the
same intravariable cluster may map to different outputs in $\mathbf{V}_{H}$
for some downstream function $f^{L}$ after applying $\tau$.

Figure 20: Visualization of the AIC (Def. 8).

Still, there are alternatives if weaker assumptions are desired. The AIC
ensures that an $\mathcal{L}_{3}$-$\tau$ consistent model exists, as
demonstrated by Alg. 1, but it is not a necessary condition in applications
that do not require full $\mathcal{L}_{3}$-$\tau$ consistency. For example, if
one only works on the interventional level ($\mathcal{L}_{2}$), one may not
necessarily care if consistencies between $\mathcal{L}_{3}$ quantities are
lost.

In the strictest form, making as minimal assumptions as needed for the task, a
query-specific AIC can be assumed. That is,

######  Definition 25.

Let $\mathbb{Q}$ be a set of queries of the form
$P^{\mathcal{M}_{H}}(\mathbf{y}_{H,1[\mathbf{x}_{H,1}]},\mathbf{y}_{H,2[\mathbf{x}_{H,2}]},\dots)$.
For all $Q\in\mathbb{Q}$, we have

$\displaystyle\sum_{\forall
i\mathbf{y}_{L,i}\in\mathcal{D}_{\mathbf{Y}_{L,i}}:\tau(\mathbf{y}_{L,i})=\mathbf{y}_{H,i}}P^{\mathcal{M}_{L}}(\mathbf{y}_{L,1[\mathbf{x}_{L,1}]},\mathbf{y}_{L,2[\mathbf{x}_{L,2}]},\dots)$
$\displaystyle=\sum_{\forall
i\mathbf{y}_{L,i}\in\mathcal{D}_{\mathbf{Y}_{L,i}}:\tau(\mathbf{y}_{L,i})=\mathbf{y}_{H,i}}P^{\mathcal{M}_{L}}(\mathbf{y}_{L,1[\mathbf{x}^{\prime}_{L,1}]},\mathbf{y}_{L,2[\mathbf{x}^{\prime}_{L,2}]},\dots),$

for all $\mathbf{x}_{L,i},\mathbf{x}^{\prime}_{L,i}$ such that
$\tau(\mathbf{x}_{L,i})=\tau(\mathbf{x}^{\prime}_{L,i})=\mathbf{x}_{H,i}$.
$\blacksquare$

In words, $\tau$ satisfies Def. 25 if and only if there exists
$\mathcal{M}_{H}$ that is $Q$-$\tau$ consistent with $\mathcal{M}_{L}$ for all
$Q\in\mathbb{Q}$, since all such variations of $Q$ have the same probability.
Still, it may be more useful to have a general criterion that is not query-
dependent.

It turns out that the study of weaker forms of the AIC is not new and is
sfirst discussed in Chalupka, Perona, and Eberhardt (2015). Broadly, the paper
studies a similar abstraction setting in which causal macro-variables are
learned from observational data from the low-level micro-variables. Similar to
the intravariable learning problem posed in Sec. 4, the paper aims to learn
intravariable clusters of a low-level high-dimensional variable $I$ (e.g., an
image), which causes some other variable $T$, possibly with unobserved
confounding. However, while the results of Sec. 4 focus on satisfying the AIC
in Def. 8, Chalupka, Perona, and Eberhardt (2015) focuses on satisfying weaker
conditions that are verifiable through data.

More specifically, consider the setting proposed in Chalupka, Perona, and
Eberhardt (2015) illustrated by the causal diagram in Fig. 21. Note that Eq.
33 in Def. 8 requires that

$\tau(f_{T}(i_{1},\mathbf{u}_{T}))=\tau(f_{T}(i_{2},\mathbf{u}_{T})),$ (102)

for any $i_{1},i_{2}\in\mathcal{D}_{I}$ that are chosen to be clustered in the
same intravariable cluster. This condition is a requirement on the SCM
function $f_{T}$, which, as demonstrated by Lem. 5, implies that one cannot
cluster together any two values $i_{1},i_{2}$ without further information
about $f_{T}$, which may be difficult to acquire due to the unobserved nature
of the SCM. Instead, the paper proposes two other properties. Notably, two
values $i_{1}$ and $i_{2}$ are part of the same _observational partition_
(Chalupka, Perona, and Eberhardt 2015, Def. 1) if

$P(t\mid i_{1})=P(t\mid i_{2}),$ (103)

and they are part of the same _causal partition_ (Chalupka, Perona, and
Eberhardt 2015, Def. 3) if

$P(t_{i_{1}})=P(t_{i_{2}}).$ (104)

Values of $\mathcal{D}_{I}$ can easily be clustered according to Eq. 103 using
the observational data. However, Eq. 104 may be a more desirable property if
the goal of downstream tasks is to perform causal inferences from $I$ to $T$.

It turns out that a clustering learned according to Eq. 103 typically also
satisfies Eq. 104 according to a major proven result of the paper, the _Causal
Coarsening Theorem_ (Chalupka, Perona, and Eberhardt 2015, Thm. 5), which
states (in informal terms) that the maximally coarse clustering that satisfies
Eq. 104 is almost always a coarser clustering of the maximally coarse
clustering that satisfies Eq. 103. This essentially implies that, under
certain faithfulness conditions, one can achieve the property of Eq. 104
simply as a consequence of achieving Eq. 103, which can be accomplished with
observational data alone.

The concepts of the partitions described by Eqs. 103 and 104 can be
generalized to the general SCM/graph case. Suppose we are given the
inter/intravariable clusters $\mathbb{C}$ and $\mathbb{D}$. Consider a variant
of the AIC applied on conditional probabilties:

######  Definition 26 (Conditional Abstract Invariance Condition).

Let $P(\mathbf{V}_{L})$ be an observational distribution over
$\mathbf{V}_{L}$, and let
$\tau:\mathcal{D}_{\mathbf{V}_{L}}\rightarrow\mathcal{D}_{\mathbf{V}_{H}}$ be
a constructive abstraction function relative to $\mathbb{C}$ and $\mathbb{D}$.
$P(\mathbf{V}_{L})$ is said to satisfy the conditional abstract invariance
condition (conditional AIC, for short) w.r.t. $\tau$ if, for all
$\mathbf{X}_{L}$ where $\mathbf{X}_{L}$ is a union of clusters of
$\mathbb{C}$, all
$\mathbf{x}_{1},\mathbf{x}_{2}\in\mathcal{D}_{\mathbf{X}_{L}}$ such that
$\tau(\mathbf{x}_{1})=\tau(\mathbf{x}_{2})$, and all
$\mathbf{v}_{H}\in\mathcal{D}_{\mathbf{V}_{H}}$, we have

$\sum_{\mathbf{v}_{L}\in\mathcal{D}_{\mathbf{V}_{L}}:\tau(\mathbf{v}_{L})=\mathbf{v}_{H}}P(\mathbf{v}_{L}\mid\mathbf{x}_{1})=\sum_{\mathbf{v}_{L}\in\mathcal{D}_{\mathbf{V}_{L}}:\tau(\mathbf{v}_{L})=\mathbf{v}_{H}}P(\mathbf{v}_{L}\mid\mathbf{x}_{2}).$
(105)

$\blacksquare$

$I$$T$ Figure 21: Problem setting in Chalupka, Perona, and Eberhardt (2015).

