Systems with Switching Causal Relations: A Meta-Causal Perspective
Authors: Moritz Willig
Tim Tobiasch
Florian Peter Busch
Jonas Seng
Devendra Singh Dhami
Kristian Kersting
Venue: ICLR 2025 Spotlight (2025)
Keywords: Meta-Causality
Meta-Causal Reasoning
Agent Behavior
System Dynamics
Primary area: causal reasoning
TL;DR: This paper introduces meta-causal states to capture changing causal dynamics emerging from agents and environmental behaviors in dynamical systems.

Abstract: Most work on causality in machine learning assumes that causal relationships are driven by a constant underlying process. However, the flexibility of agents' actions or tipping points in the environmental process can change the qualitative dynamics of the system. As a result, new causal relationships may emerge, while existing ones change or disappear, resulting in an altered causal graph. To analyze these qualitative changes on the causal graph, we propose the concept of meta-causal states, which groups classical causal models into clusters based on equivalent qualitative behavior and consolidates specific mechanism parameterizations. We demonstrate how meta-causal states can be inferred from observed agent behavior, and discuss potential methods for disentangling these states from unlabeled data. Finally, we direct our analysis towards the application of a dynamical system, showing that meta-causal states can also emerge from inherent system dynamics, and thus constitute more than a context-dependent framework in which mechanisms emerge only as a result of external factors.

Published as a conference paper at ICLR 2025
SYSTEMS WITH SWITCHING CAUSAL RELATIONS: A META-CAUSAL PERSPECTIVE
Moritz Willig' Tim Nelson Tobiasch', Florian Peter Busch 1:2 Jonas Seng', Devendra Singh Dhami", Kristian Kersting" 1,2,45
DDpartment of Computer Science, Technical University of Darmstadt, Germany Hessian Center for Al (hessian.Al), Germany
DDept. of Mathematics and Computer Science, Eindhoven University of Technology, Netherlands
"Centre for Cognitive Science, Technical University of Darmstadt, Germany
German Research Center for AI (DFKI), Germany
moritz wwillig@cee -u- rmstadt.de
ABSTRACT
Most work on causality in machine learning assumes that causal relationships are driven by constant underlying process. However, the flexibility of agents' ac- tions or tipping points in the environmental process can change the qualitative dynamics of the system. As result, new causal relationships may emerge, while existing ones change or disappear, resulting in an altered causal graph. To an- alyze these qualitative changes the causal graph, propose the concept of mela-causal states, which groups classical causal models into clusters based on equivalent qualitative behavior and consolidates specific mechanism parameteri- zations. We demonstrate how meta-causal states can be inferred from observed agent behavior, and discuss potential methods for disentangling these states from unlabeled data. Finally, we direct our analysis towards the application of dy- namical system, showing that meta-causal states can also emerge from inherent system dynamics, and thus constitute more than context-dependent framework in which mechanisms emerge only as a result external factors.
INTRODUCTION
Structural Causal Models (SCM) have become the de facto formalism in causality by representing causal relations as structural equation models (Pearl (2009); [Spirtes et al. (2000)). While SCM are most useful for representing the intricate mechanistic details of systems, it can be challenging to de- rive the general qualitative behavior that emerges from the interplay of individual equations. Talking about the general type of relation emitted by particular mechanisms generalizes above the narrow computational view and has the ability to inspect causal systems from a more general perspective.
In addition to that, causal graphs can be subject to change whenever novel mechanisms emerge or vanish within a system. Prominently, agents can 'break' the natural unfolding of systems dynamics by forecasting system behavior and preemptively intervening in the course of events. As inherent parts of the environment, agents commonly establish or suppress the emergence of causal connec- tions (Zhang & Bareinboim] 2017 Lee & Bareinboim 2018 Dasgupta et al.) (019).
Consider the scenario shown in Figure (left), where an agent A (with position Ax) follows an agent B (with position Bx) according to its internal policy Az. We are interested in answering the question 'What is the cause of the current position of agent A?' In general, the observed system can be formalized as follows: Ax := fA (Bx, AA) and Bx == f (UB). Note that, from a classical observe B + A, since A self-conditions itself to follow B, by instantiating
causal perspective, the equation fA via its policy and thus becomes dependent on B. Classical causal considerations, which only consider how relations between variables constructed, cannot give the correct answer. Only when we take a meta-causal stance and think about how equation fA came to be -how meta- causal states induce qualitative changes in behavior-, can give sufficient answer this question.

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Classical Attribution
Pata-Causal Attribution
<
Ax
Bx
A
A Agent Establiahes Distance Constraint
Classical Causal Relation Rool Cause Altribution
Ax
B
Observed Causai Direction
Figure I: Meta-Causality Identifies the Policy as a Meta Root Cause. Agent A intends to main- tain its distance from agent B by conditioning its position Ax on the position Bx. which establishes a control mechanism, Ax == f f(Bx) ). In standard causal inference, we would infer Bx - Ax and, therefore, B to be the root cause. Taking meta-causal perspective reveals however, that Az estab- lishes the edge Bx + Ax in the first place (A, + (Bx + Ax)) such that Ar is considered the root cause on the meta-level. (Best Viewed in Color)
Causal models do not exist in isolation, but emerge from the environmental dynamics of underly- ing mediation process (Hofmann al. 2020 Arnellos et al. 2006). Changes in causal relations due to intervention environmental change often assumed to be helpful conditions for identifying causal structures (Pearl [009 [Peters et al. 2016 ZZang & Bareinboim, 2017] Dasgupta et al. 2019; Gerstenberg 2024). These operations often work the individual causal graphs, but never reason about the emerging meta-causal structure that governs the transitions between the different SCM. As first formalization of metacausal models in this area, we describe how qualitative changes in the causal graphs can be summarized by metacausal models.
Contributions and Structure. The contributions of this work are follows:
To best of our knowledge, we are the first to formally introduce typing mechanisms that generalize edges in causal graphs, and the first to provide formalization of meta- causal models (MCM) that are capable of capturing the switching type dynamics of causal mechanisms.
We present approach to discover the number of meta-causal states in the bivariate
We demonstrate that meta-causal models can be more powerful than classical SCM when it comes to expressing qualitative differences within certain system dynamics.
We show how meta-causal analysis can lead to different root cause attribution than clas- sical causal inference, and is furthermore able to identify causes of mechanistic changes even when no actual change in effect can be observed.
We proceed follows: In Section 2 describe the necessary basics of Pearlian causality, mediation processes, and typing assumptions. In Sec. Fe introduce meta-causality by first formalizing meta- causal frames, and the role of types as generalization to a binary edge representations. Finally we define meta-causal models that are able to capture the qualitative dynamics in SCM behavior. In Sec. we showcase several examples of meta-causal applications. Finally, we discuss connections to related work in Sec. 5 and conclude our findings in Sec. 6
BACKGROUND
Providing a higher-level perspective on meta-causes meta-causality touches on number of existing works that leverage meta-causal ideas, even if not explicitly stated. We will highlight relations of these works the 'Related Work' of section Here, we continue to provide the necessary concepts on causality, mediation processes and typing, needed for the definitions in paper:
Causal Models. A common formalization causality is provided via Structural Causal Models (SCM; (Spirtes al. 2000 Pearl| 2009)). An SCM is defined as a tuple M : (U, V, F, Po), where U is the set of exogenous variables, V the set of endogenous variables, is the set of structural equations determining endogenous variables, and Pu is the distribution over exoge- nous variables
U. Every endogenous variable V; E V is determined via a structural equation of parent values pa(w:), consisting of endogenous and exogenous
w: = (pa(w:) that takes in
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variables, and outputs the value of U₁- The set of all variables is denoted by X = UUV with values x E X and N = XX|, Every SCM M entails a graph GM that can be constructed by adding edges X,) E X x X for each variable X; E X and its parents X₃ E Pa(X)). This can be expressed as adjacency matrix A E BNNN where A. 1 if (i E 9 and Aij 0 otherwise.
