Do Not Marginalize Mechanisms, Rather Consolidate!
Authors: Moritz Willig
Matej Zečević
Devendra Singh Dhami
Kristian Kersting
Venue: NeurIPS 2023 poster (2023)
Keywords: Structural Causal Models
Marginalization
Consolidation
Compression
TL;DR: We introduce the concept of consolidating causal mechanisms to transform large-scale SCM while preserving consistent interventional behaviour.

Abstract: Structural causal models (SCMs) are a powerful tool for understanding the complex causal relationships that underlie many real-world systems. As these systems grow in size, the number of variables and complexity of interactions between them does, too. Thus, becoming convoluted and difficult to analyze. This is particularly true in the context of machine learning and artificial intelligence, where an ever increasing amount of data demands for new methods to simplify and compress large scale SCM. While methods for marginalizing and abstracting SCM already exist today, they may destroy the causality of the marginalized model. To alleviate this, we introduce the concept of consolidating causal mechanisms to transform large-scale SCM while preserving consistent interventional behaviour. We show consolidation is a powerful method for simplifying SCM, discuss reduction of computational complexity and give a perspective on generalizing abilities of consolidated SCM.

Do Not Marginalize Mechanisms, Rather Consolidate!
Moritz Willig Technical University of Darmstadt moritz. wi willighcs. tu-darmstadt... de
Matej Zečević Technical University of Darmstadt matej wwceeviiuuu de
Devendra Singh Dhami
Kristian Kersting
Eindhoven University of Technology" Hessian Center for AI (hessian.Al) d.s. S. www.hhhattuu .ll
Technical University Darmstadt
Hessian Center for Al (hessian German Research Center for AI (DFKI)
kerstingecs tu-darmstadt.com de
Abstract
Structural causal models (SCMs) powerful tool for understanding the complex causal relationships that underlie many real-world systems. As these systems grow in size, the number of variables and complexity of interactions between them does, too. Thus, becoming convoluted and difficult to analyze. This is particularly true increasing
the context of machine learning and artificial intelligence, where amount of data demands for new methods to simplify and compress large scale SCM. While methods for marginalizing and abstracting SCM already exist today, they may destroy the causality of the marginalized model. To alleviate this, we introduce the concept of consolidating causal mechanisms to transform large-scale SCM while preserving consistent interventional behaviour. We show consolidation is a powerful method for simplifying SCM, discuss reduction of computational complexity and give perspective on generalizing abilities of consolidated SCM.
Introduction
Even complex real world systems might be modeled using structural causal models (SCM) Pearl
2009 and several methods exist for doing so automatically from data (Spirtes et al. 2000 Pearl
2009 Peters et al.] 2017]. While technically reflecting the causal structure of the systems under
consideration, SCM might not entail intuitive interpretations
the user. Large scale SCM like,
appearing for example in genomics, medical data [Squires al.) 2022 Ribeiro-Dantas et al. 2023 or machine learning (Schölkopf et al.) 2021 Berrevoets et al. 20033 may become increasingly complex and thereby less interpretable. Contrary to this, computing average treatment effects might be too uninformative given the specific application, as the complete causal mechanism compressed into a single number. Ideally user could express the factors of interest and yield a reduced causal system that isolates the relevant mechanism from the rest of the model.
In contrast to other probabilistic models, SCM model the additional aspect of interventions. Consider for example a row of dominoes and its corresponding causal graph as shown Figure [ If the starting stone it tipped over, it will affect the following stones, causing the whole row to fall. Humans usually have a good intuition about predicting the unfolding of such physical systems [Gerstenberg. 2022 Beck and Riggs| 2014 4 Zhou et al.) 2023 Second to that, it is easy to imagine what would happen, we were to hold onto a domino stone, that is, intervening actively upon the domino sequence. Alternatively, can programmatically simulate these systems to reason about their outcomes. A simulator tediously computes and updates positions, rotations and collision states all objects in
DSD contributed while being with Aessian.AI and TU Darmstadt before joining TUle.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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Causal Model
Consolidation
Marginalization
17.7.P.P
X
X
Figure 1: Consolidation vs. Marginalization. Even simple real-world systems, like this row of dominoes, composed of numerous intermediate steps. Classical structural causal models require the explicit evaluation of the individual structural equations to respect possible interventions along the computational chain and yield the final value of X5. The intermediate steps (X2,X3, X3) might be marginalized to obtain a simplified functional representation. Marginalization, however, loses some causal interpretation of the process, interventions on the marginalized variables can no longer be performed. Consolidation of causal models simplifies the graph structure (compare to Appendix [.I]] while respecting interventions on the marginalized variables. Thus, preserving the ability intervene on the underlying causal mechanisms, (Best viewed in color.)
the system. Depending on the abstraction level of our SCM, computations might be simplified to represent individual stones binary variables, indicating stone standing up or getting pushed over. Nonetheless, classical evaluation of simplified SCM is still performed step by step to be able to respect possible interventions the individual stones. Given that we might only be interested in the outcome. That is, whether or not the last stone will tip over, computing all intermediate steps seems be waste of computation, as already noted by Peters and Halpern [202T). Under these premises, we are interested in preserving the ability to intervene while also being computationally efficient. Classical marginalization [Pearl [009 Rubenstein et al.) 2017] is of no help to as it destroys the
causal aspect of interventions attached to the variables.
Consolidation Marginalization. By marginalizing do not only remove variables, but also all associated interventions, destroying the causal mechanisms of the marginalized variables. The insight of this paper, as alluded to in Figure [ (center), is that there exists an intermediate tier of consolidated models that fill the gap between unaltered SCM and ones with 'classical' marginalization applied. Consolidation simplifies the causal graph by compressing computations of consolidated variables into the equations of compositional variables that are functionally equivalent to the initial model, while still respecting the causal effects of possible interventions. As such consolidation generalizes marginalization in the sense that marginalization can be modeled by consolidating without interventions (I 0: see Def. and Sec. 3.. questions involve causality, then consolidation necessarily needs to be considered since it can actually handle interventions (all cases where I & 9). If causal questions are not of concern, then marginalization can be considered as remaining the 'standard' marginalization procedure. One perspective on our approach is to describe consolidation as 'intervention preserving marginalization
Structure and Contributions of this paper. In section two we discuss the foundation of SCM and related work. In section three formally establish the composition of structural causal equations, partitioned SCM, and finally consolidated SCM. In section four we discuss the possible compression and computational simplifications resulting from consolidating models. We present an applied example for simplifying time series data and in a second example demonstrate how consolidation reveals the policy of game agent. Finally, in section five, we discuss generalizing abilities of our method and provide a perspective on the broader impact and further research. The technical contributions of this paper are as follows:
We define causal compositional variables that yield functionally equivalent distributions to SCM under intervention.
We formalize (partially) consolidated SCM by partitioning SCM under a constraint that guarantees the consistent evaluation with respect to the initial SCM.
We discuss conditions under which consolidation leads to compressed causal equations.
We demonstrate consolidation on two examples. First, obtaining a causal model of reduced size and, secondly, revealing the underlying policy of a causal decision making process.

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2 Preliminaries and Related Work
In general we write sets variables in bold upper-case (X) and their values in lower-case (x). Single variables and their values written in normal style (X, x). Specific elements of a set are indicated by subscript index (X,). Probability distributions over a variable X or a set of variables X are denoted by x and Px respectively. A detailed list of notation can be found in Appendix E
Structural Causal Models provide a framework to formalize a notion causality via graphical models Pearl 2009] From a computational perspective, structural equation models (SEM) can be considered instead of SCM [Halpern 2000 Spirtes et al. 2000]. While focusing on computational aspects of consolidating causal equations, use Pearl's 's formalism of SCM. Modeling causal systems using SCM over SEM does not affect our freedom, as Rubeentten et al. [017] show consistency between both frameworks, Similar to earlier works of Halpem [2000]. , Beckers and Halpern [2019] and Rubenstein et al.] (2017). we do not assume independence of exogenous variables and model SCM with explicit set of allowed interventions.
Definition A structural causal model a tuple M (V, UV. forming a directed acyelic graph 9 over variables X = (Xi...., Xx} taking values in X = [I.E Ns subject to strict partial order <x. where
V = (X XNJ C N, K is the set of endogenous variables.
- =XX+V.. XK} J is the set of exogenous variables.
F is the set of deterministic structural equations, V; == Ji(X'), where the parents are
X' XX EX <x V.).
I IS (4i,. Lvex where U; is the i-th element of v, Is, indicates an intervention do(X; = vi) and such that I I - E I. I is the set of perfect interventions under consideration. A perfect intervention do = U.) replaces the unintervened f. by the constant assignment V₁:= U4.
Pu is the probability distribution over U.
As we focus on computational aspects of SCM, we do not regard exogenous variables to be latent, but rather consider them take values which are not under control of the causal system itself. As such, their values are not determined via any structural equation. By construction of I at most one intervention on any specific variable can be included in any intervention set I. The additional constraint enforces that Z is closed under subsets, i.e. that any subset of any E I is also part of I. This condition is placed to yield valid intervention sets when partitioning the SCM. Every A entails DAG structure G (X, E`) consisting of vertices X and edges E, where directed edge from X, to X, exists if 3xg, I1 E fx.f.(x', (x',x)) * fi(',, x1). For every variable X; we define ch(X,), pa(X, and an(X. as the set direct children, direct parents and ancestors respectively, according to g Additionally, every M entails observational distribution P by propagating Pu through the structural equations. Any perfect intervention / on a variable X; replaces f. with a new probability distribution P,. As a consequence M entails infinitely many intervened distributions PM--
Related Work. Several works acknowledge the need for model simplification when working with causal models at different levels of modeling detail or finding consistent mappings between two
already existing causal models [Rubenstein et al.) 2017 Chalupka et al.) 2016, Beckers et al. 2020 Zennaro et al. 2023 Brehmer al. 2022]. However, whenever providing explicit methods of mapping SCM, marginalization is considered as tool of removing variables. Several other works have been dedicated to proving consistency and identifiability results for grouping or clustering variables in general Anand et al. 2022 Squires et al. 2022). Works on abstractions by [Beckers) and Halpern, 2019 Beckers et al. 2020 focus on simplifying models by mapping between SEM of different levels of abstractions. With regard to computational aspects, Rubenstein al. (017
R
"e define ch (h) pa(X) and an(X) for sets of variables X, as the union of sets gained by individual variable evaluations, e.g., pa(X)= Uxex pa(X).
*e always reference a distribution with respect some SCM M, therefore, if we write PM then this the distribution over the full variable set, that is, Px.
3