That is, two values can be clustered together as long as their conditional
probabilities do not change in the space of $\mathbf{V}_{H}$. Note that this
condition is entirely focused on the observational distributions
$P(\mathbf{V}_{L})$ and $P(\mathbf{V}_{H})$ and make no requirements over any
distributions from the higher layers, $\mathcal{L}_{2}$ and $\mathcal{L}_{3}$.

Now consider an interventional variant:

######  Definition 27 (Interventional Abstract Invariance Condition).

Let $\mathcal{L}_{2}(\mathcal{M}_{L})$ be a collection of interventional
distributions over $\mathbf{V}_{L}$, and let
$\tau:\mathcal{D}_{\mathbf{V}_{L}}\rightarrow\mathcal{D}_{\mathbf{V}_{H}}$ be
a constructive abstraction function relative to $\mathbb{C}$ and $\mathbb{D}$.
We say that $\mathcal{L}_{2}(\mathcal{M}_{L})$ satisfies the interventional
abstract invariance condition (interventional AIC, for short) w.r.t. $\tau$
if, for all $\mathbf{X}_{L}$ where $\mathbf{X}_{L}$ is a union of clusters of
$\mathbb{C}$, all
$\mathbf{x}_{1},\mathbf{x}_{2}\in\mathcal{D}_{\mathbf{X}_{L}}$ such that
$\tau(\mathbf{x}_{1})=\tau(\mathbf{x}_{2})$, and all
$\mathbf{v}_{H}\in\mathcal{D}_{\mathbf{V}_{H}}$, we have

$\sum_{\mathbf{v}_{L}\in\mathcal{D}_{\mathbf{V}_{L}}:\tau(\mathbf{v}_{L})=\mathbf{v}_{H}}P(\mathbf{v}_{L[\mathbf{x}_{1}]})=\sum_{\mathbf{v}_{L}\in\mathcal{D}_{\mathbf{V}_{L}}:\tau(\mathbf{v}_{L})=\mathbf{v}_{H}}P(\mathbf{v}_{L[\mathbf{x}_{2}]}).$
(106)

$\blacksquare$

This states that two values can be clustered together as long as the
corresponding interventional distributions do not change in the space of
$\mathbf{V}_{H}$.

These two definitions may be useful when looking to find abstractions while
working on lower layers of the PCH. When given the C-DAG
$\mathcal{G}_{\mathbb{C}}$, these can be refined to specific queries based on
parent-child relationships in the graph. Although not shown formally here, it
is likely that a more general variant of the Causal Coarsening Theorem could
be proven for these two definitions. Nonetheless, using these definitions in
place of the full AIC from Def. 8 has its limitations; for example, it may no
longer be possible to perform counterfactual inferences on the high-level
model. We encourage further research on this topic, comparing the
relationships between variations of the AIC.

###  D.3 Representation Learning with Intravariable Clusters

Recall the following proposition from Sec. 4: See 5

When learning intravariable clusters, Prop. 5 states that without extra
information, each value needs to be put in its own cluster. However, if extra
information is available for use, this can be leveraged to improve the choice
of $\mathbb{D}$. In practice, this is common, and is generally provided in the
form of invariance assumptions. As a intuitive example, suppose that $X$ is an
image of an animal, $Y$ is the corresponding label, and $f_{Y}(x,u_{Y})$
describes the mechanism that humans use to classify $X$ (here, $u_{Y}$ is the
exogenous noise). Suppose that $g(x,\phi)$ is a function that rotates $x$ by
$\phi$ degrees. It is the case that the classification of animals is
rotationally invariant, that is, $f_{Y}(g(x,\phi),u_{Y})=f_{Y}(x,u_{Y})$ for
all $x$, $\phi$, $u_{Y}$. Then, in fact, $\mathbb{D}$ can be defined such that
values of $x_{1},x_{2}$ are clustered together if $g(x_{1},\phi)=x_{2}$, as
shown by the following result.

######  Proposition 12.

For each $\mathbf{C}_{i}\in\mathbb{C}$, suppose that there exists function
$g_{\mathbf{C}_{i}}:\mathcal{D}_{\mathbf{C}_{i}}\times\mathcal{D}_{\phi_{\mathbf{C}_{i}}}\rightarrow\mathcal{D}_{\mathbf{C}_{i}}$
(with parameters $\phi_{\mathbf{C}_{i}}$) such that

$f^{L}_{V}(\mathbf{c}_{i},\mathbf{u}_{V})=f^{L}_{V}(g_{\mathbf{C}_{i}}(\mathbf{c}_{i},\phi_{\mathbf{C}_{i}}),\mathbf{u}_{V})$
(107)

for all $V$ that are children of $\mathbf{C}_{i}$ and all
$\phi_{\mathbf{C}_{i}}\in\mathcal{D}_{\phi_{\mathbf{C}_{i}}}$. Then,
$\mathcal{M}_{L}$ satisfies the AIC w.r.t. any constructive abstraction
function with $\mathbb{D}$ such that
$\mathbf{c}_{i}^{(1)},\mathbf{c}_{i}^{(2)}$ being in the same intravariable
cluster implies that
$g_{\mathbf{C}_{i}}(\mathbf{c}_{i}^{(1)},\phi_{\mathbf{C}_{i}})=\mathbf{c}_{i}^{(2)}$
for some $\phi_{\mathbf{C}_{i}}$. $\blacksquare$

###### Proof.

Consider the proposed set of intravariable clusters $\mathbb{D}$. Let
$\mathbf{v}_{1},\mathbf{v}_{2}\in\mathcal{D}_{\mathbf{V}_{L}}$ be two values
such that $\tau(\mathbf{v}_{1})=\tau(\mathbf{v}_{2})$. The goal is to show
that for all $\mathbf{u}\in\mathcal{D}_{\mathbf{U}_{L}}$ and
$\mathbf{C}_{k}\in\mathbb{C}$, we have

$\begin{split}&\tau\left(\left(f^{L}_{V}(\mathbf{pa}_{V}^{(1)},\mathbf{u}_{V}):V\in\mathbf{C}_{k}\right)\right)\\\
&=\tau\left(\left(f^{L}_{V}(\mathbf{pa}_{V}^{(2)},\mathbf{u}_{V}):V\in\mathbf{C}_{k}\right)\right),\end{split}$
(108)

where $\mathbf{pa}_{V}^{(1)}$ and $\mathbf{pa}_{V}^{(2)}$ are the values
corresponding to $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ respectively. If
$\tau(\mathbf{v}_{1})=\tau(\mathbf{v}_{2})$, that implies that for all
$\mathbf{C}_{i}\in\mathbb{C}$ (where $\mathbf{c}_{i}^{(1)}$ and
$\mathbf{c}_{i}^{(2)}$ correspond to the values of $\mathbf{v}_{1}$ and
$\mathbf{v}_{2}$ respectively), $\mathbf{c}_{i}^{(1)}$ and
$\mathbf{c}_{i}^{(2)}$ must be in the same intravariable cluster in
$\mathbb{D}_{\mathbf{C}_{i}}$ (this includes the case where
$\mathbf{c}_{i}^{(1)}=\mathbf{c}_{i}^{(2)}$). By construction of $\mathbb{D}$,
this is only possible if
$g_{\mathbf{C}_{i}}(\mathbf{c}_{i}^{(1)},\phi_{\mathbf{C}_{i}})=\mathbf{c}_{i}^{(2)}$
for some $\phi_{\mathbf{C}_{i}}$.