Mediation Processes. Causal effects are embedded in an environment that governs the dynamics and mediates between different causes and effects. To reason about the existence of causal relations, we need to consider this sereess-ebbedding environment. We define a mediation process E = (S, a), adapted from Markov Decision Processes (Bellman (957 for our setting. Here, S is the state space of the environment and Ø S - S is the (possibly non-deterministic) transition function that takes the current state and outputs the next one. If we also have an initial state So E S, we call (E, 9o) an initialized mediation process. As we are concerned with the general mediation process, we omit the common notion of reward function r. Furthermore, we omit an explicit action space A and agents' policy π and model actions directly as part of the transition function ø, In accordance to considerations of Cohen (2022). this eases treatment of environment processes and agent actions as now both are defined on the same domain. The emergence of SCM from mediation processes can be studied under measure-theorets theory, as considered in Park et al. (2023). Similarly, Danzing & Mejia (2022) discuss the role of clementary actions towards the constitution of causal variables.
Typing. In the following section, we will make use of an identification function to determine the presence or absence of edges between any two variables. In particular, one can make use of different identification functions to identify different rypes of edges. Previous work on typing causality exists (Brouillard et al.] 2022). but primarily considers the types of variables that are causally related, rather than the type of structural relation itself. Other works in the field of cognitive science consider the perception of different types of mechanistic relations (e.g.. 'causing", 'enabling', 'preventing' 'despite') ) based on the role that different objects play in physical scenes (Chockler & Halpern] [004]] Wolff 2007] [Sloman et al., 2009; Walsh & Sloman 2011 Gerstenberg 2022). Since all objects are usually governed by the same physical equations, this assignment of types serves to provide post hoc explanations of scene, rather than to identify inherent properties of its computational aspects. Gerstenberg (2024) provides an 'intuitive psychology' example that fits well with our scenario.
3 META-CAUSALITT
Similar to Pearl & Mackenzie (0118), we define Meta- CausaCaus to be the [science ofl change in qualitative cause-effect behavior. Usually, the mediating process may be too fine gna-irri to yield interpretable models. Therefore, we consider a set of variables of interest X modeled by an SCM and introduced by an abstraction y S X. (We provide a brief discussion on the emergence of SCM from mediating processes in Appendix (A). Here, so could be defined as summarization or causal abstraction over the state space (Rubenstein et al., 2017: Beckers & Halpern 2019 Annnd ct al. 2022 Wahl et al. 2023 Kekić et al.) 2023 Willig et 2023).. In order to identify the type of causa relations from a mediating process, we need to be able to decide on what constitutes a type.
Definition (Meta-Causal Frame). For a given mediation process E =S,a) a meta-causal frame is a tuple F = (E, (,,,X(Ts))) (T) with:
rype-encoders Tzj X, X T that assign a type t E T to the functional dependence of X, on X,, induced by the underlying process (S, a). This relation is between X: (values of Xx) and the abstraction of the transition function ip
X = 1 : S X}.
an identification function I S X X T with I(s, X,, Xx) + t= (y)(p(s) that assigns a type to every pair of causal variables for any state of the environment.
Types generalize the role of edges in causal graphs, while type encoders Tıj determine the particular type of edges from properties of the underlying functional relations poor In most classical scenarios, the co-domain T of the type encoder Tiy is chosen to be Boolean, representing the existence or absence of edges. In other cases, different values t E T can be understood as particular types of edges, like positive, negative, or the absence of influence. This will help us to distinguish meta- causal states that share the same graph adjacency. The only requirement for T is that it must contain a special value 0, which indicates the total absence of an edge. catta-aausal states now generalize the idea of binary adjacency matrices. We, intentionally, do not restrict the identification function in
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any way. However, particular choices, e.g. functions identifying only direct causal effects, are more likely to result in classical SCM. We provide a short discussion on this in Appendix B
Definition 2 (Meta-Causee State), In a meta-causal frame F = (E, X, Taj, I), meta-causal state
is matrix T TNXN For given environment state E S. the actual meta-causal state Tg has the entries Te,s == I(,, = (p)(p(s),
meta-causal state T E TNN represents graph containing edges eig. of a particular type E In particular Tay indicates the presence (Ti, * 0) or absence (Tij 0) of individual edges. Our goal for meta-causal models (MCM) is to capture the dynamics of the underlying model. In particular, we are interested in modeling how different meta-causal states transition into each other. This behavior allows us to model them in a finite-state machine (Moore et al.) 1956):
Definition 3 (Meta-Causal Model). For meta-causal frame F = (E, meta-causal model is a finite-state machine defined as tuple A = (T xN 5,6), where the set of meta-causal states T NNN is the set of machine states, the set of environment states S is the inpul alphabet, and 6: T:TNNN S > T-NN is transition function.
Usually, have the objective to learn the transition function 8 for unknown state transition function a. The state transition function 5 can be approximated as 6(T, +) == I(a'(T, (T, s), x,,x), X,) Tip (i)(p)()) sood' (T, s)). In addition, o' takes the state T to better approximate transition probabilities.
As standard causal relations emerge from the underlying mediation process, the meta-causal states emerge from different types of causal effects. The transition conditions of the finite-state machine are the configurations of the environment where the quality of some environmental dynamics repre- sented by type T changes.
Inferring the Meta-Causal States. Even if the state transition function is known, may be un- clear from single observation which exact meta-causal state led to the generation of particular observation S. This is especially the case when two different meta-causal states can fit similar en- vironmental dynamics. Even in the presence of latent factors (e-g.. an agent's internal policy), the current dynamics of a system (e.g., induced by the agent's s current policy) can sometimes be in- ferred from a series of observed environment states. This requires knowledge of the meta-causal dynamics, and is subject to the condition that sequences of observed states uniquely characterize the meta-causal state. Since MCM are defined as finite state machines, the exact condition for identi- fying the meta-causal state is that observed sequences are homing sequences (Moore et al. 1956p. Note that the following example of a game of tag presented below exactly satisfies this condition, where the meta-causal states produce disjoint sets of environment states ('agent A faces agent B', or 'agent B faces agent A'), and thus can be inferred from either a single observation, when move- ment directions are observed, or two observations, when they need be inferred from the change in position of two successive observations.
'Game of Tag' Example. causal graph and two different meta-states. In general, we expect agent B to make arbitrary moves that increase its distance to A. while A tries to catch up to B, or vice versa. In essence, this is cyclic causal relationship between the agents, where both states have the same binary adjacency matrix. Note, however, that the two states differ in the type of relationship that goes from A B (and B to A). We can use an identification function that analyzes the current behavior of the agents to identify each edge. Since A can tag B, the behavior of the system changes when the directions of the typed arrows are reversed, so that the type of the edges is either 'escaping' or "chasing".
Consider idealized game of tag between two agents, with a simple
While the underlying policy of an agent may not be apparent from observation as an endogenous variable, it can be inferred by observing the agents' actions over time. Knowing the rules of the game, one can assume that the encireling agent faces the other agent and thus moves towards it, while the fleeing agents show the opposite behavior:
(tB-A= = "A Chasing") + (Apas (Bpos - Apos) > 0)
where Apos is the velocity vector of agent A (possibly computed from the position of two consecu- tive time steps); Apos and Bpos are the agent positions, and is the dot product. Once the edge types are known, the policy can be identified immediately.
Causal and Anti-Causia Meta-Causal States. Assigning meta-causal states to particular system observations can be understood labeling the individual observations. However, it is generally
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unclear whether the meta-causal state has observational controlling character on the system under consideration. One could ask the question whether the system dynamics cause the meta- causal state, whether the meta- state causes the system dynamies, (or whether they actually the same concept), similar to the well-known discussion "On Causal and Anticausal Learning" by Schölkopf et al. 2012), but from meta-causal perspective.
At this point in time, we cannot give definitive conditions on how to answer this question, but we present two examples that support either one of two opposing views. First, in Sec. 4.3 we present a scenario of a dynamical system where the structural equations of both meta-causal states are the same, since the system dynamics are governed by its seff-referential system dynamics. Intervening on the meta-causal state will therefore have no effect on the underlying structural equations, and thus can have no effect on the actual system dynamics. In such scenarios, the meta-causal is rather a descriptive label and cannot be considered an external conditioning factor. In the following, we discuss the opposite example, where a meta-causal state can be modeled as an external variable conditioning the structural equations.