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demonstrate the causal consistency of SEM, providing simplifications results for marginalizing SEM. However, their theorems (cf. Sec.5) explicitly exclude interventions on the marginalized variables.
3 Consolidation of Causal Graphical Structures
In this section, we present an approach to consolidating structural equation systems under interven- tion. This is the key contribution of this work compared to previous works that only considered marginalization of unintervened subsystems [Pearl] 2009 Peters et al. 2017 Rubenstein et al.) 2017]. The focus is on computational aspects of marginalizing intermediate variables while preserving effects of interventions. A formalization of marginalizing intervenable structural equation systems introduced in this section. Section D exxamines conditions under which consolidation leads to an actual reduction in complexity, followed by two practical examples.
We start with the definition of a Causal Compositional Variable (CCV) that has similar semantics to cluster DAGs [Anand et al. 2022]. in that both capture the causal semantics over a set of variables. In contrast to cluster DAGs, CCVs are defined over an SCM .M and moreover expose an interface for explicitly applying interventions to the individual variables inside the CCV. We define CCV with a corresponding function p. that takes the exogenous variables U as its input and outputs the values of subset E C V. Thus we write PE to denote the set of computed variables. To be able to condition on interventions, P takes the set of interventions I as it would be applied to the SCM as its second argument.
Definition 2 (Causal Compositional Variable) A variable X == := E is causal compositional variable over some subset E S V of an SCM M. if a consolidation function Pe (x 11, I) + X E exists for which Ex P for all I E I. where PL is the distribution of target variables in under some intervention set I.
Put in simple terms, PE yields the same values for E as would be determined by evaluation of the initial SCM M given any u Pu. Naturally, there always exists such a function fe for every E C V, which is computing e E E via evaluation of M itself. However, PE is not required to adhere to the computation sequence imposed by the structural causal model M. In particular, Ps is not required explicitly compute the intermediate values of any Vi V E, which gives way to simplifying internal computations. As such CCV serves as a possible stand-in for replacing whole SCM by function of possibly simpler computational complexity:
Definition (Consolidated SCM) Given a causal compositional variable X := PP (U,(U)) and some base SCM M, we cull ME (E, UM: PE, I. PUε) a consolidated SCM.
The distributions of the consolidated SCM PMr are not equal to that of the initial SCM P.M, since Me only computes a subset E S V of all endogenous variables. However, for that subset E, the initial SCM and consolidated model yield the same P1 for all I E I.
3.1 Partition of Structural Causal Models
So far, we considered constructing compositional variables from SCM such that they exhibit functional equivalent behaviour and, by doing so, are able to replace base SCM by consolidated SCMs using CCVs. However, compositional variables trade off the 'semantic' graph structure of classical SCM against a computationally simpler (refer to Sec. 4 but 'black box' function. In practice we might, therefore, only want replace certain parts of an SCM with consolidated functions. To achieve this goal, we formalize a partition of base SCM into multiple sub SCM. Multiple other works have considered the existence of joint variable clusters within SCM [[nand al. 2022] (Squires al.) 2022]. However, allowing for arbitrary clusters may induce cycles to the model, which would be undesirable. In our work we constrain the clustering by requiring partitions that enforce acyclicity and, therefore, ensure a well defined evaluation order that is consistent with that of the initial SCM.
Endogenous nodes of base SCM M can be partitioned into L mutually exclusive exhaustive components A = { [A E P(V) Ø : i E {I...., L}}} with V A,, Ay E A i i + j A, A,, = and UlE! 1 A V. We also call A the (exhaustive) partition. We can any cluster A A to
"To be precise x llver Xi, where I[ is the n-ary Cartesian product.

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B
B
B
D
C
D
X
E
Figure 2: Consolidating SCMs. (Left) The base graph of an exemplary SCM M gets deconstructed into three sub SCM using the partition set A {{A), BB, D). (C, E)). Exogenous variables are displayed with dashed circles. (Center) A subgraph g' within a larger base SCM. There exists directed path that exits and re-enters G", thus preventing self-enclosed evaluation of G'. (Right) Consolidation of a sub SCM into a multivariate compositional variable. X2 is an aspect variable chosen by the user, (X2 E). X4 and X5 are needed for further computation, thus E' = (X2, X4, XsJ. The value of X2 is computed via PE" and interventions can be performed via its parameter I. The dotted line indicates that X₂ is not an 'independent" variable. Specifically is not allowed to intervene on X2 via 'egge cutting', making it independent of Pe (and in consequence causing Pr to compute inconsistent values for X6 and X7).
form new sub SCM MA: .MA- structural equations of A, and PU, is the distribution over UA induced by the base SCM. As some intervention I I might intervene variables which are no longer part of MA, define a mapping
PU,)) where = pa(A) A. FA are the
dAA -I IA which removes those invalid interventions: e' (1) := (do(V; = v,) E I: V E V Consequently define IA == (I) I Σ}. As expected, whenever set of interventions I does not intervene on any V E A, thA maps it to the empty set. For notational brevity, we assume the implicit application of on any whenever we apply interventions to sub SCM. Figure 2 (left)
presents an exemplary construction of sub SCM from a given partition of a base SCM.
Unconstrained partitions may divide SCM in arbitrary way. To guarantee evaluation order of the individual sub SCM that is consistent with that the base SCM we need to ensure that any particular sub SCM can be evaluated in a continuous, self-enclosed manner. That is, no intermediate evaluation of external nodes, V & A, is required. Figure (center) illustrates a counter-cxample of a non-complying partition where an intermediate external evaluation to G" is required. To prevent such cases we require the partitions to yield a strict partial ordering under the following definition: the binary relation A₁ Rx A2 - 3A,- A1, Aj A₂ A, <x A, holds if at least variable in A₁ needs to be evaluated before some other variable in A according to <x of the base SCM. We call partition A "according M" iff Rx is strict partial orden over all A A.
Definition 4 (Partitioned SCM) Given an exhaustive partition A, partitioned SCM MA for some base SCM M is defined as MA A,, UUAA,UFA, UZA, UPUA,), {I...L) there exists strict partial order Rx over all A, E according M and every Ma. UA, FA. IA, PUa,) forms a valid sub SCM.
Consistency of partitioned SCM evaluation. To ensure for the consistent evaluation of all sub SCM MA within partitioned SCM MMA need ensure that the evaluation is carried out according to some Rx that is compliant according to the base SCM .M o Doing so, guarantees that the value of every exogenous variable U, of sub SCM MM,, that is not truly exogenous (U; & Mu) - is computed as an endogenous variable V. inside another MA,, that is evaluated before MA, with U, Vj. For example G2 in Fig. 2 (left) computes the values of B and D, required as exogenous variables by G3. Lastly, during evaluation, all MMA need to agree on the same set of applied interventions. This is done by fixing a particular I' ddurnng evaluation and computing the intervention set I'A == A (I') specific to every MA. An algorithm for evaluating partitioned SCM and its proof of consistency are presented in Appendix
In particular Rx is strict partial order, it asymmetric: A A1, Ao A: A1 A2 implying that the evaluation of no two sub SCM mutually depend on each other. As <x partial order, there may exist multiple total orders which comply with the partial ordering of M.
Rx A1).
5