For
$g_{V}=\\{g_{\mathbf{C}_{i}}:\mathbf{C}_{i}\cap\mathbf{Pa}_{V}\neq\emptyset\\}$,
denote $g_{V}(\mathbf{pa}_{V}^{(1)},\phi)$ as the collection of outputs of
$g_{\mathbf{C}_{i}}(\mathbf{c}_{i}^{(1)},\phi_{\mathbf{C}_{i}})$ for the
functions of $g_{\mathbf{C}_{i}}\in g_{V}$, which, as established earlier, is
equal to $\mathbf{pa}_{V}^{(2)}$. Then we have

$\displaystyle\tau\left(\left(f^{L}_{V}(\mathbf{pa}_{V}^{(1)},\mathbf{u}_{V}):V\in\mathbf{C}_{k}\right)\right)$
$\displaystyle=\tau\left(\left(f^{L}_{V}(g_{V}(\mathbf{pa}_{V}^{(1)},\phi),\mathbf{u}_{V}):V\in\mathbf{C}_{k}\right)\right)$
$\displaystyle=\tau\left(\left(f^{L}_{V}(\mathbf{pa}_{V}^{(2)},\mathbf{u}_{V}):V\in\mathbf{C}_{k}\right)\right),$

completing the proof. ∎

For intuition, consider the following simple example.

######  Example 17.

Consider a situation where $Y$ is some binary variable that is a (noisy)
bitwise AND of two other binary variables $X_{1}$ and $X_{2}$. For example,
perhaps $Y$ denotes whether a law is enacted, and $X_{1}$ and $X_{2}$ denotes
the votes of the two branches of government, like in Ex. 11. Formally, let
$\mathcal{M}_{L}$ be defined as follows:

$\displaystyle\mathbf{U}_{L}$
$\displaystyle=\\{U_{X_{1}},U_{X_{2}},U_{Y}\\},\text{ all binary}$
$\displaystyle\mathbf{V}_{L}$ $\displaystyle=\\{X_{1},X_{2},Y\\},\text{ all
binary}$ $\displaystyle\mathcal{F}_{L}$
$\displaystyle=\begin{cases}X_{1}\leftarrow
f^{L}_{X_{1}}(u_{X_{1}})&=u_{X_{1}}\\\ X_{2}\leftarrow
f^{L}_{X_{2}}(u_{X_{2}})&=u_{X_{2}}\\\ Y\leftarrow
f^{L}_{Y}(x_{1},x_{2},u_{Y})&=(x_{1}\wedge x_{2})\oplus u_{Y}\end{cases}$
$\displaystyle P(\mathbf{U}_{L})$
$\displaystyle=\begin{cases}P(U_{X_{1}}=1)&=P(U_{X_{2}}=1)=0.5\\\
P(U_{Y}=1)&=0.1\end{cases}$

Now suppose we want to create an abstraction of $\mathcal{M}_{L}$ using the
intervariable clusters
$\mathbb{C}=\\{\mathbf{C}_{1}=\\{X_{1},X_{2}\\},\mathbf{C}_{2}=\\{Y\\}\\}$.
The domain $\mathcal{D}_{\mathbf{C}_{1}}$ has four values as $X_{1}$ and
$X_{2}$ can both be either 0 or 1. However, as emphasized by Prop. 5, the only
intravariable clustering we can choose without additional information is the
one where each value is in its own cluster. In other words,
$\mathbb{D}_{\mathbf{C}_{1}}=\\{\\{(X_{1}=0,X_{2}=0)\\},\\{(X_{1}=0,X_{2}=1)\\},\\{(X_{1}=1,X_{2}=0)\\},\\{(X_{1}=1,X_{2}=1)\\}\\}$.

However, suppose we are given additional information that $f_{Y}$ is
_permutation invariant_ to its endogenous inputs, i.e.,

$f_{Y}(x_{1},x_{2},u_{Y})=f_{Y}(x_{2},x_{1},u_{Y}).$ (109)

In the notation of Prop. 12, we can define $\phi_{\mathbf{C}_{1}}$ to be a
binary variable such that 0 means original order and 1 means reversed. Then we
can define $g(x_{1},x_{2},\phi_{\mathbf{C}_{1}})=(x_{1},x_{2})$ if
$\phi_{\mathbf{C}_{1}}=0$ or $(x_{2},x_{1})$ if $\phi_{\mathbf{C}_{1}}=1$.
Then, this implies that
$f_{Y}(x_{1},x_{2},u_{Y})=f_{Y}(g(x_{1},x_{2},\phi_{\mathbf{C}_{1}}),u_{Y})$
for any choice of $\phi_{\mathbf{C}_{1}}$.

By Prop. 12, this implies that $(X_{1}=0,X_{2}=1)$ and $(X_{1}=1,X_{2}=0)$ can
be placed in the same intravariable cluster without violating the AIC. Indeed,
the function $f^{L}_{Y}(x_{1},x_{2},u_{Y})=(x_{1}\wedge x_{2})\oplus u_{Y}$
does not change when $x_{1}$ and $x_{2}$ are swapped, so we have

$\begin{split}&\tau\left(\left(f^{L}_{Y}(X_{1}=0,X_{2}=1,\mathbf{u}_{V}):V\in\mathbf{C}_{k}\right)\right)\\\
&=\tau\left(\left(f^{L}_{Y}(X_{1}=1,X_{2}=0,\mathbf{u}_{V}):V\in\mathbf{C}_{k}\right)\right),\end{split}$

confirming that the AIC still holds when $(X_{1}=0,X_{2}=1)$ and
$(X_{1}=1,X_{2}=0)$ can be placed in the same intravariable cluster.
$\blacksquare$

In practice, this invariance information can be incorporated in the process of
learning the intravariable clusters while training $\widehat{\tau}$ in an
RNCM, as described in Def. 11. To take into account Prop. 12, the training
objective could include a term to enforce the invariance specified by
$g_{\mathbf{C}_{i}}$ (e.g. through a penalty on
$\tau_{\mathbf{C}_{i}}(\mathbf{c}_{i},\bm{\theta}_{\mathbf{C}_{i}})-\tau_{\mathbf{C}_{i}}(g_{\mathbf{C}_{i}}(\mathbf{c}_{i},\phi_{\mathbf{C}_{i}}),\bm{\theta}_{\mathbf{C}_{i}})$).

In fact, it turns out that the idea of intravariable clusters works in tandem
with techniques in deep representation learning when it comes to incorporating
invariances in the data. In the case of image data for example, many works in
computer vision have already leveraged general patterns found in images (e.g.
rotations, crops, flips, etc. do not affect classification) to achieve faster
training with less data (Shorten and Khoshgoftaar 2019). Permutation
invariance concepts (Zaheer et al. 2017; Murphy et al. 2019) have been used as
pooling functions for convolutional neural networks (LeCun and Bengio 1998).
Existing frameworks for representation learning such as through contrastive
methods (Chen et al. 2020) can be used for learning $\tau$.

##  Appendix E Additional Examples

This section contains examples to improve the clarity of concepts in the
paper.