Role of Contextual Independencies. Changes in the causal graph are often attributed to changes in the environment modeled by some exogenous variables Z, as, for example, leveraged with the invariant causal prediction (Peters et al., 2016). In this case, we get MCM where the transition function only contains self-loops. This makes it clear that MCM are more general tool in analyzing meta-causal structures. If we introduce a 'no-edge" type L = 0 for this scenario, a meta-causal model can describe the condition that X; and X, are contextually independent for some Z 2. For suitable sype-encoder with a surjective mapping 2 + TNxN and a given family of compatible (see Appendix C) decomposed structural equations (Jit)zez we have fz == OieI..N Z and
f [<<+)
if (1)())))) #0
ful
otherwise
where J7 are the structural equations of the edge egj that are active under the environment Z and (in a slight imprecision in the actual definition) (< *) is the function that carries on the previous function of the composition and discards Xj. While the 'no-edge' type 0 could be handled like any other type, we have listed it for clarity such that individual variables X, become contextually independent whenever fajl Z = 0. We provide conditions for the reduction of MCM in Appendix D
4 APPLICATIONS
In this section, we discuss several applications of the meta-causal formalism. First, we revisit the motivational example and consider how meta-causal models can be used attribute responsibility. Next, we identify the presence of multiple mechanisms for the bivariate case from sets of unlabeled data. Finally, we analyze the emergence of meta-causal states from a dynamical system, highlighting that meta-causal states are more expressive than simple conditionings of the adjacency matrix. Code made available at httpttts:///tm / com Mor ttw! wetiii/mmeaaaussss Models
4.1 ATTRIBUTING RESPONSIBIL.TY
Consider again the motivational example of Figure D where agent A with position Ax follows an agent B with position Bx as dictated by its policy Ar. In this scene, we can imagine a counter- factual scenario in which we replace the 'following' policy of agent A with, e.g., a 'standing still' policy and find that the Bx + Ax edge vanishes. As a result, we infer the meta-causal mecha- nism AT + (Bx + Ax) for the system and thus An as the root cause of for values of Ax. In conclusion, while A is conditioned on B on low level, the meta-causal reason for the existence of the edge B A is caused by the A,. Both attributions, tracing back causes through the structural equations Ax= == fBx) or our meta-causa approach, are valid conclusions in their own regard.
Note that in this scenario, classic counterfactual consideration, AT -standing still would also have inferred an effect of Ax on Ax from a purely value- perspective. Attributing effects via observed changes in variable values is a valid approach, but it fails to explain preventa- tive mechanisms. For example, consider the simple scenario where two locks prevent a door from opening. Classical counterfactual analysis would attribute zero effect to the opening of either lock.
A A= following
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d
0.0
d 0.1
d
= 0.2
k'
2
3
4
2
4
1
2
3
4
k-
1
2
81
3
7
7
2
85
3
7
3
1
83
3
8
5
2
41
54
4
0
43
48
0
49
1
47
3
22
6
63
77
2
0
13
4 92
0
0
1
7
89
0
9
87
0
1
5
7
Table 1: Confusion Matrices for Identifying Meta-Causal Mechanisms. The table shows iden- tification results for predicting the number of mechanisms for the bivariate case for 100 randomly sampled meta-causal mechanisms. is the maximum sample deviation from the average mechanism probability. Rows indicate the true number of mechanisms, while columns indicate the algorithms' predictions. :.' indicates setups where the algorithm did not make decision. In general, the algo- rithm is rather conservative in its predictions. In all cases where a decision is made, the number of correct predictions along the diagonals dominate. The first and second most frequent predictions marked in green and orange, respectively. (Best Viewed in Color)
meta-causal perspective would reveal that the opening of the first lock already changes the meta- causal state of the model by removing its causal mechanisms that condition the state of the door. In a sense, our meta-causal causal perspective is similar to that of the actual causality framework (Halpemn 2016 Chockler & Halper/ 2004). However, actual causality operates at the 'actual', i.e. value-based, level and does not take the mechanistic meta-causal view into perspective. Meta-causal models already allow us reason about causal effects after opening the first lock due to change in the causal graph, without us having to consider the effects of opening the second lock at some point to reach that conclusion.
4.2 DISCOVERING CCUSAL STATES IN THE BIVARIATE CASE
Our goal in this experiment is to recover the number of meta-causal states E [1..4] from data that exists between two variables X, Y that are directly connected by a linear equation with added noise. We assume that each meta-causal state gives rise to a different linear equation fa == og + 8k N E N, where ag, 8& are the slope and intercept of the respective mechanism and N is zero-centered, symmetric, and quasiconvex noise distributior" Without loss of generality, we apply Laplacian noise, for which an I.--regression can estimate the true parameters of the linear equations (Hoyer et al. (2088). The causal direction of the mechanism is randomly chosen between different meta-causal states. The exact sampling parameters and plots of the resulting distributions described in Sec. (and plots of sample distributions are shown in Fig. in the Appendix). In general, this scenario corresponds to the setting described above of inferring the values a latent variable Z (with K [Z) that indicate particular meta-causal state of the system. Our goal is to recover the number of parameterizations the causal mechanisms, and as a consequence to be able to reason about which points
generated by which mechanism.
Approach. The problem we are trying to solve twofold : first, we are initially unaware of the underlying meta-causal state that generated particular data point (æ,, Ui), which prevents us from estimating the parameterization (ou, (x)) of the mechanism. Conversely, our lack of knowledge about the mechanism parameterizations (ax,, 3x ) prevents us from assigning class probabil- ities to the individual data points. Since neither the state assignment nor the mechanism parameteri- zations are initially known, we perform an snteetattion-aiiiiiiie procedure to iteratively estimate and assign the observed data points to the discovered meta-causal state parameterizations. Due to the local convergence properties of the EM algorithm, we further embed it into a locally optimized local random sample consensus approach & Bolles (981)) Chum et al.] (2003)). RANSAC approaches repeatedly sample initial parameter configurations to avoid local minima, and successively perform several steps of local optimization here the EM algorithm- to regress the true parameters of the mechanism.
(EM: Dempster et al. (1977)
(O-RANSAA,(Fischi
''mplying unimodality and monotonic decreasing from zero allows to distinguish the noise mean and intercept and recover the parameters from simple linear regression

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Analysis Type
Mechanism
Mech.
Mech.
4 Mech.
Max. Class
Theoretical
I
23
363
8,179
Deviation = 0.0
Empirical
2
8
24
173
Max. Class
Theoretical
1
26
429
10,659
Deviation = 0.1
Empirical
2
8
25
177
Max. Class
Theoretical
30
526
14,859
Deviation = 0.2
Empirical
2
8
26
177
Estimated Number of Required Resamples for Obtaining a 95% Convergence Rate with the LO-RANSAC Algorithm per Number of Mechanisms and Maximum Class Deviation. The empirical observed convergence rates of the EM algorithm drastically reduce the theoretical derived bound of required samples. This reveals that convergence assumptions where chosen to be quite conservative, and attests good fit of the EM algorithm for regressing mechanism parameters.
Table 2:
Assuming for the moment that the correct number of mechanisms k- has been chosen, we assume that the EM algorithm is able to regress the parameters of the mechanisms, a whenever there exists a pair of points for each of the mechanisms, where both points of the pair are samples gener- ated by that particular mechanism. The chances of sampling such an initial configuration decrease rapidly with an increasing number of mechanisms (e.g. 0.036% probability for k 4 and equal class probabilities). Furthermore, we assume that the sampling probabilities of the individual mech- anisms in the data can deviate from the mean by up to a certain factor d. In our experiments, we consider setups with d E {0.0, 0.1, 0.2}. Given the number of classes and the maximum sample de- viation of the mechanisms, one can compute an upper bound on the number of resamples required to have 95% chance of drawing at least one valid initialization. The bound is maximized by assigning the first half of the classes the maximum deviation probability Pk-max = (1 + d)/k and the other half the minimum deviation probability Pa-min = (1 d)/k. We provide the formulas for the upper bound estimation and in Sec. Elin the Appendix and provide the calculated required resample counts in Table 2 In the worst case, for scenario with & 4 mechanisms and d = 0.2 maximum class probability deviation, nearly 14, 900 restarts of the EM algorithm are required, drastically increasing the potential runtime.