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Algorithm Consolidation of Structural Causal Models
1: procedure CONSOLIDATE(M,AL E)
2: for all A, in A do
3:
Filter aspect variables for the current A; E. E, U (pa(V) | A,) Add variables that required by other sub SCM.
E, f A
4:
5:
UA. < -P((AA) A;
b Define exogenous variables and interventions.
6:
IA. (T) IEI) (fdo(X;
u) E X; E A,} I I}
7:
PE (UA,, + [F): XG; E Aij
> Define a causal compositional variable via PE
8:
PE, f- argmin
Peg k(pe;)
b Minimize representation (see Sec. 4).
s.t. AR: (UA.) PE: (UA.)
9:
MA.E c (E,, U UA PE: IA, PUA,) D Define the sub SCM resulting from A, and E.
10:
end for
11:
MAE < UUE, UPB: UTA. ii
> Merge all MA. E-
12: return MA.E
D Return the consolidated SCM.
13:
end procedure
Figure 3: CONSOLIDATE Algorithm. The above pseudo-code summarizes the consolidation algorithm as described in this paper by utilizing causal compositional variables and partitioned SCM obtain simplified SCM. Depending on the use-case Step 1l might be skipped and the partitioned SCM might be returned instead.
Partial consolidation of SCM. Having defined partitioned SCM allows to selectively swap out arbitrary sub SCM by their consolidated SCM. In Def. 3 we placed no constraints on to allow for arbitrary consolidation of variables. For sub SCM MA that appear within a partitioned SCM MA we need to constrain E to additionally include all variables V E UA such that evaluation of MA additionally computes all variables needed exogenous by other sub SCMs. Fig. (right) shows an exemplary sub SCM with X₂ (green) chosen as a relevant aspect variable by the user, and X4, Xs. being required by evaluations of subsequent SCM. Thus E' = (X2, X4, Xs). Whether consider E or E' depends on the standpoint of the user. From a computational perspective E' is important as it holds all variables that need to be computed by p. On the other hand, the set E captures aspects of the SCM important to the user i.e., variables of interest. We will therefore refer to sub SCM with Me (and in the same breath write (gr) but use MMA,E (see the following Det. to retain the initial set of variables chosen by the user. Having defined consolidated SCM ME. partitioned SCM MA and the required constraint on E we are now equipped with the tools define a partially consolidated SCM that yields a consistent Pe with the base SCM.
Definition (Partially Consolidated SCM) A partially consolidated SCM MA.FF is a partitioned SCM M. such that subset of sub SCM MA are being replaced by consolidated SCM Me where E = (V. E VA: (V, E E) (3MA = (V', U', " P.P. U'))
Algorithm [uummarizes all considerations of this chapter. starting out from subset E and partition A up to a (partially) consolidated SCM MA..E. An exemplary step-by-step application of the algorithm can be found in Appendix [.3] The purpose of the argmin operation in Line 8 is minimize complexity of P':; by finding minimal encoding. We discuss this step in more detail in the following
section. After formally introducing consolidation, we are ready to illustrate its applicability.
4 Compression of Causal Equations
Model consolidation can lead to compression by reducing the model's graph structure and leveraging redundant computations across equations. This may result in smaller, simpler models that are computationally more efficient and easier to analyze. Compressing structural equations a minimal representation is highly dependent on the equations under consideration and probably incomputable for most problems. As there ultimately no way of measuring compressibility of SCM by only considering their connecting graph structure, provide a discussion with regard to some of the

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information-tien teerrriicaaereiccaall implications. Specifically, we discuss compression properties for some of the basic structures appearing within SCM; namely chains, forks and colliders. In this section, we, first, analyze how consolidated models may leverage redundant computations for reducing complexity within chained equation in general. Second, we give a condition under which equations, and their interventions can be dropped from the consolidation model altogether. Thirdly, we analyse how interventions within the consolidated model affect our ability to compress equations. Lastly, we will walk through two examples of model compression.
General compression of equation systems. Using our formalization of (partially) consolidated SCM, we now have the chance to replace certain parts of an SCM with computationally simpler expressions. The notion of what a 'simple' expression may be, varies depending on the application and is subjective to the user. To define measurable metric, we reside to simplified notion of complexity by measuring the representation length of our consolidated equations. We assume that all structural equations of an SCM can be expressed in terms of elementary operators, where each term contributes the same amount of complexity. As such, we can apply Kolmogorov complexity IC Kolmogorov 1963] Then desirable minimal representation of a structural equation fī is one that minimizes X (J): J == argmin /; (((()) s.t. (pp(((()))..)) = (x))(X))).
Classical marginalization reduces the number of variables in a graph. To keep the model consistent after marginalization, all children B == ch(A) of marginalized variable A additionally need to incorporate the values of pa(A) to accommodate for the causal effects that where previously flowing through A into B. This modifies the structural equations of any B E B, f'z == fB o fA. where f' and f'z are the structural equations of B before and after marginalization, respectively. Evaluation of the separate equations JA provides an upper bound on the complexity of the composed representation k(f')) S ((({a) ) + K((f)) [Zvonkin and Levin 1970 Since the consolidated system is not required to compute A explicitly, the encoding length of J might resort to directly computing B from the values of pa(A). Also, the chain rule for Kolmogorov complexity only considers the case of reproducing, SA and [g in their initial forms. In addition to that, we might also use semantic rules to reduce equation length, c.g. by collapsing consecutive additions Yo, E R.3c E R.a + b = c and so Whether consolidation actually leads simplified equations depends strongly on the specific equations and their connecting graph structure. No simplification effects occur in cases of already minimal systems, while strong cancellation occurs in the case of fB- fa being inverses to each other (see Appendix II..). Lastly, we want to refer to Appendix [B.2] where we showcase the insufficiency of matrix composition to obtain minimal function representations in the case of linear systems.
Marginalizing child-less variables. Regardless of the particular causal graph structure, all equations which do not affect PEr can be removed from the model to reduce its overall complexity. In particular we point out that Pg. invariant to all X & an(E'.. By the following deduction we infer that we always consolidate all child-less variables (if not part of E' themselves) from M: V.X e X X\E [(ch(c)( = 0) (VX' E X.X & pa(X')) = (VX' E X.X & an(X') (x')))=XX & an(E)), Since child-less variables do not affect PE. we can not only consolidate but marginalize them. (Reducing to the same scenario as in Rubenstein et al. 2017 Thm. 9]). Therefore, we are allowed to drop interventions do(X, c) with X, E an(E') from the set of allowed interventions. This process can be applied repeatedly until we have pruned the SCM from all child-less variables irrelevant to E'.
4.1 Simplifying Graphical Structures
In contrast to marginalization, consolidation preserves the effects of interventions for consolidated variables. This effectively adds conditional branching to every structural equation fi if some I I with do = c) E I exists:
V=== fc (((pa(V))
if do. c) E else
(1)
While conditional branching might prevent us from compressing equations, we consider that not all variables might be affected by interventions. As such, we might be able to utilize local structures within the graph to simplify equations. In the following we briefly discuss the possibilities of simplifying chains, forks and collider structures within the graphs of SCMs:
Simplifying Chains. Consolidating chains of consequent variables corresponds to 'stacking" struc- tural equations and computing the last non-consolidated variable directly. In the general case,
7

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Consolidation
Marginalization
Lo0 dofSip 1.0 dol = 1.0)
1000
tocl sharmes (%)
0.75
dhargmos
0.70
0.25
0.00
0.00
timon (dayys)
time (dass)
Figure 4: Consolidating real world mechanism. (Left) The causal time-series model of a milling machine, representing tool length L, utilization U. sharpness and accuracy A. (Center-Lefft) Removing child-less nodes L, and A, and controlling for the parents U, yields a simplified causal structure. (Center-Right) Plots for the consolidated structural equation of S. Colored areas show the effects of varying U by one and two sigma (+0005, =0.1) respectively. Dashed grey lines indicate interventions, which are respected truthfully by the consolidated function. (Right) Marginalization, likewise, simplifies the model, but does not allow us to investigate the effects of interventions.
conditional branching complicates the simplification of the stacked equations into single closed- form representation. When considering the case of marginalization, that is without considering interventions, as done in Rubenstein al. [2017]. composition of equations turns into direct function composition X; == fa o fi-1 o fi-2 ..... To this end, a complexity bound on chained equations over finite discrete domains is discussed in Appendix B.3 well consolidation of the motivating dominoes example in Appendix D.I
Simplifying Forks Consolidating the parent node B of fork structure, + B - C, might lead duplication of fB into the equations of both child nodes, f'A := fA o fB, fCc := fc fB. If pa(B) C U, then A and C' will be confounded by exogenous variables. This is the reason why we did not require independence of exogenous variables in Def. [ Still, consistency with the initial SCM is guaranteed, since we require all structural equations to be deterministic. As a consequence, every evaluation of the duplicated structural equations fB inside S'A and I'c yields the same value when given the same inputs. While determinism of structural equations is formally required, illustrate consistent reparameterization non non-deterministic models in Appendix
Simplifying Colliders. Colliders are the most promising graphical structures for simplifying equa- tions. When consolidating and C' of a collider A + B t C, we might leverage mutual information between fA and fc to simplify fg. Especially in the case of pa(A) = pa(C), consider A + X - C for example, we might be able discard fA and fc altogether and compute B directly from X.
4.2 Time Series Example: Tool Wear
We will now demonstrate a simple application of consolidation for a possibly more applied scenario. Imagine that we want create causal model of industrial unit under continuous use, e.g. milling machine. At the end of every work day the length L and sharpness S of the milling cutter are measured. From these measurements other metrics such the cutting accuracy A can be derived. Interventions on the process are performed by grinding the cutter, 'resetting' it to a certain sharpness. While every intervention grinds away some material, the weight and size changes the considered aspect of accuracy. Throughout our recordings we might encounter multiple such interventions. From the data we fit a 'classical' SCM that models the time series on day-to-day basis, V,-1 + V.. We observe the tool to loose some percentage of its sharpness per day depending on its utilization U,. The intervention do (S, 1) resets the sharpness to a constant value, while do(S; = 1, L, = 1) models a tool replacement. As by Def. [ I needs to include do = 1). which might be recalibration of the machine. Figure 4 (left) shows the initial causal graph of the time series model as defined by the following SCM:
negligible for
U {U, \/.50 0.05²))
V
(L,
I
P((do(S, = 1), dL, = 1.0), do(S, = L, 1)})
M
fi(l,u)
:= =(1.0- - 0.002u)l
F
for (8)-1,
= =(1.0- 0.3u)st-1
fa(s)
=0.8
8