###  E.1 Sec. 2 Examples

We provide more examples of the concepts in Sec. 2, namely, abstractions
constructed through inter/intravariable clusters. In addition to Example 1
within the section, we provide a more involved example below.

######  Example 18.

Suppose an economist is studying the effects of implementing a new type of
government policy on recession prevention. The economist records data on
several variables of interest: whether the policy is implemented ($X$);
whether the policy is lobbied ($W$); economic spending in terms of consumption
($C$), investment ($I$), government spending ($G$), imports ($M$), and exports
($E$); and whether or not there is a recession ($Y$). Out of these variables,
$Z$, $X$, and $Y$ are binary, and $C$, $I$, $G$, $M$, and $E$ are numerical
values representing how much the spending has .changed relative to the
previous year (in billions of dollars).

Suppose the true SCM $\mathcal{M}^{*}=\mathcal{M}_{L}$ is defined as follows:

$\displaystyle\mathbf{U}_{L}$
$\displaystyle=\\{U_{W},U_{X},U_{XY},U_{C},U_{I},U_{G},U_{M},U_{E},U_{IG},$
$\displaystyle U_{CM},U_{EM}\\}$ $\displaystyle\mathbf{V}_{L}$
$\displaystyle=\\{W,X,C,I,G,M,E,Y\\}$ $\displaystyle\mathcal{F}_{L}$
$\displaystyle=\\{$ $\displaystyle W\leftarrow f_{W}(u_{W})=u_{W}$
$\displaystyle X\leftarrow f_{X}(w,u_{X},u_{XY})=w\oplus u_{X}\oplus u_{XY}$
$\displaystyle C\leftarrow f_{C}(x,g,u_{C})=5x-0.2g+u_{C}+u_{CM}$
$\displaystyle I\leftarrow f_{I}(x,u_{I})=-5x+u_{I}+u_{IG}$ $\displaystyle
G\leftarrow f_{G}(x,u_{G})=10x+u_{G}+u_{IG}$ $\displaystyle E\leftarrow
f_{E}(u_{E})=u_{E}+u_{EM}$ $\displaystyle M\leftarrow
f_{M}(c,u_{M})=0.2c+u_{M}+u_{CM}-u_{EM}$ $\displaystyle Y\leftarrow
f_{Y}(c,i,g,e,m,u_{XY})=$ $\displaystyle\quad\mathbf{1}\\{c+i+g+e-m\leq
0\\}\oplus u_{XY}$ $\displaystyle P(\mathbf{U}_{L})$
$\displaystyle=\begin{cases}U_{W},U_{X}&\sim\operatorname{Bernoulli}(0.5)\\\
U_{XY}&\sim\operatorname{Bernoulli}(0.1)\\\ U_{C},U_{I},U_{G},U_{M},U_{E}&\sim
N(0,10)\\\ U_{IG},U_{CM},U_{EM}&\sim N(0,2)\end{cases}$

To summarize, the policy has some impact on the consumption, investment, and
government spending. Whether or not there is a recession depends on imports
and exports in addition to all of these factors. Indeed, despite the fact that
this is a toy example, the SCM $\mathcal{M}^{*}$ is already quite complex to
describe with this many variables and functions, and perhaps this level of
detail is not needed to achieve the inference we desire. We will see how
utilizing abstractions can help with this.

We first note that perhaps $\mathbf{V}_{L}$ contains too many micro-level
variables that can be summarized with a smaller set of abstract higher-level
variables. Further, perhaps using real-valued variables is overly complex, and
the same phenomena can be described without loss of generality using a lower-
dimensional space. For entertaining these considerations, we utilize the idea
of intervariable and intravariable clusters.

Suppose we are only interested in the causal effect of the policy $X$ on
recession $Y$. We could study the same phenomenon under a simpler set of
variables. To do so, we can cluster variables of $\mathbf{V}_{L}$ to form a
new set of macro-level variables. First, note that although we may have data
on $C$, $I$, $G$, $E$, and $M$, it may make more sense to simply cluster them
together and consider them as one variable (e.g. GDP). Further, perhaps we may
decide that $W$ is not relevant to the analysis and exclude it from the study.
We leave these clustering decisions at the discretion of the data scientist
(e.g. the economist in this example).

By the definition of intervariable clusters from Def. 5, we can choose
clusters
$\mathbb{C}=\\{\mathbf{C}_{1}=\\{X\\},\mathbf{C}_{2}=\\{C,I,G,E,M\\},\mathbf{C}_{3}=\\{Y\\}\\}$.
We leave $X$ and $Y$ in their own clusters while grouping all of $C$, $I$,
$G$, $E$, $M$ into one cluster $\mathbf{C}_{2}$. $W$ is excluded from all of
the clusters and is effectively projected out of the system. We can then treat
$\mathbf{C}_{1}$, $\mathbf{C}_{2}$, and $\mathbf{C}_{3}$ as our new variables.
Let us relabel them $X$, $Z$, and $Y$ respectively, where $Z$. We can define
our higher level variables as $\mathbf{V}_{H}=\\{X,Z,Y\\}$.

Further, it is not immediately clear what the domain of $Z$ is. Certainly, it
could be left as a tuple $(C,I,G,E,M)\in\mathbb{R}^{5}$. However, this would
be an overly complex representation of $Z$, and we do not need to retain all
of the joint information of $C,I,G,E,M$. In other words, $\mathcal{D}_{Z}$
does not have to be equal to $\mathcal{D}_{\mathbf{C}_{2}}$ and can be
represented more compactly. Instead, we may choose to define $Z$ as the GDP,
or $Z=C+I+G-E-M$. This can be described through the use of intravariable
clusters.

By the definition of intravariable clusters from Def. 5, we must choose
$\mathbb{D}=\\{\mathbb{D}_{\mathbf{C}_{1}},\mathbb{D}_{\mathbf{C}_{2}},\mathbb{D}_{\mathbf{C}_{3}}\\}$,
where $\mathbb{D}_{\mathbf{C}_{i}}$ is a partitioning of the domain
$\mathcal{D}_{\mathbf{C}_{i}}$. Since $X$ and $Y$ are already binary
variables, we will not be able to group their values together, so we simply
define $\mathbb{D}_{\mathbf{C}_{1}}$ and $\mathbb{D}_{\mathbf{C}_{2}}$ as
$\\{\\{0\\},\\{1\\}\\}$. For $\mathbf{C}_{2}$ however, we must cluster values
of $(C,I,G,E,M)$ such that
$(c_{1},i_{1},g_{1},e_{1},m_{1}),(c_{2},i_{2},g_{2},e_{2},m_{2})\in\mathcal{D}_{\mathbf{C}_{2}}$
are in the same cluster if and only if
$c_{1}+i_{1}+g_{1}+e_{1}-m_{1}=c_{2}+i_{2}+g_{2}+e_{2}-m_{2}$. In other words,
we can, for example, define
$\mathbb{D}_{\mathbf{C}_{2}}=\\{\mathcal{D}_{\mathbf{C}_{2}}^{j}:(c,i,g,e,m)\in\mathcal{D}_{\mathbf{C}_{2}},c+i+g+e-m=j\\}$.