In our experiments, we find that our assumptions about EM convergence are rather conservative. Our evaluations show that the EM algorithm is still likely to be able regress the true parameters, given that some of the initial points are sampled from incorrect mechanisms. We measure the empirical convergence rate by measuring the convergence rate of the EM algorithm over 5,000 different setups (500 randomly generated setups with 10 parameter resamples each). We perform 5 EM steps for setups with For each initialization, we count the EM algorithm as converged if the slope and intercept of the true and predicted values do not differ by more than an absolute value of 0.2.
and l mechanisms, and increase to 10 EM iterations for and 4 mechanisms.
Determining the Number of Mechanisms. The above approach is able to regress the true parame- terization of mechanisms when the real number of parameters k: is given. However, it is still unclear how to determine the correct k.. Computing the parameters for all k E K and comparing for the best goodness of fit is generally a bad indicator for choosing the right k since fitting more mecha- nisms usually captures additional noise and thus reduces the error. In our case, take advantage of the fact that we assumed the noise to be Laplacian distributed. Thus, the residuals of the samples assigned partieular mechanism can be tested against the Laplacian distribution.
The EM algorithms return the estimated parameters a'e. Sy and the mean standard deviation bg of the Laplacian] 2 This allows one to compute the class densities k( yı ale 3%,,, b1) for a pair of data point may be ambiguous due to the overlap in the estimated PDFs, we normalize the density values of all mechanisms per data point and consider only those points for which the probability of the dominant mechanism is 0.4 higher than the second class: (xl, ) := ((z,, y4) #1 = j: f < p##1 x (1 0.4)), where #n indicates the class with the n-th highest density value. Finally, the residuals y - f (xx) y²; o B6. B/)) are
values (x;, yt). Since the assignment of mechanisms
"Since mechanisms can go in both directions, X + Y and Y + X (cyclic relations are not considered), we repeat the regression for both directions and use an Anderson-Darling test (Anderson & Darling 1952) on the residual test which of the distrihutions more closely resembles Laplacian distribution at each step.
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computed for all data points (x)xx) assigned to particular predicted mechanism f We choose the parameterization that best fits the data for each & E [1..4] and, make use of the Sadlerson-Daliing
test (Anderson & Darling [952) to test the empirical distribution function of the residuals against the Laplacian distribution with an a = 0.95 using significance values estimated by Chen (2002). If all residual distributions of all mechanisms for a given k' pass the Anderson-Darling choose that number as our predicted number of mechanisms. If the algorithm finds that none of the k [1..4] setups pass the test, we refrain from making a decision. We provide the pseudo code for our method in Algorithm. in the Appendix.
test,
Evaluation and Results. We evaluate our approach over all k [1..4] by generating 100 different datasets for every particular number of mechanisms. For every dataset we sample 500 data points from each mechanism (x* ) = fe(ara + Bg +1) where I ~ [(,, by), using the same sampling method as before (c.f. Appendix G Finally, the algorithm recovers the number of mechanisms.
We compare theoretically computed and empirical in convergence results in Table In practice, we observe that the convergence of the EM algorithm is more favorable than estimated, reducing from 23 to required examples for the simple case of k = 2, d = 0.0, and requiring up to 83-times fewer 0.2, reducing from theoretical of 14,859
samples for the most challenging setup of k = 4, d to an empirical estimate of 177 samples. The actual convergence probabilities and the formula for and Sec. of the Appendix.
deriving to sample counts are given in Table and
Table shows the confusion matrices between the actual number of mechanism and the predicted number for different values of maximum class imbalances. In general, we find that approach is rather conservative when in assigning a number of mechanisms. However, when only considering the cases where the number of mechanisms is assigned, the correct predictions along the main di- agonal dominate with over 60% accuracy for k = 4 and d E (0.0, 0.1} and rising above 80% for * 4 for. In the case of d = 0.2, the results indicate higher confusion rates with 41.6% accuracy for the overall worst case of K = 4.
Extension to Meta-Causal State Discovery on Graphs. Our results indicate that identifying meta- causal mechanisms even in the bivariate case comes with an increasing number of uncertainty when it comes to increasing numbers of mechanisms. Given that the number of mechanisms could be reliably inferred from data for all variables, the meta-causal states could be identified as all unique combinations of mechanisms that are jointly active at a certain point time. To recover the full set of meta-causal states, one needs to be able to simultaneously estimate the triple of active parents for every mechanism, the mechanisms parameterizations, and the resulting meta-state assignment of all data points for every meta-state of the system. All of the three components can vary between each meta-causal state (e.g. edges vanishing or possibly switching direction, thus altering the parent sets). Note that whenever any of the three components is known, the problem becomes rather trivial. However, without making any additional assumptions on the model or data and given the results on the already challenging task of identifying mechanisms in the bivariate case, the extension unsupervised full-fledged meta-causal state discovery is not obvious to us at the time of writing and we leave to future work come up with a feasible algorithm.
4.3 A META-STATE ANALYSIS ON INDUCED FATIGUE
Stress is a major cause of fatigue and can lead other long-term health problems (Maisel et al. 2021 Franklin et al. 2012 Dimsdale 2008
Bremner 2006 While short-term exposure may be
helpful in enhancing individual performance, long-term stress is detrimental and resilience may
decline over time (Wang & Saudino, 2011 Maisel et al. 2021). While actual and perceived stress
levels (Cohen et al.) 1983 Bremner 2006 Schlotz et al. 2011 Giannakakis et al., 2019) may vary between individuals (Calkins 1994. Haggerty] (196). the overall effect remains the same. We want to model such system as an example. For simplicity, we present an idealized system that radically reduced to the only factors of external stress, modeling everyday environmental factors, and the self- influencing level of internal/perceived stress.
While being rather simple in setup, the example serves good demonstration on how dynamical systems induce meta-causal states that exhibit qualitative different behavior, while employing the same set of underlying structural equations. As such, the inherent behavior of the system is not only due to a conditioning external factor. We use an identification function to distinguish between two different modes of operation of a causal mechanism. In particular, we interested whether
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1.0
0.3
0.9
1.
E
11.5
11.7
0.5
Syes
0.4
E
0.3
02
0.3
0.2
6.1
0.0
01 022 0.3 0.4 0.5 0.6 0/7 0.3 1.0 Stress Level::
0.0
0.2
0.4
0.5
0.8
1.0
External Stressors
Figure 2: Mechanistic Decomposition as Meta-Causal States. (left) The effect of the stress level on itself (orange) plotted against the identity (blue: corresponding to a non-self-reinforcing effect). Once a certain threshold is reached, the function switches its behavior from self-suppressing to self-reinforcing effect. (center) Contribution of the stress level mechanism for varying external stressors. Red arrows indicate self-reinforcing effect, while green arrows indicate suppressive effect. The gray area highlights the system configuration without external stressors. Although all states governed by the same structural equations, our meta-causal analysis identifies the mech- anistic difference and decomposes the corresponding initial conditions into different meta-causal states. (right) The standard SCM gets decomposed into different meta-causal states. While the graph adjacency remains the same, the different starting conditions identify different behavioral types of the causal mechanism. (Best viewed in color.)
the inherent dynamics of the internal stress level are self-reinforcing or self-suppressing. For easier analysis of the system we decompose the dynamics of the internal stress variable into a 'decayed stress' and 'resulting stress' s computation. The first term are the previous stress levels decayed over time with external factors added. The resulting stress is then the output of a Sigmoidal function (Fig. [ left) that either reinforces or suppresses the value (Fig. center). The structural equations are defined as follows and we assume that all values lie in the interval of [0, 1]:
Ja = 0.95 o.. (s' 0.5 ext.Stress) and Ss 1.01 exp(-15x + 7.5) 0.5) + 0.5
where s and d are the resulting and decayed stress levels, and s' is the previous stress level of s.