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p8| =
'coin' if do(target_com coin =0) Ielse 'flag'
P8g == 'finished'
(2)
282 =
"finished' if do do(target_conl _oiit 0) & Telse 'flag'
D84 = "finished'
ps 'coin'
Рsg 'finished'
p82 = 'finished'
4sa == 'finished"
Figure 5: Complex situations can be easy to understand using consolidation. Encoding the behaviour of agents acting in game environments (left) often results complex causal graphs (right: indeed unreadable due to complexity. A readable version is contained in the Appendix). Even very simple levels with a single agent, coin, power-up and enemy, entail causal graphs that are intuitively non-interpretable. Especially the intertwining of game mechanics and agent behaviour complicates the inference of the agents' actual policy and makes it impossible to judge its performance. In our example suboptimal greedy policy is embedded within the causal graph, which can be made visible using consolidation (bottom, Eq. ( Please note that ps abbreviates 'planning sequence'. In contrast to marginalization (Eq. one can still intervene on the consolidated system.
Now, might be interested in extracting formula for the total tool sharpness S, at arbitrary point in time f. Thus, our consolidation set consists of all S. E = {St). Since U2 is exogenous, we make the additional assumption that the utilization follows normal distribution and simplify to the expected value = 0.5 (Figure g center-left). As laid out before, we can marginalize all child-less variables Lt and A, not part of E'. As all are no longer part of ME VE maps interventions do(S; = 8r, L, = 42) + do(S; = st). Considering the unintervened case, structural equation are now simplified via function composition, fs, = fs, ....Js. which results in the following equation fs. := (1 0.3 0.5) = 0.85'. According to Eq. ( we now have to inspect interventions as potential candidates for conditional branching. All remaining interventions are of the form do(S; = 1.0). Applying an intervention at time to equals shifting the following equations by the time of that last intervention t' == tg. Finally, we arrive at the following consolidated equation:
fs(t) 0.85
where to = max, (i (s) EIAiSH
Fig. (center-right) shows the resulting plot of the consolidated model under interventions do(S12 1) and do 821 1). We successfully demonstrated the power of consolidation models for dynamical system while preserving the ability to intervene. In theory more complex dynamical systems could be consolidated. However, as these kind of self-referential models require more involved discussion, we kindly refer the reader to Bongers et al. 2018 2021] Peters et al. [2022 for further considerations.
4.3 Revealing Agent Policy
In our second example we apply consolidation to a more complex causal graph relating the game state of simple platformer environment to the actions of an agent. See Appendix D.4 for the full causal graph and structural equations. Throughout the level the agent ((1) in Fig, 5) can interact with coin ((2) in Fig.), power-up (3), enemy (4) and the finish flag (5) to accumulate a certain reward (6) by doing so. The power-up is required to interact with the enemy. During play, the agent takes the state of the environment as its input and outputs the state of 'towards coin' 'towards powerupp etc. The order of the agent actions is then recorded via four 'planning sequence i' for i E (1...4 .] variables. A causal graph, like the one presented Appendix D.4] might be extracted automatically from observational data, or designed by an expert. Due to the sheer number of variables and edges, dependencies the obtained SCM are hard to trace. To get a better understanding, we use consolidation to reveal the policy of our agent. We consolidate all endogenous variables except player-