Now, with the intravariable clusters $\mathbb{D}$ defined, we can choose the
domains of $\mathbf{V}_{H}$ as simply their corresponding clusters in
$\mathbb{D}$. That is, each value of $Z$ corresponds to some
$\mathcal{D}_{\mathbf{C}_{2}}^{j}$. In fact, we can simply set $Z=j$, where
$j=C+I+G+E-M$, which intuitively corresponds to the idea that $Z$ represents
the annual change in GDP. Note that the domain of $Z$, $\mathcal{D}_{Z}$
becomes smaller than $\mathcal{D}_{\mathbf{C}_{2}}$ in some sense. First, it
is lower dimensional ($\mathbb{R}$ instead of $\mathbb{R}^{5}$), and second,
there are clearly values of $(C,I,G,E,M)$ that are mapped to the same value of
$Z$. For example, $(C=1,I=2,G=3,E=4,M=2$ maps to the same value as
$(C=3,I=2,G=1,E=5,M=3)$ because $1+2+3+4-2=3+2+1+5-3=8$.

However, perhaps this level of dimensionality reduction is insufficient. In
terms of cardinality, $\mathcal{D}_{Z}$ is the same size as
$\mathcal{D}_{\mathbf{C}_{2}}$ as the cardinality of $\mathbb{R}$ and
$\mathbb{R}^{5}$ are the same. Perhaps the domain of $Z$ could be compressed
further. Indeed, we could turn $Z$ into a binary variable by defining
$\mathbb{D}_{\mathbf{C}_{2}}=\\{\mathcal{D}^{0}_{\mathbf{C}_{2}},\mathcal{D}^{1}_{\mathbf{C}_{2}}\\}$,
where $(c,i,g,e,m)\in\mathcal{D}^{0}_{\mathbf{C}_{2}}$ if $c+i+g+e-m>0$ or it
is in $\mathcal{D}^{1}_{\mathbf{C}_{2}}$ otherwise. In other words, we drop
all information about the annual change in GDP except whether it is positive
or negative. We will use this clustering for the rest of the examples, and we
will see in later examples why this clustering is allowed and makes sense.
Sec. 4 expands on the general discussion of which intravariable cluster
choices are allowed.

With the idea of inter/intravariable clusters, the concept of constructive
abstraction functions can be established as in Def. 6. From Example 18, the
function $\tau$ constructed from $\mathbb{C}$ and $\mathbb{D}$ is clear. For
example, consider the value
$\mathbf{v}_{L}=(W=0,X=1,C=2.6,I=-1.2,G=10.2,E=0.4,M=1.2,Y=0)$. We can compute
$\mathbf{v}_{H}=\tau(\mathbf{v}_{L})$ and find that
$\mathbf{v}_{H}=(X=1,Z=0,Y=0)$. In this case,
$\tau=(\tau_{\mathbf{C}_{1}},\tau_{\mathbf{C}_{2}},\tau_{\mathbf{C}_{3}})$,
where $\tau_{\mathbf{C}_{1}}$ and $\tau_{\mathbf{C}_{3}}$ are the identity
function on $X$ and $Y$ respectively, and
$\tau_{\mathbf{C}_{2}}(C=2.6,I=-1.2,G=10.2,E=0.4,M=1.2)=\mathbf{1}\\{2.6-1.2+10.2+0.4-1.2\leq
0\\}=0$. That is, for $\mathbf{C}_{2}$, the intravariable clusters are defined
such that values of $(c,i,g,e,m)$ are divided into two categories depending on
whether $c+i+g+e-m\leq 0$, which we can arbitrarily choose as the binary
values $0$ and $1$. This provides a mapping from every value of
$\mathcal{D}_{\mathbf{V}_{L}}$ to some value of
$\mathcal{D}_{\mathbf{V}_{H}}$. $\blacksquare$

Note that $\tau$ alone does not define an abstraction. While $\tau$ provides a
well-defined mapping from $\mathbf{V}_{L}$ to $\mathbf{V}_{H}$, not every SCM
defined over $\mathbf{V}_{H}$ can be considered an abstraction of
$\mathcal{M}^{*}$. Consider the following example.

######  Example 19.

Consider the SCM $\mathcal{M}_{H}$ defined as follows:

$\mathcal{M}_{H}=\begin{cases}\mathbf{U}_{H}&=\\{U_{X},U_{Z},U_{Y}\\}\\\
\mathbf{V}_{H}&=\\{X,Z,Y\\}\\\ \mathcal{F}_{H}&=\begin{cases}X\leftarrow
f_{X}(u_{X})&=u_{X}\\\ Z\leftarrow f_{Z}(u_{Z})&=u_{Z}\\\ Y\leftarrow
f_{Y}(u_{Y})&=u_{Y}\end{cases}\\\
P(\mathbf{U}_{H})&=U_{X},U_{Z},U_{Y}\sim\operatorname{Bernoulli}(0.5)\end{cases}$

One could argue that $\mathcal{M}_{H}$ is defined over $\mathbf{V}_{H}$,
constructed via $\tau$ from Example 18. However, it is trivial to see that
$\mathcal{M}_{H}$ retains none of the meaning of the variables of
$\mathbf{V}_{H}$ intended by $\tau$ and is clearly oblivious of
$\mathcal{M}_{L}$. After all, there is not even any causal relationship
between the variables defined in $\mathcal{M}_{H}$. Intuitively,
$\mathcal{M}_{H}$ is clearly not an abstraction of $\mathcal{M}^{*}$, and this
is reinforced by the fact that $\mathcal{M}_{H}$ does not match common
definitions of abstractions, such as Def. 15.

However, assuming that we do not have access to $\mathcal{M}_{H}$ or
$\mathcal{M}^{*}$, and can only observe them through their distributions of
the PCH, how could we tell that $\mathcal{M}_{H}$ is not an abstraction of
$\mathcal{M}^{*}$? Certainly, there are inconsistencies in the distribution
too. For example, note that

$\displaystyle P^{\mathcal{M}^{*}}(C+I+G+E-M>0\mid X=1)$
$\displaystyle=P(5-0.2g+u_{C}+u_{CM}-5+u_{I}+u_{IG}+10+u_{G}$
$\displaystyle+u_{IG}+u_{E}+u_{EM}-0.2c-u_{M}-u_{CM}-u_{EM}>0)$
$\displaystyle=P(5-0.2(10+u_{G}+u_{IG})+u_{C}+u_{CM}-5+u_{I}$
$\displaystyle+u_{IG}+10+u_{G}+u_{IG}+u_{E}+u_{EM}$
$\displaystyle-0.2(5-0.2(10+u_{G}+u_{IG})+u_{C}+u_{CM})$ $\displaystyle-
u_{M}-u_{CM}-u_{EM}>0)$
$\displaystyle=P(7.4+0.84u_{G}+1.8u_{IG}+0.8u_{C}-0.2u_{CM}$
$\displaystyle+1.04u_{I}+u_{E}-u_{M}>0)$
$\displaystyle=P(N(0,7.056)+N(0,6.48)+N(0,6.4)+N(0,0.08)$
$\displaystyle+N(0,10.816)+N(0,10)+N(0,10)>-7.4)$
$\displaystyle=P(N(0,50.832)>-7.4)\approx 0.85.$

On the other hand, we see that $P^{\mathcal{M}_{H}}(Z=0\mid
X=1)=P^{\mathcal{M}_{H}}(Z=0)=P(U_{Z}=0)=0.5$. We would expect that if
$\mathcal{M}_{H}$ were an abstraction of $\mathcal{M}^{*}$, then
$P^{\mathcal{M}_{H}}(Z=0\mid X=1)$ should match
$P^{\mathcal{M}^{*}}(C+I+G+E-M>0\mid X=1)$, which is evidently not the case.