We now define the identification function to be I := (f.). where f. is the second order deriva- tive of the Sigmoidal f.. with either positive negative effect on the stress level. The described system has two stable modes of dynamics. Note how the second-order inflection point at 0.5 of the Sigmoid acts as a transition point on the behavior of the mechanism. Stress values below 0.5 get suppressed, while values above 0.5 are amplified. Transitions between the two stable states can only be initiated via external stressors. Effectively this results three possible meta-causal states which are governed via the following transition function:
o (t,
0 with a == sign(fa) = sign(s - 0.5) E (-1,0,1}
Role of Latent Conditioning, A key takeaway of this example is that the current meta-state persists due to the inherent stress level and dynamics of the system. contrast to other examples where system dynamics where purely due to the meta-causal state, variable values play role in the overall system dynamics. As a result, the stressed state of a person would persist even when the initiating external stressors disappear. Intervening as both positive and negative reinforcing effects are governed by the same equation. Thus, creating disparity between the intervened meta-causal state and the systems' identified functional behavior.
the meta-causal state of the system is now ill-defined,
RELATED WORK
Causal Transportability and Reinforcement Learning. Meta-causal models cover cases that duce to conditionally dependent causal graphs due to changing environments (Peters et al. 2016 Heinze-Deml et al.] 2018). but also extend beyond that for dynamical systems. In this sense, the work of Talon et al. 2024) takes a meta-causal view by transporting edges different causal effects between environments. In general, the transportabilityy of causal relations (Zhang & Bareinboim

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2017
Lee & Bareinboim 2018 Correa et al. 2022) be thought of as learning identification functions that identify general conditions of the underlying processes to transfer certain types of causal effects between enviromments [Sonar al. (201): Yu et al. (2024): Brouillard et al. (2022). This has been studied to some extent under the name of meta-reinforcement learning, which attempts to predict causal relations from the observations of environments (ooonmets//SStterrr et al. (2023); Daaaguptaaa al..
(2019). Generally, transferability has been considered in reinforcement learning, where the efficient use of data is omnipresent and causality provides guarantees regarding the transferability of mecha-
nisms between environments (Sæmundsson et al., 2018 Dasgupta et al., 2019 Zeng et al. 2023).
Gating Models. Modeling switching causal relations (Liu al., 2023) via so called 'gates' has been considered in prior works Minka & Winn (2008); Winn (2012). While MCMs extend beyond context-specific independencies, gates pose a practical way of modeling switching relations cases
where MCMs can be reduced to context-conditionel SCM.
Large Language Models (LLMs). Meta-causal representations are an important consideration for LLMs and other foundation models, since these models are typically limited to learning world dynamics from purely observational textual descriptions. LLMs need to learn meta-causal models that allow them to simulate the consequences of interventions (Lampinen et al.) 2024 Li et al. 2021 2022). To the best of our knowledge, Zečević et al. (2023) made the first attempt to define explicit meta-causal models that integrate with the Pearlian causal framework. However, their MCM are purely defined as adjacency matrix memorization, such that causal reasoning in L.LMs equals a simple knowledge recall (X + Y) + (exy Mem) and thus fails to generalize to novel scenarios.
Actual Causality, Attribution and Cognition. Our MCM framework can be used to infer and at- tribute responsibility, as shown in Sec. 4.1 and therefore touches on the topics of fairness, actual causation (AC) and work on counterfactual reasoning in cognitive science. (Von Kügelgenn et al. 2022 Karimi al. 2021 Helper] 2000; 2016 Chockler Halpern 2004; Gerstenberg et al.] 2014 Gerstenberg 2024 The similarities also extend to how MCM encourage reasoning about the de- pendence between actual environmental contingencies and qualitative types of causal mechanisms. In this sense, MCM allow for the direct characterization of actual causes of system dynamics types, but a rigorous formalization open for future work. While AC in combination with classical SCM only describes relationships between causal variable configurations and observable events, there are cases where we can take an MCM and derive an SCM that encodes the meta-causal types with instrumental variables, thus allowing similar meta-causal analysis within the AC framework.
6 CONCLUSION
We formally introduced meta-causal models that are able to capture the dynamics of switching causal types of causal graphs and, in many cases, better express the qualitative behavior of the system under consideration. Within MCM, types generalize the notion of specific structural equations and abstract away unnecessary detail. We presented a motivating example of how a classical causal and meta-causal inference might disagree on the attribute of root causes. We extended claims by considering that MCM are still able attribute changes in mechanistic behavior of a system, even when no actual changes becomes apparent. apparent. We presented first approach to recover meta-causal states in the bivariate case. Although our experimental results only represent a first preliminary approach, we find that MCM are powerful tool for modeling, reasoning, and inferring system dynamics. We demonstrated how MCM can be deployed dynamical systems and proved that they extend beyond conventional SCM.
Limitations and Future Work. While we have formally introduced meta-causal models, there remain several open directions to pursue, which we briefly touch on Appendix [H] We have been able to provide examples that illustrate the differences between standard causal, and meta-causal attribution. In particular, the combined application of standard and meta-causal explainability will allow for the joint consideration of actual and mechanistic in future attribution methods. However, our approach to recovering meta-causal states from unlabeled data is open to extension. Discovery on the full causal graphs is desirable goal that is difficult to achieve for the reasons discussed in this paper. Finally, we made a first attempt to present examples for and against the controlling or observational role of the presented observational perspective on MCM is mainly of interest for analytical applications, the application to agent systems and reinforcement learning might open up further fields of application.
models, which we briefly discuss further Appendix While,
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ACKNOWLEDGMENTS
The Technical University of Darmstadt authors received funding by the EU project EXPLAIN, funded by the German Federal Ministry of Education and Research (BMBF) (grant 01-S22030D), They acknowledge the support of the German Science Foundation (DFG) project "Causality. Ar- gumentation, and Machine Learning" (CAML2, KE 1686/3-2) of the SPP 1999 "Robust Argumen- tation Machines" (RATIO). This work supported by the Hessian Ministry of Higher Education, Research, Science and the Arts (HMWK; projects "The Third Wave of AI"). It was funded by the European Union. Views and opinions expressed are, however, those of the author(s) only and do not necessarily reflect those of the European Union or the European Health and Digital Executive Agency (HaDEA). Neither the European Union nor the granting authority can be held responsible for them. Grant Agreement no. 101 120763 TANGO. This work supported from the National High-PPeformance Computing project for Computational Engineering Sciences (NHR4CES).
The Eindhoven University of Technology authors received support from their Department of Math- ematics and Computer Science and the Findhoven Artificial Intelligence Systems Institute.
REFERENCES
Tara Anand, Adèle H Ribeiro, Jin Tian, and Elias Bareinboim. Effect identification in cluster causal diagrams. arXiv preprint arXiv:2202.1222 2022.
Theodore W Anderson and Donald A Darling, Asymptotic theory of certain" goodness of fit" criteria based on stochastic processes. The annals of mathematical statistics, pp. 193-212, 1952.
Argyris Arnellos, Thomas Spyrou, and John Darzentas. Dynamic interactions in artificial environ- ments: causal and non-causal aspects for the emergence of meaning. Systemics, Cybernetics and Informatics, 3:82-89, 2006.
Sander Beckers and Joseph Y Halpern. Abstracting causal models. In Proceedings of the aaai conference on artificial intelligence, volume 33, pp. 2678-2685, 2019.
Richard Bellman. A markovian decision process. Journal of mathematics and mechanics, pp. 679- 684, 1957.
J Douglas as Bremner. Traumatic stress: effects on the brain. Dialogues clinical neuroscience, 8(4): 445 461, 2006.