---

entity 'distance' and the "planning sequence" variables. To be able to modify the agents behaviour, we allow interventions by forbidding the agent to target certain entities: I P(fdo coin 0), footarget_enemy = 0), Ltttarrgetpporppp==---
Like before, we consolidate equations considering the unintervened case, and then add back in conditional branching for interventions to yield equations (2) in Figure [ Contrary to the very complex structure of the SCM, the consolidated equation reveals the actually very simple policy of the agent. We find from the consolidated equation that the agent only collects the coin, if not intervened upon, and then heads directly towards the flag. This insight might not be obvious from the initial SCM and, at least, is difficult to spot a priori by looking at the unconsolidated equations. When inspecting the original SCM more closely we find, that a constant factor is added the calculation of "targeting cost powerup." This factor might serve to accommodate for the time lost when speeding up or slowing down towards a target. Furthermore, we see that the agent pursues greedy policy, thus, never considering the overall higher reward of the power-up and enemy together. Instead the policy ignores the power-up, due to its low reward and in consequence also never targets the enemy. This behaviour not only leads to strong simplification of the SCM, but also allows us to discard the imperfect policy, without the need to run possibly costly trials, just to come to the same conclusion.
To summarize, consolidation is a strictly more powerful operation than marginalization. More examples and domains can be found in Appendix D
Conclusions
Consolidation is a powerful tool for transforming SCMs, while preserving the causal aspect of interventions. In addition, consolidating SCMs can lead to more general models. For example, recall the tool wear example of Sec. 4.2. While the initial causal graph operates on discrete time steps our consolidated function provides a continuous relaxation of the causal process. We can evaluate at any point in time E IR and no longer dependent on the day-to-day basis which was modeled by the initial graph. Additionally, the consolidated equation of our motivating example of rows of dominoes (compare Appendix D.II. yields a generalized formula that is independent of the actual number of dominoes that compose the row, by making use of first-order quantifiers. In our discussion of Sec. J saw that can further benefit from consolidation in all cases where the initial SCM does not already represent the smallest possible causal model. Lastly, these simplifications align with our goal of making SCMs more interpretable. Consider that the initial domino SCM only provides a 'local' view on the system, by only providing equations for every individual stone, "If stone A falls it pushes over stone B. except in the case of an intervention. If stone B falls, ... and on. The equation of the consolidated SCM can be directly translated into a single natural language sentence, e.g. "The last domino will fall, if the first domino is pushed over, except in the case of holding onto or pushing over a stone along the way", capturing the causal mechanisms of the system much more intuitively. Our last example of Sec. 4.3 strikingly revealed the sub-optimal, greedy agent behaviour in a game setting. While we illustrated examples that are well suited for consolidation, we are positively inclined to expect consolidation to be helpful towards broad range of applications.
Limitations and Broader Impact. Throughout the paper we considered exact consolidations, in that Def. 2 requires strict equality between Pe (U,,II) and Pt- This assumption might be met in logic and idealized scenarios, but may hinder consolidation of SCM in other applications due to noise inherent to the system. The definition might be relaxed by allowing for small deviations of PE from the distribution of the unconsolidated SCM. Thus, relaxing the strict equality P Xxg = P with
[Pxg- - Pt < e for some small e > O, provides a relaxed consolidation constraint for noisy systems. SCM constitute a well suited framework for representing causal knowledge in the form of graphical models. The ability to trace effects through structural equations that yield explanations about the role of variables within causal models is required to make results accessible to non-expenn non-experts. Actual causation, and only recently, causal abstractions and constrained causal models have come attention in the field of causality [Halpern 2016 Zennaro et al.) 2023 Beckers and Halpern 2019 Blom et al. 2020) and might be beneficial for future works consolidation. Apart from computational advantages, consolidation of SCMs presents itself as a method that enables researchers to break down complex structures and present aspects of causal systems in a broadly accessible manner. Without such tools, SCMs run the danger of being only useful to specialized experts.
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Acknowledgments and Disclosure of Funding
The authors acknowledge the support of the German Science Foundation (DFG) project "Causal- ity. Argumentation, and Machine Learning" (CAML2, KE 1686/3-2) of the SPP 1999 "Robust Argumentation Machines" (RATIO). The work was supported by the Hessian Ministry of Higher Education Research, Science and the Arts (HMWK) via the DEPTH group CAUSE of the Hessian Center for AT (hessian.ai). This work was partly funded by the ICT-48 Network of Al Research Excellence Center "TAILOR" (EU Horizon 2020, GA No 952215) and by the Federal Ministry of Education and Research (BMBF; project "PlexPlain" FKZ O11S1908 1). It benefited from the Hessian research priority programme LOEWE within the project WhiteBox, the HMWK cluster project "The Third Wave of AI." and the Collaboration Lab "Al in Construction" (AICO) of the TU Darmstadt and HOCHTIEF. We thank the anonymous reviewers for their valuable feedback and constructive criticism, which significantly improved this paper. We sincerely appreciate their time and expertise in evaluating research. We acknowledge the of ChatGPT for restructuring some sentences while writing the paper.
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Supplementary Material "Do Not Marginalize Mechanisms, Rather Consolidate!"
A Evaluation of Partitioned SCM
partitioned SCM M,r consists of several sub SCM MA, that, in sum, cover all variables and structural equations of an initial SCM M. Thus, evaluation of partitioned SCM yields the same of values v E V as the original M. Similar to the evaluation of structural equation in the initial M, sub SCM need to be evaluated in specific order guarantee all É My exist. As such, sub SCM can be considered multivariate variables that establish another high-level DAG. The evaluation order determined via the relation Rx defined in Sec. 3.1 and depends on the graph partition A and the order of X imposed by the the initial SCM.
Algorithm Evaluation of partitioned SCM
1: procedure PARITTIONEDSCMVVAAMMAA u, I)
2: 1--
D x will gradually collect all values E X of M Sort clusters by strict partial order imposed by M
3:
for A sort(AA, Rx) do M' t -M' E MA where A' A u' f for EX X E eMu
4:
5:
6:
I'+ (I)
7:
== -MA (u')
8:
=xv
9:
end for
10: {2ę E X; Eyy}
Filter all E to get v V
11: return v
12: end procedure
2 shows the evaluation of partitioned SCM, where MA is the partitioned SCM we want to evaluate, are the values of exogenous variables the initial model M and is the set of applied interventions. The outcomes of sub SCM that are not related via Rx are invariant to the evaluation order among each other. Even though Rx defines the ordering of sub SCM only up to some partial order, SArt(A, Rx) can pick any total ordering that is valid with Rx.
Proof (Consistency of Partitioned SCM Evaluation) Evaluations of M' every, in step 7 com- pute all variables Ve A by evaluating fi of the original SCM, yielding the same values the evaluation of A in M. Therefore P MM, = PM- By Def. every variable V E V contained within some sub SCM M' The evaluation of PartitionedSCMEval is complete, in the sense that all V = UA= = UAEA A are evaluated, as the evaluation of all M' E -MA is guaranteed by iterating over all in step Finally PMM, UAEA PM -AAA PMA -P
B Complexity reduction in function composition
Reduction of encoding length might vary depending on the type and structure of the equations under consideration. No compression of structural equation is gained when the system consolidated equations is already minimal. Compression of equation to an identity function is showcased in the following.
B.1 Compression chained inverses
Reduction to constant complexity for the unintervened system reached in the case of fB f-1 Consider the equation chain of X + A + B with A getting marginalized. Immediately f'z
IB /A = f 1 o ʃa Id follows. Therefore, B = X, which is a single assignment of the value(s) of into B. Remaining complexity within the consolidated function then only due to conditional branching in cases of do(AA a), do b) e I.
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B.2 Matrix composition is not sufficient for compressing equations
The operation of matrix multiplication, as a way of expressing composition of linear functions, stays within the class of matrices. Matrix multiplication, therefore, serves as possible candidate to be considered when consolidating equations and reducing the encoding length of a linear structural systems. When written down an "high-level" view, matrices can expressed in terms of single variables A, E TRMN and matrix multiplication x : MRM x R.NxO - RMxo Assuming equations fy A X and fz == B X, we can reduce the length of the composed equation f/2 : A x B X X by multiply the matrices A and together, fi = C X with = A B. While we effectively reduced the number of high-level symbols written in the equation, we are hiding computational complexity in the structure of the matrix C. The following simple counterexample demonstrates a situation where the size, as well as, the number of non-zero entries even increases:
C
A
B
1 17
0
17
0
0
0
0
1 1
=
0
1
X
0
1 1
0
1
[
Thus, proving that pure matrix multiplication, is not suitable to keep, or even minimize, the size of composed function representations.
B.3 Compression over Finite Discrete Domains
Consolidation may reduce the number of variables within graph, but burdens the remaining equations with the complexity of the consolidated variables. Without the need to explicitly compute values of consolidated variables, we might leverage cancellation effects to simplify equations, as outlined in the main paper. In terms of compression, no guarantees can be given in the general case. However, we will now show, that the often considered case of chained maps between finite discrete domains simplifies or at least preserves complexity.
The cardinality of the image of deterministic function f : 3 between two finite discrete sets X. y' is bounded by the cardinality of its domain: Img(f)l Dom(f) S ( where Img(f) is the image and Dom(f) the domain of f. In particular, the strict inequality Img(f)l < [Dom(f)l holds for all non-injective maps. Function composition may further reduce the 'effective' domain of function, by only considering values of the image of the previous map as Domvetsetive (f) (f)
inputs to the next function. In contrast considering to all possible values of X' in the case of the non-composed map, the image of the previous function may only be subset of X. Therefore, = (mg(fi)l < Dom(fi)l. In particular, the effective image of a composition chain fr o... fi is bounded by the function with the smallest image: Imgeffective (fn o ... o fi)l &. min Img(f.)l. Thus, equation chains over finite discrete domains strictly preserve or reduce the effective size of the image, allowing for possibly simpler combined representation in comparison to representing the functions individually.
fa fi + Imgentective (f2) < [Domeficctve(]]]]
C Reparameterization of non-deterministic structural equations.