The intuition behind this connection is that $\tau$ directly maps cases of
$C+I+G+E-M>0$ to cases of $Z=0$, so their corresponding probabilities should
be the same. If $Q=P^{\mathcal{M}^{*}}(C+I+G+E-M>0\mid X=1)$, we would say
that $\mathcal{M}_{H}$ is not $Q$-$\tau$ consistent with $\mathcal{M}^{*}$,
via Def. 7. $\blacksquare$

Not all functions mapping between two spaces of variables are constructive
abstraction functions. Some abstraction functions, while not being
constructive, may still have qualities that are expected from abstractions.
However, the lack of structure in these functions make certain features of the
abstraction models difficult to define, such as the concept of $Q$-$\tau$
consistency. The following example describes a few such cases.

######  Example 20.

To start, we note that when fixing the intervariable and intravariable
clusters $\mathbb{C}$ and $\mathbb{D}$, the corresponding constructive
abstraction function $\tau$ is deterministic, and the corresponding
$\mathbf{V}_{H}$ is fixed, up to a bijective mapping of its domain. For
example, suppose $\mathbb{C}=\\{\mathbf{A},\mathbf{B},\mathbf{C}\\}$, and
$\mathbb{D}_{\mathbf{A}}=\\{\mathbf{a}_{1},\mathbf{a}_{2}\\},\mathbb{D}_{\mathbf{B}}=\\{\mathbf{b}_{1},\mathbf{b}_{2}\\},\mathbb{D}_{\mathbf{C}}=\\{\mathbf{c}_{1},\mathbf{c}_{2}\\}$.
However, let $\mathbf{V}_{H}=\\{X,Y,Z,W\\}$, where
$\mathcal{D}_{X}=\mathcal{D}_{Y}=\mathcal{D}_{Z}=\\{0,1\\}$ and
$\mathcal{D}_{W}=\\{0,1,2\\}$. While a function could be constructed from
$\mathbb{D}$ and $\mathbb{C}$, it could not be a valid mapping to
$\mathbf{V}_{H}$. First of all, there are four variables in $\mathbf{V}_{H}$
but only three clusters in $\mathbb{C}$, so the mapping could not be one-to-
one.

Even if $X$ were removed, all of the intervariable clusters have values
clustered into two sets, but $W$ is a ternary variable, so a one-to-one
mapping is not possible with any of the intervariable clusters. However, if
$W$ were removed from $\mathbf{V}_{H}$, then $\tau$ could be a valid
constructive abstraction mapping between $\mathbf{V}_{L}$ and
$\mathbf{V}_{H}$, since the other three variables are binary. The number of
variables of $\mathbf{V}_{H}$ and their domain sizes match the number of
clusters of $\mathbb{C}$ and the number within each cluster of $\mathbb{D}$,
respectively. In fact, at least without considering the model distributions,
any cluster of $\mathbb{C}$ could map to any of the variables of
$\mathbf{V}_{H}\setminus\\{W\\}$ as they are isomorphic.

In other words, $\tau$ fails to be a constructive tau abstraction for any
choice of $\mathbf{V}_{H}$ that is incompatible with $\mathbb{C}$ and
$\mathbb{D}$. Intuitively, this means that when given $\mathbb{C}$ and
$\mathbb{D}$, the function $\tau$ and high level variables $\mathbf{V}_{H}$
are already fixed, which is how the rest of this paper is framed. However,
there can exist cases with some function $\tau$ mapping from $\mathbf{V}_{L}$
to $\mathbf{V}_{H}$ such that $\tau$ cannot be considered a constructive
abstraction function for any choice of $\mathbb{C}$ and $\mathbb{D}$. Some
cases may even appear to be valid abstractions intuitively.

Consider an example of a company board setting. Suppose in a company, the
board consists of the CEO, Alice ($A$), and two vice presidents, Bob ($B$) and
Charlie ($C$). When voting on company policies, each board member can choose
to vote for ($+1$), vote against ($-1$), or abstain ($0$). In other words,
$A$, $B$, and $C$ are ternary variables with domain $\\{-1,0,+1\\}$. While
data can be collected on each of the members’ voting behaviors
($V_{L}=\\{A,B,C\\}$), it may be more sensible to aggregate the votes into a
more useful quantity. Suppose two high level variables are computed
($\mathbf{V}_{H}=\\{X,Z\\}$) through some abstraction function
$\tau=(\tau_{X},\tau_{Z})$, defined as follows.

$\displaystyle\tau_{X}(a,b,c)$ $\displaystyle=\begin{cases}+1&a+b+c>0\\\
0&a+b+c=0\\\ -1&a+b+c<0\end{cases}$ $\displaystyle\tau_{Z}(a,b,c)$
$\displaystyle=\begin{cases}a&a\neq 0\\\ +1&a=0,b+c>0\\\ 0&a=0,b+c=0\\\
-1&a=0,b+c<0\end{cases}$

In words, $X$ is the aggregate vote that is simply the majority vote of all
three members. On the other hand $Z$ is an aggregate vote that prioritizes
Alice’s vote, as she is CEO. If she abstains, then it is an aggregate of the
votes of the vice presidents. For example, $\tau(A=+1,B=-1,C=-1)=(X=-1,Z=+1)$.

Perhaps in company matters, it is more useful to use the variables $X$ and $Z$
over the individual votes of the board members. However, $\tau$ cannot be
considered a constructive abstraction function for any choice of clusters
$\mathbb{C}$ and $\mathbb{D}$. This is because both $X$ and $Z$ change values
depending on all of the values of $A$, $B$, and $C$, so the variables of
$\mathbf{V}_{L}$ cannot be cleanly separated into two different clusters.

Even so, this choice of $\tau$ seems like it could result in a valid
abstraction. The difficulty lies with the analysis of causal quantities after
$\tau$ is fixed. Without the notion of clusters, the definition of $Q$-$\tau$
consistency fails to work, so it is no longer clear what causal quantities
correspond to what. For example, what would an intervention of $A=+1$ imply on
the $\mathbf{V}_{H}$ level? Or, what would an intervention on $Z=+1$ imply on
the $\mathbf{V}_{L}$ level? The answer might change depending on the setting,
or there may not even be an answer that makes sense at all. This is why prior
works like Rubenstein et al. (2017) define the mapping between interventions
separately from $\tau$, which maps the variables.

In this particular case, provided that queries do not require separation of
$X$ and $Z$ (e.g. queries like $P(X_{Z=z}=x)$ are not needed), then it may be
sensible to cluster $A$, $B$, and $C$ together into one intervariable cluster
and then simply have one variable in $\mathbf{V}_{H}$ used for downstream
tasks. This allows for the theory in this paper to be applied, significantly
reducing the complexity of defining the abstractions. $\blacksquare$

See the following for a negative example of an abstraction of the drug example
in Ex. 3.

######  Example 21.

Continuing Example 3, consider the following SCM $\mathcal{M}_{H}$.