Philippe Brouillard, Perouz Taslakian. Alexandre Lacoste, Sébastien Lachapelle, and Alexandre Drouin. Typing assumptions improve identification in causal discovery. In Conference on Causal Learning and Reasoning, pp. 162-177. PMLR, 2022.
SD Calkins. Origins and outcomes of individual differences in emotion regulation. The Development of Emotion Regulation: Biological and Behavioral Considerations, Monographs of the Society for Research in Child Development, 59(2-3), 1994.
Colin Chen. Tests for the goodness-of-fit of the laplace distribution. Communications in Statistics- Simulation and Computation, 31(1):159 174, 2002.
David Maxwell Chickering. Optimal structure identification with greedy search. Journal of machine learning research, 3(N0v):507-55,, 2002.
Hana Chockler and Joseph Y Halpern. Responsibility and blame: A structural-model approach.
Journal of Artificial Intelligence Research, 22:93-115, 2004.
Ondřej Chum, Jiří Matas, and Josef Kittler. Locally optimized ransac. In Pattern Recognition: 25th DAGM Symposium, Magdeburg, Germany, September 10-12, 2003, Proceedings 25, pp. 236-243. Springer, 2003.
Sheldon Cohen, Tom Kamarck, and Robin Mermelstein. A global measure of perceived stress.
Journal of health and social behavior, pp. 385-396, 1983.
Taco Cohen. Towards a grounded theory of causation for embodied ai. arXiv preprint arXiv:2206. 13973, 2022.
11

---

Published as a conference paper at ICLR 2025
Juan D Correa, Sanghack Lee, and Elias Bareinboim. Counterfactual transportability: formal approach. In International Conference on Machine Learning, PP. 4370 4390. PMLR, 2022.
Ishita Dasgupta, Jane Wang, Silvia Chiappa, Jovana Mitrovic, Pedro Ortega, David Raposo, Edward Hughes, Peter Battaglia, Matthew Botvinick, and Zeb Kurth-Nelson. Causal reasoning from meta-reinforcement learning. arXiv preprini arXiv: 1et11199110000 2019.
Arthur P Dempster, Nan M Laird, and Donald Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the royal statistical society: series B (methodological), 39(1): 1-22, 1977.
Joel E Dimsdale. Psychological stress and cardiovascular disease. Journal of the American College of Cardiology. 51(13):1237-1246 2008.
Martin A Fischler and Robert C Bolles. Random sample consensus: a paradigm for model fitting with applications image analysis and automated cartography. Communications of the ACM, 24 (6):381-395 1981.
Tamara B Franklin, Bechara J Saab, and Isabelle M Mansuy. Neural mechanisms of stress resilience and vulnerability. Neuron, 75(5):747-761, 2012.
Tobias Gerstenberg. What would have happened? counterfactuals, hypotheticals and causal judge- ments. Philosophical Transactions of the Royal Society B, 377(1866):22103333, 2022.
Tobias Gerstenberg. Counterfactual simulation in causal cognition. Trends in Cognitive Sciences, 2024.
Tobias Gerstenberg, Noah Goodman, David Lagnado, and Josh Tenenbaum. From counterfactual simulation causal judgment. In Proceedings of the annual meeting of the cognitive science society, volume 36, 2014.
Giorgos Giannakakis, Dimitris Grigoriadis, Katerina Giannakaki, Olympia Simantiraki, Alexandros Roniotis, and Manolis Tsiknakis. Review on psychological stress detection using biosignals. IEEE transactions on affective computing, (3(1):440-460 2019.
Robert Haggerty. Stress, risk, and resilience in children and adolexcents: Processes, mechanisms, and interventions. Cambridge University Press, 1996.
Joseph Y Halpern. Axiomatizing causal reasoning. Journal of Artificial Intelligence Research, 12: 317-337, 2000.
Joseph Y Halpern. Actual causality. MiT Press, 2016.
Christina Meinze-Deml, Jonas Peters, and Nicolai Meinshausen. Invariant causal prediction for nonlinear models. Journal of Causal Inference, 602):20170016, 2018.
Stefan G Hofmann, Joshua Curtiss, and Steven C Hayes. Beyond linear mediation: Toward
dynamic network approach to study treatment processes. Clinical Psychology Review, 76:101824, 2020.
Patrik Hoyer, Dominik Janzing, Joris M Mooij, Jonas Peters, and Bernhard Schölkopf. Nonlinear causal discovery with additive noise models. Advances in neural information processing systems, 21, 2008.
Dominik Janzing and Sergio Hernan Garrido Mejia. Phenomenological causality. arXiv preprint wwXiv:2211.0902. 2022.
Amir-Hossels Karimi, Bernhard Schölkopf, and Isabel Valera. Algorithmic recourse: from coun- terfactual explanations to interventions. In Proceedings of the 2021 ACM conference on fairness, accountability, and transparency. pp. 353-362, 2021.
Armin Kekić, Bernhard Schölkopf, and Michel Besserve. Targeted reduction of causal models. In ICLR 2024 Workshop A14Differennialiiuuttoo In Science, 2023.
12

---

Published as a conference paper at ICLR 2025
Andrew Lampinen, Stephanie Chan, Ishita Dasgupta, Andrew Nam, and Jane Wang. Passive learn- ing of active causal strategies in agents and language models. Advances in Neural Information Processing Systems, 36, 2024.
Sanghack Lee and Elias Bareinboim. Structural causal bandits: Where to intervene? Advances in neural information processing systems, 31, 2018.
Belinda Li, Maxwell Nye, and Jacob Andreas. Implicit representations of meaning in neural language models. arXiv preprint arXiv:2106.00773 2021.
Kenneth Li, Aspen K Hopkins, David Bau, Fernanda Viégas, Hanspeter Pfister, and Martin Watten- berg. Emergent world representations: Exploring a sequence model trained on a synthetic task. arXiv preprint arXiv:2210.133882 2022.
Yongtuo Liu, Sara Magliacane, Miltiadis Kofinas, and Efstratios Gavves. Graph switching dynami- cal systems. In International Conference on Machine Learning, pp. 21867-21883, PMLR, 2023.
Peter Maisel, Erika Baum, and Norbert Donner-Banzhoff. Fatigue as the chief complaint: epi- demiology, causes, diagnosis, and treatment. Deutsches Ärzteblatt International, 118(33-34): 566, 2021.
Tom Minka and John Winn. Gates. Advances in neural information processing systems, 21, 2008.
Edward F Moore al. Gedanken-expprrmeents on sequential machines. Automata studies, 34:129- 153, 1956.
Junhyung Park, Simon Buchholz, Bernhard Schölkopf, and Krikamol Muandet. measure-theoretic axiomatisation of causality. Advances in Neural Information Processing Systems, 36:28510- 28540, 2023.
Judea Pearl. Causality. Cambridge university press, 2009.
Judea Pearl and Dana Mackenzie. The book of why: the new science of cause and effect. Basic books, 2018
Jonas Peters, Peter Bühlmann, and Nicolai Meinshausen. Causal inference by using invariant pre- diction: identification and confidence intervals. Journal of the Royal Statistical Society Series B: Statistical Methodology, 78(5):977-0012, 2016.
Paul K Rubenstein, Sebastian Weichwald, Stephan Bongers, Joris M Mooij, Dominik Janzing. and Bernhard Schölkopf. Causal consistency of structural equation mod- cls. arXiv preprint arXiv:1707.00879 2017.
Moritz
Steindór Sæmundsson, Katja Hofmann, and Mare Peter Deisenroth. Meta reinforcement learning with latent variable gaussian processes. arXiv preprint arXiv: 1803.07551, 2018.
Andreas WM Sauter, Erman Acar, and Vincent Francois-Lavet. A meta-reiifforeement learning algorithm for causal discovery. In Conference Causal Learning and Reasoning, pp. 602-619. PMLR. 2023.
Wolff Schlotz, Ilona S Yim, Peggy M Zoccola, Lars Jansen, and Peter Schulz. The perceived stress reactivity scale: measurement invariance, stability, and validity in three countries. Psychological assessment, 23(1):80, 2011.