Consolidation of structural equations might lead to duplication of -on-deteriiiistic terms within consolidated systems. For example when consolidating fork structures (compare Sec. 14.p. Without further precautions, different values might be sampled from the duplicated non-deterministic equa- tions. An example where consolidating a variable B with non-deterministic equation I' (indicated by a squiggly line) leads to inconsistent behaviour is shown in 6 In M1, C' and D both copy on the value of B. Therefore, e = d yields always. My shows a graph where B is consolidated from M1. As result the non-deterministic equation fB is duplicated into the equations of C' and D, such that fc := Bern(A) and fn == Bern (A). Within the consolidated model MM different values might be be sampled from the different noise terms Bern( (A) in fc and fD- Consequently * d might occur in My. To obtain consistent behaviour with the initial M1. we need to ensure agreement about the value of Bern(A) across all instances of the duplicated equation. To do so, we reparameterize Mj and explicitly store fixed value, sampled from Bern(A) (A), into new exogenous variable R. The equation In is then reparameterized into a deterministic structural equation taking the variable R as an additional argument, resulting in M2- When consolidating B within M2, all instances of In now yield the same value, the noise term is fixed via R and finally PMC; PM
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M
Mr
M2
M2
A
R
A
R
D
D (4) Consistent Model Behaviour: C-D
(1) Non-Determistti Model. Yields C-D
(2) Inconsistent Behaviour: OC+D
(3) Reparameterization to Deterministic Eqs.
Figure 6: Reparamcterization of non-deterministic models. The SCM M, contains a non- deterministic equation B := (ern(A) (marked with a squiggly line). With C == B and := B, M1 always yields C = D. Simply consolidating (or marginalizing) B creates model MM with C' := (ern(A) and := Bern(A),, such that possibly C' * D. Reparameterizing fg by introducing an exogenous random variable R U(0 1) and B := A < R. yields the SCM M2 with only deterministic equations. Consolidating (or marginalizing) B in M2 leads to M2 where C == < R and D == A R, thus always C = D.
D Consolidation Examples
In this section we show further detailed applications of consolidation. Section D. presents the worked out consolidation of the dominoes motivating example of the paper, with regard to generalizing abilities of consolidates models. Section D.2 considers consolidation of the classical firing squad example. contrast to other examples, we focus on consolidating graphs with multiple edges in the causal graph. Lastly we provide the causal graph and structural equations of the game agent policy discussed in the main paper, in Section D.4
D.1 Motivating Example: Dominoes
While we applied consolidation to a particular SCMs in the main paper. we will discuss the motivating example with focus on obtaining representations that cover generalize over populations of SCM. We demonstrate this on the particular example of rows of dominoes, as a simple SCM with highly homogenous structure. Regardless of whether the SCM is obtained by using methods for direct identification of causal graphs from image data, as presented by Brehmer et al. 2022], or abstracting
physical simulation using r-abstractions [Beckers and Halpern| 2019]; we assume to be provided with a binary representation of the domino stones. The state of every domino S, indicates whether it is standing up or getting pushed over. In this case, the structural equations for all dominoes the same: fi := S4-1. As a result tipping over the first stone in a row will lead to all stones falling. Also, we are only interested in the final outcome of the chain. That is, whether the last stone will fall or not (E = {S,)) }). Again, we use consolidation to collapse the structural equations in the unintervened case: Sn Jn holding up any of the dominoes or tipping it over, I = (do(S, = 0). do(S; = 1))- Upon evaluation, the unconsolidated model needs check for every domino if it is being intervened or not, requiring conditional branches. Using the fact that perfect interventions 'overwrite' the variable state for the following dominoes, we introduce a first order quantifier that handles all intervention in a unified way. Finally, by combining the formulas of the intervened and unintervened case, we find the following simple equation:
Sj. We consider single active allowed intervention of
[x
if 3do = xx; E
S, = [
else
The resulting equation no longer has notion of the actual number of dominoes and, in fact, is invariant to it. We realise that introducing the first-order for-all V and exists 3 quantifiers allows for unified representation of arbitrary chains of dominoes. Similar observations are discussed in (Peters) and Halpern (2021] and Halpem and Peters [022] which introduce generalized SEM (GSEM). As intermediate the equations are no longer computed explicitly, the structural equations of consolidated models for different row lengths only differ in the set of allowed interventions I. That is, for row of three domino stones I (d = v1), do(V22 = v1), do33 = un v1)). while for four stones the additional do(V, -i) is defined. As set out in the introduction of this paper, consider
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if
SE(A) = A%5 =
fB(A) = [1
if A > 5
[r(A) = A%10 =
B
F
true if B = if S B 10
fo(E,F) =- EAF
fc(B) =
false
SD(C) =
true
otherwise
Jn(C, CVG
Figure 7: Example SCM for the Application of CONSOLIDATE, The figure shows toy SCM for demonstrating application of the CONSOLIDATE algorithm. Consider the following SCM with its structural equations and resulting graph (endogenous variables are B, C, D, E, F, G, H with only one exogenous A with each structural equation highlighted on the r.h.s., note that the subscript on f_ denotes the variable to be determined e.g. B < (A))).. In the first step, the algorithm' user decides on a partition. Let's consider for instance the following partition i.e., allowed intervention and consolidation sets: A {{E, F, G}, CBC ;;;; E true),, {do(D D = false)), ) {do(G = false)
{C, F, {{do(D
consolidation as tool for obtaining more interpretable SCM. Towards this end, consolidation might help us in detecting similar structures within an SCM. Doing so eases understanding of causal systems, as the user only has to understand the general mechanisms of particular SCM once and is then able to apply the gained knowledge to newly appearing SCM of the same type.
D.2 Firing Squad Example
While the dominoes and tool wear examples where mainly considering the consolidation of sequential structures, we want to briefly demonstrate the consolidation of structural equations that are arranged in a parallel fashion. We consider variation of the well known firing squad example [Hopkins) and Pearl 2007| with variable number N of rifleman. A commander (C) gives orders to rifleman (R; (I...N)) which shoot accurately and the prisoner (P) dies. For the sequential stacking of equations found that interventions exert 'overwriting" effect. That is, every intervention fixes the value of a variable, making the unfolding of the following equations independent of all previous computations. To yield similar effect for parallel equations paths between the cause and effect. In this scenario, this can easily be expressed by using an all-quantifier. When consolidating the SCM, consider only the captain C and prisoner P. E = {C, P), while allowing for any combination interventions that prevent the rifleman from shooting I P({do(R, O) After consolidation, obtain the following equation:
need to block all
p:= lives if C 0 do(, = =0) e dies
else
As with the dominoes example, again in situation where the consolidated equation intuitively summarizes the effects of individual: "The prisoner lives if the captain does not give orders, or all riflemen are prevented from shooting".
D.3 Step-by-step CONSOLIDATE Application
In this section provide step-by-step application of the CONSOLIDATE algorithm given in Algorithm[1 [ Consider the SCM shown in Figure 7 with its structural equations and resulting graph. The endogenous variables are B, C, D, E. F, G. !f with only one exogenous A. Structural equation are highlighted on the right-hand side. Note that the subscript on fa denotes the variable to be determined e.g. B < (A)).
In a first step, the algorithm's user has to decide on suitable partition. Consider for instance the following partition (indicated by dashed lines in the figure), the following allowed intervention and consolidation set:
A = [[E,F,), F.
I [[do[D= = true),, (do(D false)), (do(G false))]
E = (C
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The following example presents step-by-step application of the CONSOLIDATE algorithm for the cluster A (E, G.:
Step 3:
E1 + [B
Step 4:
E + (E
Step 5:
UA. +
ZA. B
{A]
Step
[[do(G false)}
Step 7:
PE + (fE(A) A mod 0;
fF(A) = A mod 10 = 0; Sa(E, F)
F}
Step 8:
PE + argminX(pp: =
Pr(A) A mod 10 = 0; PG(F, IA,) == (do(G = false) & IA.)} MA (LF, G), (F), PE; (Ido(G false)
Step 9:
Note how computing fE is no longer required. In similar fashion, equations in A₂ resemble a chain that can be composed: fc fa (previously called 'stacked'; cf. Sec, 4.1). Since [Img(/n) = 2, at least one of the three conditions of Se (since Sc is 3-case function) will be discarded. (Eventually is not in E and not required by any other sub SCM it can be marginalized. Ag then reduces to Pig. + (pH(C, G) == CV
yielding PEg + (pc(A):=A < 5}).
D.4 Revealing Agent Policy: Causal Graph and Equations
In this section we explicitly list the structural equations representing observed interactions between platformer environment and a possible rule based agent. The resulting causal graph is shown in Fig at the end of the appendíx. Except for the parentless variables goin_rewards "powerup_rewe reward" "player_position', rewardd' 'player_ position'. 'position_cone conion" 'position_ powerup', 'posi-
"enemy_ rewaddd, tion_eeneet enemy". 'position flag' and "target_flag" flag', which are exogenous and determined by the en- vironment, all variables are considered endogenous:
player_position, position_coin. position_ portiipp_ppwrrpppppp posppoon_sseeeeeee enemmeeeee______eeeeeeeee position_flal E [0..1]²
== 3; powerup_reware == 1; manemy_rewart := 9; flag reward :=
With X in (coin, powerup, enemy, flag):
distance X : ||position_xx X - player_posstion_XX]]
near X distance_ < 3.0
targeting_costl ctarc_cos_______ = 1.0 + 0.5 x distance_A
target_coin == targeting srttrinng_cot_coorre < enemy_reward
larget_powerup == targeting geeeting_cost_powreee < powerup_rewart
target_enemy == targeting_cost_eneet < enemy_reward A powered_up
target_flag- := True
powered_u up powwetp
towards_coin target_coin A coin_reward > max(((__rward]|argee J. E (pooerrruppnnn enemy, (lag))
towards target_enemy A
fungetppoeerup A powerup_reward > max( ( reward target X) x (cnin, flag)
towards_enemy :=
> max [X_ _ew__eeardrrarrett...
(enemy powerup,lla)
towards_flag == target_ faagtfflg/// A Beward > max( [ex_eeerd|taargett..... _X x www.inooweerpeeeeeeec.o
jump == near near_enemy A -powered_up
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finished if towards_ffae flag
A (flag
planting_sequenc__])
C CCC C
coin
towards__cmm coin
A (coin
Manning_sequence,
powerup if towards_powerup A (powerup
paaning_sequencee
planning sequence,
enemy
if towards_enemy
A (enemy
planiing_sequenc__]
i-1
flag
towards_ffaa flag
A (flag
$ U
j=1
finished
else
20
coin_rewand coin E planning_sequence,
powerup_reward if powerup planning_sequeneee
enemy_reward enemy E plamiing_sequence, A powerup E planning_sequence
reward if flag E ppanniing_sequence,,
E
Mathematical symbols and notation
The following table contains mathematical functions and notation used throughout the paper.
Notation
Meaning
X;
X
(set of) variable(s).
an;
Value(s) of X; X.
X,
The i-th variable of X.
Xs
The subset (X, : i E $} Xf
Px
A probability distribution over variables X.
Pxxx
value x sampled from a distribution over X.
P())
The power set.
fog
Function composition, (f o (x) = fg(xx)). N-ary Cartesian product over the domain of X. vector norm.
Ix, X
1112
bla,b)
Uniform Distribution.
N(H,
Normal Distribution.
Bern(p)
Bernoulli distribution; Takes value with probability and otherwise. Probability distribution over the SCM M.
PM
PL
Probability distribution over the SCM M under intervention I. An endogenous variable of an SCM M. An exogenous variable of SCM M. Structural equation of the variable X,.
M
V
U
f
18