$\displaystyle\mathbf{U}_{H}$ $\displaystyle=\\{U_{X},U_{Y}\\}$
$\displaystyle\mathbf{V}_{H}$ $\displaystyle=\\{X,Y\\}$
$\displaystyle\mathcal{F}_{H}$ $\displaystyle=\begin{cases}X\leftarrow
f^{H}_{X}(u_{X})&=u_{X}\\\ Y\leftarrow f^{H}_{Y}(u_{Y})&=u_{Y}\end{cases}$
$\displaystyle P(\mathbf{U}_{H})$
$\displaystyle=\begin{cases}P(U_{X}=1)&=0.5\\\ P(U_{Y}=1)&=0.2\end{cases}$

Indeed, this choice of $\mathcal{M}_{H}$ is defined over $X$ and $Y$. However,
it does not seem like there is any connection between $\mathcal{M}_{H}$ and
$\mathcal{M}_{L}$ from Example 3, even if
$\mathbf{V}_{H}=\tau(\mathbf{V}_{L})$. To verify this, we can compare the
distributions induced by the two models. Note that while
$P^{\mathcal{M}_{L}}(Y=1\mid A=1,B=1)\approx 0.853$, we see that
$P^{\mathcal{M}_{H}}(Y=1\mid
X=1)=P^{\mathcal{M}_{H}}(Y=1)=P^{\mathcal{M}_{H}}(U_{Y}=1)=0.2$. It seems that
these two quantities should be related, as $\tau$ maps $(A=1,B=1)$ to $X=1$,
yet they are clearly not equal in the two models. This is similar for the
causal effect $P^{\mathcal{M}_{L}}(Y_{A=1,B=1}=1)$. Computing
$P^{\mathcal{M}_{H}}(Y_{X=1}=1)=P^{\mathcal{M}_{H}}(Y=1)=0.2$ actually yields
the same result, which is clearly incorrect. In fact, it even seems that the
causal relations are incorrect, as $f^{H}_{Y}$ does not use $X$ as an input.

$\blacksquare$

Example 21 shows an example of $\mathcal{M}_{H}$ that is a poor abstraction of
$\mathcal{M}_{L}$ despite the fact that it is defined over the space of
$\mathbf{V}_{H}$ mapped by $\tau$. From the example, intuition tells us that a
proper abstraction of $\mathcal{M}_{L}$ should match $\mathcal{M}_{L}$ in
certain quantities, including observational, interventional, and
counterfactual quantities. Specifically, there are quantities induced by
$\mathcal{M}_{L}$ that appear to have matching counterparts in
$\mathcal{M}_{H}$ based on $\tau$. This notion is made concrete through the
concept of $Q$-$\tau$ consistency (Def. 7).


Recent CMS results on soft QCD physics

Rajat Gupta$\star$ (on behalf of the CMS Collaboration)

Panjab University, Chandigarh

*<EMAIL_ADDRESS>
50th International Symposium on Multiparticle Dynamics
(ISMD2021)
12-16 July 2021 10.21468/SciPostPhysProc.?

## Abstract

Soft quantum chromodynamics (QCD) measurements play an important role in
fundamental QCD studies as well as in tuning of corresponding Monte Carlo
generator models for a good description of experimental data. Recent results
of soft QCD measurements with the CMS experiment, such as minimum
bias/underlying event physics, double parton scattering and forward jet
production production are presented.

###### Contents

  1. 1 Introduction
  2. 2 DPS study using inclusive four jets process
  3. 3 DPS study using inclusive Z+jets process
  4. 4 Hard color singlet exchange in dijet events
  5. 5 Central exlusive $\pi^{+}\pi^{-}$ production in pp collisions
  6. 6 Summary

##  1 Introduction

The dominant component of proton-proton (pp) collision comes from the
processes involving soft and semi-hard quantum chromodynamics (QCD). Soft
interactions cannot be described in terms of perturbative QCD, so it requires
phenomenological models for their description. In this document, latest CMS
[1] and TOTEM [2] measurements on soft QCD physics and their comparisons with
predictions of various theoretical models are presented.

##  2 DPS study using inclusive four jets process

Because of the complex structure of nucleons, many parton-parton interactions
can occur in a single pp collision. Double parton scattering (DPS) refers to
events in which two hard parton-parton interactions occur in single pp
collision. A study of inclusive production of four-jets in pp collisions at a
center-of-mass energy ($\sqrt{s}$) of 13 TeV is presented [3].

The study is performed using the data, which correspond to an integrated
luminosity of 0.042 $pb^{-1}$, collected using the CMS detector at the CERN
LHC in 2016, during a special data taking period with a low probability for
several pp interactions occurring within the same or neighbouring bunch
crossings (hence referred to as ‘pileup’). Differential cross section is
measured as functions of jet $p_{T}$, jet pseudorapidity ($\eta$), and several
other observables that describe the angular correlations between the jets.

Two phase space regions are choosen, each determined by choices on jet
$p_{T}$. Region I is defined by four leading jets which are required to lie
within $|\eta|<$ 4.7, with $p_{T}$ thresholds of 35, 30, 25, and 20 GeV.
Asymmetric selection is chosen over symmetric ones, since the symmetric
threshold tends to reduce the DPS contribution with respect to the single
parton scattering (SPS) fraction, according to higher-order calculations or
calculations performed in the $k_{T}$-factorization framework. In region II,
the $\Delta S$ distribution is obtained with $p_{T}$ thresholds of 50, 30, 30,
and 30 GeV. Since there are no jet triggers with $p_{T}$ thresholds below 30
GeV, this selection is required to obtain the cross sections $\sigma_{A}$ and
$\sigma_{B}$ used in the extraction of $\sigma_{eff}$. The measured
distributions are corrected for detector effects with the TUnfold program and
compared with pythia 8, herwig7, herwig++ and multijet models.

Different parton shower implementations have less impact on the $\Delta S$
distribution. The DPS tune CDPSTP8S1-4j describes the shape within
uncertainty, whereas all other models predictions underestimate the data at
low $DeltaS$, indicating a possible need for more DPS contribution.

The contribution of the DPS is extracted using a template fitted to the data,
SPS distributions derived from Monte Carlo event generators, as well as a DPS
distribution obtained from inclusive single-jet events in data. The results of
$\sigma_{\rm eff}$ derived with the models based on the current CP5 and CH3
tunes and where the hard MPI has been eliminated are shown in figure 2.

All results, except for the values obtained with the NLO $2\to 2$ models,
agree with the measurement performed by the ATLAS collaboration at a
$\sqrt{s}$ of 7 TeV, where a $\sigma_{\rm eff}$ equal to
$14.9^{+1.2}_{-1.0}\mathrm{(stat)}^{+5.1}_{-3.8}\mathrm{(syst)}~{}\mathrm{mb}$
was found, while none agree with the value of
$21.3^{+1.2}_{-1.6}~{}\mathrm{mb}$ from the CMS measurement at a $\sqrt{s}$ of
7 TeV, which is more in line with the results obtained with some of the models
based on older underlying event (UE) tunes.

##  3 DPS study using inclusive Z+jets process

Another DPS measurement is performed to explore various observables which are
sensitive to the DPS using Z+jets process with the CMS detector at $\sqrt{s}$
= 13 TeV, where the Z boson decays into two oppositely charged muons [4]. In
order to be consistent with previous DPS measurements, jets are considered to
have a $p_{T}$ threshold of 20 GeV.