Bernhard Schölkopf. Dominik Janzing, Jonas Peters, Eleni Sgouritsa, Kun Zhang, and Joris Mooij. On causal and anticausal learning. arXiv preprint arXiv: /206.6471, 2012.
Steven A Sloman, Philip M Fernbach, and Scott Ewing. Causal models: The representational in- frastructure for moral judgment. Psychology of learning and motivation, 50:1-26, 2009.
Anoopkumar Sonar, Vincent Pacelli, and Anirudha Majumdar. Invariant policy optimization: To- wards stronger generalization in reinforcement learning. In Learning for Dynamics and Control, pp. 21-33. PMLR, 2021.
13

---

Published as a conference paper at ICLR 2025
Peter Spirtes, Clark N Glymour, Richard Scheines, and David Heckerman. Causation, prediction, and search. MIT press, 2000.
Davide Talon, Phillip Lippe, Stuart James, Alessio Del Bue, and Sara Magliacane. Towards the reusability and compositionality of causal representations. arXiv preprint arXiv:2403.0988 2024.
Julius Von Kügelgen, Amir- Karimi, Umang Bhatt, Isabel Valera, Adrian Weller, and Bern- hard Schölkopf. On the fairness of causal algorithmic recourse. In Proceedings of the AAAI conference on artificial intelligence, volume 36, pp. 9584-9594, 2022.
Jonas Wahl, Urmi Ninad, and Jakob Runge. Foundations of causal discovery groups of variables. arXiv preprint 2rXx:22300000477,, 2023.
Clare R Walsh and Steven A Sloman. The meaning of cause and prevent: The role of causal mech- anism. Mind & Language. 26(1):21-52, 2011.
Manjie Wang and Kimberly Saudino. Emotion regulation and stress. Journal of Adult Develop- ment, 18:95-103, 2011.
Moritz Willig, Matej Zečević, Devendra Singh Dhami, and Kristian Kersting. Do not marginalize mechanisms, rather consolidate! In Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS), 2023.
John Winn. Causality with gates. In Artificial intelligence and statistics, pp. 1314-1322. PMLR, 2012.
Phillip Wolff. Representing causation. Journal of experimental psychology: General, 136(1):82, 2007.
Zhongwei Yu, Jingqing Ruan, and Dengpeng Xing. Learning causal dynamics models in object- oriented environments. arXiv preprint arXiv:2405. 12611, 2024.
Matej Zetević, Moritz Willig, Devendra Singh Dhami, and Kristian Kersting. Causal parrots: Large language models may talk causality but are not causal. Transactions on Machine Learning Re- search, 2023.
Yan Zeng, Ruichu Cai, Fuchun Sun, Libo Huang, and Zhifeng Hao. A survey on causal reinforce- ment learning. arXiv preprint arXiv:2302.05209, 2023.
Junzhe Zhang and Elias Bareinboim. Transfer learning in armed bandit: causal approach. In Proceedings of the 16th Conference on Autonomous Agents and MultiAgent Systems, pp. 1778- 1780, 2017.
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APPENDIX FOR "SYSTEMS WITH SWITCHING CAUSAL RELATIONS: A META-CAUSAR PERSPECTIVE"
The appendix is structured follows: In Sec. A we describe the general emergence of causal lations in SCM from the underlying mediating process. In Sec. B we discuss desirable classes of identification functions. In Sec. Bec.Crun we consider the existence and compatibility of decomposed equa- tions. In Sec. De prove condition for the reduction for meta-causa models to conditioned SCM. In Sec. B we derive the formula for the theoretical upper bound of the 95% confidence interval.
Sec F present the sample statistics and convergence results of the applied LO-RANSAC algorithm.
we present the exact parameters for sampling bivariate meta-causal mechanism data. In Sec. H we discuss further practical applications of meta- causal models. Finally, in Sec. we discuss
In Sec.
G
possible assertive properties of meta-causal models.
A EMERGENCE OF CAUSAL EFFECTS FROM MEDIATING PROCESSES
The definition of a meta-causal frames (Def. grounds the emergence of standard SCM to an under- lying mediating process E. In particular, for any meta-causal frame, particular causal interactions are read via the identification function. While the underlying mediation process is dime-dependent, the resulting causal graph is the projection of all causal interactions within the process onto a graphical structure. Note, however, that the resulting SCM still preserves the sequence of causal interactions through the DAG induced partial ordering of variables. This 'logical' time ordering (induced via the partial order) can be seen as an abstraction of the underlying process.
B DESIRABLE CLASSES OF IDENTIFICATION FUNCTIONS
Particular choices of different identification functions will result in different identified meta-causal states. However, from a classical causal perspective it might be desirable to choose particular classes of functions to identify faithful SCM. In particular, one could ask the question whether indirect effects are of interest and should also be identified by the identification function. Consider a scenario where three variables X, Y, Z are considered part of an to- to-be-identified SCM from an underlying process. In our scenario we identify the direct causal effects + Y and Y + Z and -under the assumption that no further direct edges can be identified-, assume the graph be faithful and do not identify the indirect causal relation - Z. This however changes, once variable Y is dropped of the variable set X of the SCM. Now, with the same underlying mediating process we want our identification function to identify X + Z, which was omitted before. Generally, we assume that in most cases identification function that identify direct causal effects with regard to X are the most common.
Following this rather high-level discussion, we frame the previous procedure in terms of the iden- tification function and functional relation. Given all functional relations S the type decoder can determine whether the whole functional relation X, between variables X, is mediated by some other (set of) intermediate mechanism(s) X In situations where X, + X2 - X, the type en- coder needs identify whether the relation X; + X, is purely mediated via X (meaning that all computational paths from X; Xj in X55 that does not passes through X3 or some additional direct effect exists that are not 'shielded' by X Conversely, whenever some Xg E X₂ of the intermediate set is dropped//maginnalized from the causal variables, the mechanism X X is no longer identified as the mechanism of a causal variable within the overall X5 and (given that the remaining S (A) does not shield X; from X,) direct arrow X, X, can be inferred.
Given our meta-causal framework we, however, have the freedom to identify the indirect relations. In that case one might assign these edges a particular '[1-hop] indirection' type, indicating it to
'Note in this context, that assume structural equations to be uniquely identifiable. This might be im- plemented via type-theoretic considerations which every variable the underlying process gets assigned a unique type. such that for every s, E S., Vi, j. o S. + S, hold. Even though two functions might be isomorphic they can be thus be differentiated via their domain and codomain. In practice, one might addition- ally consider the computations graph of the mediation process uniquely differentiate hetween isomorphic functions
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be the result of an indirect relation via (possibly unobserved) intermediate variable. Consider, how this notion of edge types over unobserved variables is already in use in CPDAGs as for ex- ample produced by the PC GES algorithms (Spirtes al. 2000: Chickering 2002 to indicate undirected edges for which the orientation is currently unknown.
C COMPATIBILITY OF DECOMPOSED EQUATIONS
In this section, we provide brief discussion on the existence and compatibility of functions X as considered in Equation
Since the conditioned SCM in Eq. i modeled after an the particular causal model that exists under meta-causal state indexed by z, follows that particular composition of functions J has exist (and that the functions are compatible), since the full function fj exists for each meta-causal state. In a naïve approach, the order of composition i E {I..N} enforces a particular evaluation order of the functions, and particular requires this order to be the same for every meta-causal state.
Generally, the presented function composition might not work, in the case that the individual func- tions get chosen badly from the start. A simple solution to overcome such problem is assume compositions of lifted functions and assume their signatures to be compatible (which is always permitted due to the known existence of the composed f7)) Note that functions fij. fiy. whose functional type 4ij did not change under change of to z', make signatures compatible.
Common handling of compositionality in SCM: The adjustment of signatures in fact often considered in the case of compositional SCM, c.g. additive noise models, where the signature of an outer 'merging function' f. e.g. Σiepa(j) fiz, is fact adjusted based on the number parents (or, otherwise, zero weight edges are incorrectly excluded from the parent set).