---

Figure 8: Causal graph of an agent policy. The causal graph of greedy agent inside platformer environment. The parentless variables are exogenous. Their value determined via the game environment. The final 'score' variable is left out for clarity.
targeting cost flag
targot Dlag
bear flag
distance Blag
slag reward
position
targeting cost enemy
jamp
planning sequence 44
tanget enemy
fing
emeny
distance ememmy
planning sequent sequence.
enemy
enemy
player position
sagreing cost. powerup
powerup
planning sequence. 2
Lape powerup
near, powerrppppppp
coin
planning sequence
dis-atcc powerng
reward
powered
hageting cost. coin
coln
Bear coin
distance colD
19


Reviews:
[{"review_id": "6mcnbn9DUT", "reviewer": "Reviewer_CKKq", "summary": "An operation of consolidation on SCMs is defined and its merits laid out in numerous examples. The operation amalgamates variables while preserving aspects of the causal structure. As opposed to the similar operation of marginalization that comes from probability theory and is well-known in causal abstraction, the consolidation operation is capable of representing interventions on those variables that were abstracted since the consolidated variable is of a special kind. It also accommodates compressions since other compatible functions can be assigned to the consolidated variable; this also opens the possibility of widening the domains of the original variables.", "questions": "- The title seems inappropriate (especially imperative mood). Are you suggesting that consolidation is always superior to marginalization?\n- Line 4: \"Thus, ... analyze.\" is not a full sentence\n- Lines 35--36: \"Given that ... outcome.\" is not a full sentence. Awkward transition follows in \"That is, ...\"\n- Line 48: \"intervention preserving\" -> \"intervention-preserving\"\n- Line 69: it is inappropriate to cite do-calculus specifically here, rather, this is talking about the foundations of the SCM framework as a whole.\n- Line 116: \"Section 4 four\" -> \"Section 4\"\n- Is it really necessary to include the causal graph in the right of Figure 4?", "limitations": "Yes, they have.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The problem is relatively well-situated within the literature, I believe the results to be sound, and the examples are good.", "weaknesses": "I am not convinced of the utility of the approach. What open problems does this contribute to solving?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you reviewer CKKq for your detailed review and appreciating our work and good examples. Your comments have helped us improve the paper. We hope to answer all points you made in the following:\n\n* *Regarding the title and its implications of \"superiority\" of consolidation to marginalization:* \n\n  The short answer is: Yes. The more involved answer is: all authors have discussed this thoroughly in advance and settled for the present title suggesting that we would not claim consolidation to \"always [be] superior\" due to its normative implications. Consolidation generalizes marginalization in the sense that marginalization can be modeled by consolidating with $\\mathcal{I} = \\emptyset$. Therefore, if we consider questions that involve causality / causal models, then consolidation necessarily needs to be considered since it can actually handle interventions (all cases where $\\mathcal{I} \\neq \\emptyset$). Thus, if causal questions are not of concern, then marginalization can be considered as remaining the standard procedure. On another note we'd like to point out that marginalization does not require minimality of the resulting representation. Thank you for pointing this out since we are able to see how this is not immediately obvious from the paper and now we've added a corresponding statement to the refined version.\n\n* *Regarding Line 69 regarding the do-calculus citation:*\n\n  We fully agree and have corrected it in the paper.\n\n* *Regarding the necessity of the causal graph in the right of Figure 4:*\n\n  Our intention here was to showcase an example of consolidation for larger graphs. We think that having the graph included in Figure 4 makes it more clear that even domain experts in a certain area (here the specific CoinRunner game) might struggle to reduce the graph to extract (humanly) interpretable / meaningful descriptions of the agent behavior. Consolidation proved to work for this example as efficient as with some more elementary examples. Also the graph in Fig.4 helps to emphasize how consolidation can reduce overall complexity. Nonetheless, to give all the details we had added the full graph (fully readable) to the appendix as mentioned in the paper.\n\n* *Regarding the utility of consolidation and open problems that might be tackled:*\n\n  Thank you for pointing this out since this is a key aspect to why we think consolidation is important. In the following our perspective on this matter: as with marginalization, the operation of consolidation can help with simplifying causal models and therefore remove / abstract away variables that are irrelevant to the users' specific analysis. To include the effects of interventions (which is arguably the key operation in Pearl's causality framework and in more general, philosophical accounts of interventionist causality), the operations of marginalization and causal effect estimation would need to be repeated for every mutilated graph. With consolidation however we can retain the effects of interventions. As an example recall the domino example from Figure 1 / Appendix: we do not have to apply every possible intervention and summarize their outcomes, but can simply read from the consolidated formula, that every possible intervention will prevent the effect from arriving at the last domino. While the domino example specifically is not interesting for any particular analysis, it serves as a conceptual argument that highlights the implications to more important examples such as more classical causal analysis or our second example with agents interacting with an environment (robotics scenarios). More broadly speaking we intend for future work to follow up with researching applications of consolidation after having set the foundations of the operation in this work. Consolidation lays the foundation for justifying interventions onto high-level objects (modeled by consolidated SCM) where interventions can be applied externally but the inner structure of $\\rho$ might not be known. Thanks again for highlighting this, we've added a discussion of this to the new paper version. \n\nAlso thank you for pointing out various grammar and spelling mistakes. We have corrected all of them now.\n\nWe look forward to any further discussion with you, thanks again.\n"}, {"review_id": "IPEnOvaVji", "reviewer": "Reviewer_tWPL", "summary": "This work introduced a concept of consolidating causal mechanisms to transform large-scale Structural causal  models (SCMs) while preserving consistent interventional behaviour. The author shows consolidation is powerful for simplifying SCM, disscuss the complexity and give a perspective on generalization.", "questions": "1. Can author make a detail derivation on how to extract a consolidating causal mechanism from a toy SCM?\n2. In what scale of SCMs can this method handle, and what is the complexity? Can author give some larger scale examples?", "limitations": "see the weakness and question parts.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The author builds a solid framework of consolidating causal mechanisms. The expression of notations is adeque. The spirit of compression the causal equation is interesting and straight-forward.", "weaknesses": "1. The author should spend more efforts to describe the compression of causal equations clearly, as this is the key contribution of this work.\n2. The author did not propose a concrete algorithm to compress the causal equation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you reviewer tWPL for detailing some important aspects that helped improve our paper. Here are some brief comments on what we've improved and the questions you've raised.\n\n* *Regarding discussing of compressibility*:\n\n  Compressing structural equations to a minimal representation is highly dependent on the equations under consideration and probably incomputable for most problems. As there is ultimately no way of measuring compressibility of equations by only considering their connecting graph structure, we opted to provide an discussion with regard to some of the information-theoretic implications that can be inferred in section 4.1. Thanks for highlighting this point. We have modified section 4.1 to more critically discuss this limitations of our approach.\n\n* *Regarding an algorithm for causal consolidation*:\n\n  Nonetheless, we've added the requested algorithm to the paper. Please find it attached PDF in the global response that shows the algorithm with an additional toy SCM for a step-by-step worked-out demonstration at the end of this reply. The presented concrete example consolidates a collider structure (consider consolidation of $\\mathbf{A}_1$) and a chain (consider $\\mathbf{A}_2$ of the example) matching the contents of section 4.1.\n\n  Thanks for pointing out both points, we consider especially the addition of the algorithm as a strong contribution to our paper.\n\n* *Regarding larger SCMs*:\n\n  In our example for the CoinRunner game where we consolidate agent behavior captured previously in a large graph as depicted in Figure 4, we can see that consolidation also works in larger structures. We've now added a paragraph to further discuss the general limitations one faces when performing consolidation on larger graphs. Thanks for pointing this out.\n\nWe also look forward to more discussions with you.\n\n\n**An Example Application of the 'Consolidate' Algorithm**\n\nConsider the SCM provided in the general answer with its structural equations and resulting graph (endogenous variables are $B,C,D,E,F,G,H$ with only one exogenous $A$ with each structural equation highlighted on the r.h.s., note that the subscript on $f_x$ denotes the variable to be determined e.g.\\ $B\\leftarrow f_B(A)$). \n\nIn the first step, the algorithm's user decides on a partition. Let's consider for instance the following partition i.e., allowed intervention and consolidation sets:\n\n$\\mathcal{A} = \\\\{\\\\{E,F,G\\\\}, \\\\{B,C\\\\}, \\\\{D,H\\\\}\\\\}; \\mathbf{E} = \\\\{ C, F, H \\\\};$\n$\\mathcal{I} = \\\\{ \\\\{do(D = \\text{true})\\\\}, \\\\{do(D = \\text{false})\\\\}, \\\\{do(G = \\text{false})\\\\} \\\\}$\n\nTo finalize our example, a step-by-step application of \\textsc{Consolidate} for the cluster $\\mathbf{A}_1 = \\{E,F,G\\}$:\n\n*Step 3:* $\\mathbf{E}_1 \\gets \\\\{E,F,G\\\\} \\cap \\\\{C, F, H\\\\} = \\\\{ F\\\\}$\n\n*Step 4:* $\\mathbf{E}'_1 \\gets \\\\{F\\\\} \\cup (\\text{pa}(\\textbf{V} \\setminus \\\\{ E,F,G \\\\}) \\cap \\\\{ {E,F,G} \\\\}) = \\\\{F\\\\} \\cup (\\\\{A,B,C,G\\\\} \\cap \\\\{E,F,G\\\\}) = \\\\{F,G\\\\}$\n\n*Step 5:* $\\textbf{U}_{\\mathbf{A}_1} \\gets \\text{pa}(\\\\{E,F,G\\\\}) \\setminus \\\\{E,F,G\\\\} = \\\\{A,E,F\\\\} \\setminus \\\\{E,F,G\\\\} = \\{A\\}$\n\n*Step 6:* $\\mathcal{I}_{\\mathbf{A}_1} \\gets \\\\{\\\\{ do(X_i = v) \\in \\mathbf{I}\\ : X_i \\in \\\\{ E,F,G\\\\}\\\\}: \\mathbf{I} \\in \\mathcal{I}\\\\} = \\\\{\\\\{ do(G = \\text{false}) \\\\}\\\\}$\n\n*Step 7:* $\\rho_{\\mathbf{E}'\\_1} \\gets \\\\{ f_E(A) := E \\text{ mod } 5 = 0; f_F(A) := E \\text{ mod } 10 = 0; f_G (E,G) := A \\land B  \\\\}$\n\n*Step 8:* $\\rho^\\star_{\\mathbf{E}'_1} \\gets \\text{argmin} \\mathcal{K}(\\rho\\_{\\mathbf{E}'\\_1} ) = \\\\{\\rho_F(A) := A\\text{ mod } 10=0; \\rho_G(\\rho_F, \\mathbf{I}\\_{\\mathbf{A}_1}) := \\rho_F \\land (do(G = \\text{false}) \\notin \\mathbf{I}\\_{\\mathbf{A}_1})  \\\\}$\n\n*Step 9:* $\\mathcal{M}\\_{\\mathbf{A}_1,\\mathbf{E}} \\gets (\\\\{F,G\\\\}, \\\\{F\\\\}, \\rho^\\star\\_{\\mathbf{E}'_1}, \\\\{\\\\{ do(G = \\text{false}) \\\\}\\\\}, P_A)$\n\nNote how computing $f_E$ is no longer required. In a similar fashion, equations in $\\mathbf{A}_2$ resemble a chain that can be composed: $f_C \\circ f_B$ (previously called '\\textit{stacked}'; cf. Sec. 4.1). Since $|\\text{Img}(f_B)|=2$, at least one of the three conditions of $f_C$ (since $f_C$ is a 3-case function) will be discarded. (Eventually yielding $\\rho^{\\star}\\_{\\mathbf{E}'_2}{\\gets}\\\\{ \\rho_C(A){:=} A \\leq 5\\\\}$). As $D$ is not in $\\mathbf{E}$ and not required by any other sub SCM it can be marginalized. $A_3$ then reduces to $\\rho^{\\star}\\_{\\mathbf{E}'_3}{\\gets}\\\\{ \\rho_H(C,G) := C \\lor G\\\\}$.