The measured production cross section in the fiducial region is found to be
58.5 $\pm$ 0.3 (stat) $\pm$ 7.0 (syst) $\pm$ 1.2 (theo) $\pm$ 4.0 (lumi) pb
for Z + $\geq$ 1 jet events and 44.8 $\pm$ 0.4 (stat) $\pm$ 3.7 (syst) $\pm$
0.5 (theo) $\pm$ 1.1 (lumi) pb for Z $+\geq$ 2 jets events. The cross section
is well described by various simulation within uncertainties, except MG5_aMC +
pythia 8 (with DPS specific CDPSTP8S1-WJ tune), which overestimates the
measurement by 10%.

The differential cross section as a function of $\Delta\phi$ between the Z
boson and the leading jet is shown in figure 2. The prediction of MG5_aMC +
pythia 8 without MPI underestimates the measurement by 50% at lower
$\Delta\phi$, which indicates the sensitivity of this distribution to the
presence of MPI. Different MC predictions describes the the differential cross
section distribution within uncertainties, except for the MG5_aMC + pythia 8
with DPS-specific tune CDPSTP8S1-WJ, which deviates up to 10–20%, but
describes the shape of the observable (not shown in Figure). The results
presented will be an important input for the development of DPS-specific tunes
and global tunes combined with other soft QCD measurements in pp collisions on
the TeV scale.

Figure 1: Comparison of the values for $\sigma_{\rm eff}$ extracted from data.
The results from four-jet measurements performed at lower $\sqrt{s}$ are shown
alongside the newly extracted values. Reproduced from Reference [3] (CC BY
4.0).

Figure 2: Differential cross sections as a functiona of $\Delta\phi$ between
the Z boson and the leading jet for Z + $\geq$ 1 jet events. In the bottom
panels, the total uncertainty for data is indicated by the solid yellow band
centred at 1. Reproduced from Reference [4] (CC BY 4.0).

##  4 Hard color singlet exchange in dijet events

One of the processes sensitive to Balitsky-Fadin-Kuraev-Lipatov (BFKL)
dynamics [5] is the production of two jets separated by a large rapidity
interval devoid of particle activity, known as Mueller-Tang jets [6] or jet-
gap-jet events. The CMS detector with its unprecedented $\sqrt{s}$ and large
detector coverage in rapidity provides an ideal tool for testing BFKL dynamics
and understanding the role of diffraction at large momentum transfers in
strong nuclear interactions. The study [7] is performed with the low
instantaneous luminosity data collected in pp collisions at $\sqrt{s}=$ 13 TeV
by the CMS and TOTEM experiments. The fraction of jet-gap-jet events to events
where the two jets have similar kinematics, $f_{\rm CSE}$, is measured as a
function of the $\eta$ difference between the leading two jets,
$\Delta\eta_{\rm jj}$, and the transverse momentum ($p_{\rm T}$) of the sub-
leading jet ($p_{\rm T}^{\rm jet2}$) (Figure 3). An increase with
$\Delta\eta_{\rm jj}$ and a weak dependency on $p_{\rm T}^{\rm jet2}$ are
observed. The present analysis sets a constraint on the theoretical treatment
of rapidity gap survival probability.

Figure 3: Gap fraction, $f_{\rm CSE}$, measured as a function of
$\Delta\eta_{\rm jj}$ and $p_{\rm T}^{\rm jet2}$ in inclusive dijet events and
in dijet events with a leading proton. Reproduced from Reference [7] (CC BY
4.0).

##  5 Central exlusive $\pi^{+}\pi^{-}$ production in pp collisions

Central exclusive and semiexclusive production of $\pi^{+}\pi^{-}$ pairs is
measured with the CMS detector in pp collisions at the LHC at $\sqrt{s}$ of
5.02 and 13 TeV [8]. Exclusive events are selected by vetoing additional
energy deposits in the calorimeters and by requiring two oppositely charged
pions identified via their mean energy loss in the tracker detectors.

Figure 4: Differential cross sections as functions of mass compared with
generator-level simulations for the 5.02 (left) and 13 TeV (right) data sets.
Reproduced from Reference [8] (CC BY 4.0 ).

These events are used together with correction factors to obtain invariant
mass, $p_{T}$, and rapidity distributions of the $\pi^{+}\pi^{-}$ system. The
total and differential cross sections of exclusive and semi-exclusive central
$\pi^{+}\pi^{-}$ production are measured as functions of invariant mass,
$p_{T}$, and rapidity of the $\pi^{+}\pi^{-}$ system in the fiducial region
defined as $p_{T}$ ($\pi$) $>$ 0.2 GeV and pseudorapidity $|\eta(\pi)|$ $<$
2.4.

The observed mass spectrum (shown in Figure 4 exhibits resonant structures,
which can be fitted with a simple model containing four interfering Breit-
Wigner functions, corresponding to the $f_{0}$ (500), $\rho^{0}$(770), $f_{0}$
(980), and $f_{2}$ (1270) resonances, and a continuum contribution modeled by
the DIME MC. The exclusive production cross sections are extracted from this
fit. The measured total exclusive $\pi^{+}\pi^{-}$ production cross section is
32.6 $\pm$ 0.7 (stat) $\pm$ 6.0 (syst) $\pm$ 0.8 (lumi) and 33.7 $\pm$ 1.0
(stat) $\pm$ 6.2 (syst) $\pm$ 0.8 (lumi) mb for 5.02 and 13 TeV, respectively.
The obtained cross sections of $\rho^{0}$ (770) production are higher than the
STARLIGHT model prediction, which can be explained by the presence of
semiexclusive production which is not modeled by the STARLIGHT generator.

##  6 Summary

An outline of various soft QCD and diffractive measurements have been
presented. An accurate modelling of soft QCD processes is vital for various
complex analysis such as precision mass measurement of top quark. Although
existing measurements already challenge different soft QCD models, there are
still many possibilities to direct theory with new measurements.

## References

  * [1] CMS Collaboration, “The CMS Experiment at the CERN LHC,” JINST 3 (2008) S08004.
  * [2] TOTEM Collaboration, “The TOTEM experiment at the CERN Large Hadron Collider,” JINST 3 (2008) S08007.
  * [3] CMS Collaboration, “Study of double-parton scattering in the inclusive production of four jets with low transverse momentum in proton-proton collisions at $\sqrt{s}$ = 13 TeV,” arXiv:2109.13822 (Submitted to JHEP).
  * [4] CMS Collaboration, “Measurements of Z bosons plus jets using variables sensitive to double parton scattering in pp collisions at 13 TeV,” arXiv:2105.14511 (Accepted by JHEP).
  * [5] Andreas Ekstedt and Rikard Enberg and Gunnar Ingelman, “Hard color singlet BFKL exchange and gaps between jets at the LHC,”https://arxiv.org/abs/1703.10919.
  * [6] A. Mueller and W.-K. Tang, “High energy parton-parton elastic scattering in QCD," Physics Letters B 284 (1992) 123.
  * [7] CMS Collaboration, “Hard color-singlet exchange in dijet events in proton-proton collisions at $\sqrt{s}=$ 13 TeV, Phys. Rev. D 104 (2021) 032009.
  * [8] CMS Collaboration, “Study of central exclusive $\pi^{+}\pi^{-}$ production in proton-proton collisions at $\sqrt{s}=$ 5.02 and 13 TeV, Eur. Phys. J. C 80 (2020) 718.