\n"}, {"review_id": "XWWaUsrnCa", "reviewer": "Reviewer_ZzBw", "summary": "The paper presents a framework for causal reasoning that supports the simplification of large structural causal models (SCMs).  One key operation that supports this simplification is consolidation of a SCM such that only a subset of endogenous variables are explicitly modelled, but in such a way that all possible interventions are still supported. Another key operation is the partitioning of an SCM into sub-models, in such a way that endogenous variables from one sub-SCM act as exogenous variables in another. \n\nPotential reductions in complexity are then discussed by the use of these operations and associated constraints. For instance, replacing equations within SCMs with computationally simpler expressions,  and dropping equations which do not relate to variables of interest. Throughout there is a focus on retaining information about the effects of interventions. The paper concludes with two examples: one on modelling tool wear on a milling machine and another, more complex example, exploring planning policies for a simple platformer game. \n", "questions": "My questions have mostly been articulated in the **Weaknesses** field. \n", "limitations": "The authors discuss limitations and the possible relaxations of these effectively.", "rating": 7, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "* The paper very clearly presents its contribution and relates this well to existing work, justifying the relevance of the contribution to the field.\n* The paper builds its arguments formally, and with intuition, and presents meaningful relevant results and constraints.\n* The examples illustrate well the benefits of the proposed framework.\n* The contributions appear meaningful to me and likely to be of relevance to other researchers, particularly those concerned with large scale causal modelling.\n\n", "weaknesses": "* At times the arguments are a little vague or the explanations incomplete.\n* Although mostly clear, the formal notation sometimes leaves a little to be desired.\n* There are a few places where the authors appear to make errors or omissions in their explanations.\n\nIssues with understanding:\n1. In definition 1, the set of possible interventions is a little unclear. There appears to be 1 possible intervention, $I_i$ per endogenous variable, $X_i$. Or can there be multiple potential, but mutually exclusive, interventions per endogenous variable? What exactly is $I_i$?\n2. On line 94 $f_i(\\textbf{x}, x_0)$ is used to indicate the structural equation for variable $X_i$ under the intervention that $X_j$ is set to value $x_0$, but this seems a little under-defined to me.\n3. On line 98, the authors state that $\\mathcal{M}$ entails infinitely many intervened distributions, but this seems to conflict with earlier notation (see 1.).\n4. Definition 2 could do with an explanation of $P^{\\textbf{I}}_{\\textbf{E}}$. I am assuming this means the distribution of target variables in $\\textbf{E}$ under some intervention $\\textbf{I}$ but I can't see this stated anywhere.\n5. The notation $\\rho(\\textbf{U},\\textbf{I})$ first introduced in Definition 2 does not refer to the subset of variables $\\textbf{E}$ to which it refers. This would be good practice anyway, but becomes more problematic when partially consolidated SCMs are considered (Def 5) as the variables of interest $\\textbf{E}$ are augmented with additional variables that act as exogenous variables in other sub-SCMs.\n6. The caption for Figure 2 (right) states that the dotted line indicates explicit computation for $X_2$ but it isn't clear what is meant by this.\n7. In section 3, in a number of places (lines 159, 165, 187), there appears to be repeated errors in the notation, e.g. $\\textbf{V} \\in \\textbf{A}$. I think that $\\textbf{V}$ is the complete set of endogenous elements while $\\textbf{A}$ is a subset of $\\textbf{V}$\n8. In lines 175-183, the partitioning of an SCM appears to  require that exogenous variables $\\textbf{U}_i$ of a sub SCM must be endogenous variables of another sub-SCM in the same partitioned SCM. But could they be truly exogenous variables of the whole system? Also, there appears to be notational irregularities in this paragraph relating to what is a sub-SCM and what is a partitioned SCM.\n9. Things get a bit messy around definition 5 with respect to $\\textbf{E}$ and $\\textbf{E}'$. The distinction between these two sets of variables could be clearer. For instance, if I am considering partially consolideted SCM $\\mathcal{M}_{\\mathcal{A},\\textbf{E}'}$ then how do I know what the set $\\textbf{E}$ is that is used to define $\\textbf{E}'$.\n10. In section 4, there is a chain of inference on lines 234-235 that is difficult to follow (what is the scope of the universal and existential quantifiers?) and some entity $D$ appears without being defined.\n11. I got a little lost in section 4.1. In particular, the discussion of conditional branching and stacking was a little vague.\n\n## Post rebuttal\n\nAfter reading the rebuttal and individual responses to the above points, I am raising my recommendation to \"accept\".", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you reviewer ZzBw for the detailed comments in your review which have led to improving our paper to its new version. In the following we go over each of the highlighted points one-by-one.\n\n* *Regarding the set of possible interventions:*\n\n  Thank you for pointing this out, as it helped improve as to avoid future confusion for other readers. In our paper we refer to an *intervention* $I$ as a specific instantiation of the do operator on a variable, e.g. $do(X_0 = 5)$. While sets of multiple interventions $\\textbf{I}$ (note the different formatting) can be applied over the whole SCM. In order to restrict notational clutter, we assumed that the set of allowed interventions is countable. Such that we get something like $I_0 = do(X_0=0); \\dots; I_5 = do(X_0=5); I_6 = do(X_1=0); \\dots$ for instance. While reviewing your comment we found that our formalism might break down for SCM containing uncountable (e.g. real-valued) variables (and did not restrict the actual sets to only contain one intervention per variable). To fix our mistake (and keep notational brevity), we have switched to writing $I_{i,v}$, indicating the intervention as $do(X_i = v)$. In consequence Def 1. bullet point 4 is rewritten as:\n\n  \"$\\mathcal{I} \\subseteq \\bigcup_{i \\in [1\\dots N]} \\bigcup_{v \\in {\\mathcal{X}\\_i}}$ $\\{I_{i,v}\\}$ such that $\\forall \\mathbf{I} \\in \\mathcal{I}. \\forall i \\in [1\\dots N].(\\exists! v\\in\\mathcal{X}\\_i. \\mathbf{I}\\_{i,v} \\in \\mathcal{I}) \\lor (\\lnot \\exists v \\in \\mathcal{X}\\_i. \\mathbf{I}\\_{i,v} \\in \\mathcal{I})$ and $\\mathbf{J} \\subset \\mathbf{I} \\in \\mathcal{I} \\rightarrow \\mathbf{J} \\in \\mathcal{I}$. $\\mathbf{I}$ is the set of perfect interventions under consideration. A perfect intervention $\\text{do}(V_i = v_i)$ replaces the unintervened $f_i$ by the constant assignment $V_i := v_i$.\n\n* *Regarding the \"explicit computation\" in Fig.2:*\n\n  With the figure we want to express that the value of $X_2$, as an element in $\\textbf{E}'$, is computed by $\\rho_{\\mathbf{E}'}$ (and there would be no way to depict the variable in the figure otherwise). In contrast to `normal' variables one can not directly intervene on this variable by simply cutting the edge to $\\rho_{\\mathbf{E}'}$ via an intervention. As stated in the caption, $X_2$ is the output of $\\rho_{\\mathbf{E}'}$ and not a normal variable. With 'explicit' we indicate that its value is still visible to the user. In contrast, $\\rho_{\\mathbf{E}'}$ might also compute the value of $X_1$ internally. However, this depends on the compression inside $\\rho_{\\mathbf{E}'}$ and $X_1$ is not visible to the user. Thanks for the pointer, we've added a brief statement on this to the new paper version.\n\n* *Regarding (7). Notational error on $\\mathbf{V},\\mathbf{A}$*:\n\n  Thanks for spotting that! Correct, we've fixed this.\n\n* *Regarding (8): Truly exogenous variables*\n\n  Indeed, they could also be truly exogenous variables. We've added a comment on this.\n\n* Regarding (8). Sub-SCM and partitioned SCM:\n\n  As defined in Definition 4 a partitioned SCM $\\mathcal{M}\\_{\\mathcal{A}}$ consist of multiple sub SCM $\\mathcal{M}\\_{\\mathbf{A}_i}$. For better readability we discarded the index $i$ wherever it was undefined and wrote $\\mathcal{M}\\_{\\mathbf{A}}$ instead to mean any (unspecific) $\\mathbf{A} \\in \\mathcal{A}$. We've added a clarifying statement.\n\n* *Regarding (9). Clarification on $\\mathbf{E},\\mathbf{E}'$*:\n\n  To consider either $\\mathbf{E}$ or $\\mathbf{E}'$ actually depends on the standpoint of the user (of the consolidation operation). From a computational perspective, $\\mathbf{E}'$ is important as it contains all variables that need to be computed by $\\rho$. While $\\mathbf{E}$ rather captures important aspects of the SCM to the user i.e., variables of interest. However, inferring for instance that $\\mathbf{E}'$ can be deduced from $\\mathbf{E}$, but not the other way around would pose a strong argument. We therefore choose to refer to sub SCM with $\\mathcal{M}\\_{\\mathbf{E}'}$ (and in the same breath write $\\rho_{\\mathbf{E}'}$) but use $\\mathcal{M}\\_{\\mathcal{A},\\mathbf{E}}$ to retain the initial set $\\mathbf{E}$ in the overall notation. Thanks for pointing out this nuanced discussion, we've added a short paragraph.\n\n* *Regarding (11): General considerations on compressibility*\n\n  Computing minimal representation is generally not possible (as outlined in 'General compression of equation systems' of Sec. 4). Nonetheless we intended on giving a more involved perspective on some of the basic structures (chains, forks and colliders) that appear in the SCM's implied graph structure to allow for some local optimizations using our approach (even if they do not allow to arrive at the true minimal representation $f^\\star_i$). Ultimately, there is no way of measuring compressibility of equations by only considering their connecting graph structure. However, even considering local substructures of graphs might improve compression. We have added this discussion to section 4.1. Thanks for raising this. In conjunction with other reviewers' comments, we furthermore present a concrete example of consolidating a collider and chain which will be added to the final paper. (Consider consolidation of $\\mathbf{A}_1$ and $\\mathbf{A}_2$ respectively in the provided PDF. Actual steps are provided in the answer to reviewer tWPL)\n\n**Disclaimer:** our actual answer covered all and thus more of your points. This is due to the 6000 character limit by OpenReview for NeurIPS this year. Therefore, we chose to delete points we deemed that you would maybe find less important than what we kept (We incorporated all of your comments into the paper). We can maybe provide them during the author-reviewer discussions.\n\nWe would like to once again sincerely thank you for thoroughly checking and commenting on our work!\nIt has greatly improved the overall soundness and quality of our paper in its new version and we look forward to some possible further discussions with you.\nKind regards, your authors."}]

Author rebuttal:
Thanks again to all reviewers for thorough checking and commenting on our work, helping us improve it to its current form. In addition to the individual comments, please see attached the PDF with the generally requested consolidation algorithm. The algorithm summarizes the construction of causal compositional variables and partitioned SCM - as described in the paper - which we have added to the paper as a separate subsection right before the applied examples (sec. 4.2). We have also included a more generic worked-out example show-casing the actual steps of the algorithm and featuring the consolidation of a collider and a chain. Since NeurIPS this year restricts the PDF to only containing tables and figures we provided the actual worked out example in the answer of reviewer tWPL who explicitly requested this addition.

We look forward to any further discussion with you.

Kindly, your authors.

Meta-review: The paper introduces and formalizes a process leading from a causal graph to an abstraction thereof, called "consolidation", meant to support interventions. 

The contribution of the paper mostly is to lay the foundations for defining the consolidation operation; the discussion with the reviewers concluded to the relevance of the approach to operationalize large structural causal models.
Decision: Accept (poster)