D REDUCTION OF META-CAUSAL MODELS TO CONDITIONED SCM
In this section, we provide a condition under which meta-causal models can be reduced to ordinary SCM with structural equations conditioned some external variable related to Eq. T
Definition 4 (MCM Reducability). For given mediation process E (S, and abstraction
Pp S - X call meta-causal model A (TNxN ,5,5) reducible to conditioned SCM if there is some SCM over X [UZZ} where the types of functional dependencies between X,, Xj X can be fully determined by Z.
Theorem D.1 (Specific Criterion for Meta-Causal Reducability). ffr given mediation process E E (S, o) and abstraction S - X, a meta-cuusal model A = (TNXN ,8,8) all its transitions are loops, then it is reducible to a conditioned SCM.
Proof. If meta-causal model only has loops as transitions, the meta-causal types are independent of the modeled mediation process. While types can differ for different starting conditions in the environment, its transition process never results in a type change. Hence, meta-causal types induce an equivalence relation S, compatible with and z T NNN or, alternatively, with values Z for each connected component in È (S,
introduce exogenous variable Z with
We also see some potential to weaken this criterion and, therefore, find more general condition for reducibility by further examination of the abstraction that links the mediating process with the causal variables for future work. The primary obstacle in this regard is that meta-causal models abstract away some information about structural equations, such that interventions on might lead to mismatch between the resulting SCM and the underlying mediation process.
E PROBABILITIES FOR SAMPLE COMPUTATION AND UPPER BOUNDS
Consider dataset D m of samples two variables where want to separate different functions. We assume that the data distribution contains a uniform number of samples from each function, where each class could be under- or overrepresented offset of d. Specifically,
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assume that each function is represented by (1=d) |ID] samples, i.e., the probability of encountering To identify all functions between
L+d and [X)( particular class is I-d I P(X) I " these two variables, we assume linearity and apply EM with R RÁNSAC on random pairs of samples (see Section 4.2). By selecting n pairs, there is chance that one pair is chosen from each function (we will refer to such set of pairs "correct" set of samples). In this section, we derive the probability of a correct pair being chosen at random, so that we can estimate how many times pairs need to be sampled to reliably encounter correct set of samples.
n
We denote as the event "correctly" sampling all n pairs from all n different functions. If all classes have the same number of samples, the chance of randomly selecting pair from new class 1 for the first pair samples, a-1 - for the second... .... and 1 for the last; short:
n2! 12
1 11
n!
([)
n2
If the data distribution is not perfectly uniform, i.e., d * O, we can also calculate a lower bound for the same probability. Consider two probabilities per sample: the probability of selecting class Pn (S and the probability of selecting a second sample of the same class (n((Stame) afterwards. Across all samples, these correspond to n! n and (1)) in (S)), respectively.
Let us first consider the probability of selecting one new class can be selected (probability of 1). If this sample was taken from the largest class first, the probability that subsequent samples will be taken from new classes decreases, since the space of "unsampled" classes is smaller. For this lower bound, we therefore assume that maximally large classes sampled from as much and as early as possible. According our assumptions, the largest classes each take up a fraction of L+d of the data. Therefore, the probability of selecting
class. When the first sample is taken, only
First, consider the case where is even. Here, after all the znz largest classes have been selected, only the small classes remain. For the last, second to last, ... classes, this probability represented by (1-d) 2(1-d) Overall, we get the probability
new class for successive samples has the following probabilities e. --(1+d) n-(1+d)2
12 (1 + d) i)
i(1 d)
Pa Swen
II
I
n
n
n is uneven, an average size class between the largest and smallest classes must be included:
4 -11
n (1 d)(n2 -i)
n (m- (1 + d) 0.5)
Cmew Pa (S
II +0.5
II
The probability of selecting a second sample of the same class is easier to calculate. Instead of constant probabilities in the expectation with 1 we now have two different probabilities in the even case and three in the odd case. In the even case, we have large batches and the same number of small batches, so the probability of choosing the right batch each time is
d
Pa (
Gsaae
72
72
In the uneven case, we also have to also consider the batch that has an average size
1
1 n
7
(
Cisnne rdi
Note that for both P. (Snew) and P. (SSame), the distribution of the data into the largest and smallest possible batches (according to our assumptions) results in the smallest possible probabilities; hence, the computed probability is lower bound. If the batches were more evenly sized, the probability would be larger. We can also see that a deviation of up to results a probability O, since is impossible sample from class that not represented in the data.
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Published as a conference paper at ICLR 2025
Class Deviation
1 Mechanism
2 Mechanisms
Mechanisms 592 (11.84%)
4 Mechanisms
0.0
4219 (84.38%
1740 (34.80%)
86 (1.72%
0.1
1702 (34.04%)
577 (11.54%)
84 (1.68%)
0.2
1567 (31.3%%)
555 (11.10%)
84 (1.68%)
Table 3: Empirical estimated convergence percentage of the EM algorithm for different class imbalances and number of mechanisms. The table shows the number of samples converged and the convergence rates (in parentheses) for single random initialization and for estimating the pa- rameterization of the underlying system for a given number of mechanisms. All results are reported per 5000 samples.
In total, the probability of selecting of correct sample set is the product of the two probabilities above, i.e.
( (nn(Seen Pnn(S (Seven)) P. (SncH) odd P. (Stam)
if n is even
(s(
(3)
if n is odd.
(S) is the (lower bound on the) probability of selecting correct set of samples. We can calculate the number of trials k needed to find such a set of samples with at least 95% probability. The opposite probability, of never finding it with less than 5% probability, is easier to calculate:
(1 P(S))* < 0.95 0.05
1 In(1 P(S)) - ln(0.05)
ln.055))
k - In(1 - P(S))
This allows us to determine how many attempts might be necessary. Note that while this would leave 5% chance of picking the right samples, there various practical reasons why the actual probability of finding a working set of samples will be higher, e.g., if the number of samples from each class is not as uneven assumed, or if some samples distributed in such way that even picking a sample from the "wrong" class might still lead to the identification of the correct mechanisms.
For example, n = and d = 0.2, we have an expected probability of E[P(S)) =
2: - 0.125 and
lower bound of P2(5) P(S P (Stame) 2 0.8 2 1.2 0.8 = 0.096. Larger deviations decrease the probability while a deviation of d = 0 results in the same probability as with ([)() For the lower bound, this results in & 30 samples for confidence of 95% using the above calculation steps. All resulting sample counts can be found Table 2
F EM CONVERGENCE RESULTS
The required number of resamples for 95% success rate of the RANSAC algorithm calculated by 1og(0.00))/logt)) CC)), where S1 are the convergence rates for the individual samples computed in Sec. E
Empirical convergence probabilities and resulting resampling counts for the LO-RANSAC algo- rithm are shown Tables 2 and 3 Table 4 lists the goodness of fit for all converged samples. In general, find that in cases where the approach is able to converge, it undereuts the required parameter convergence boundary of 0.2 by factors of 4.8 and 3.5 for the slope and intercept, respec- tively.
G MECHANISM SAMPLING
For experiments in Sec. 4.2 uniformly sample the number of mechanisms be in K (1..4.. The slopes of the linear equations are uniformly sampled between a E +(0.2..5) and the intercepts are in the range 3 [-5,5], We add Laplacian noise L(xlu, ju, b) 25 exp(- - (ا-يع] with = and E [0.1, 4.0]. X values are uniformly sampled in the range [-5, 5] and y: az; + [(x10, b). The average number of samples per class is set 500. Throughout the experiments,
18

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Published as a conference paper at ICLR 2025
K=1
-12
K=2
10
K=3
-00
10
10
K=4
-2
12
15
Figure 3: Sampled Mechanisms. The figure shows selection of different randomly sampled mechanism distributions, ranging from one to up to four simultaneously present mechanisms. The gray dotted lines represent the generating ground truth mechanisms. (Best Viewed In Color)
vary the class probabilities maximize the class deviation by assigning K/2 classes the maximum probability 1/K (1 + D) and K K/2 classes the minimum probability 1/K * (1-1) D). If K is odd, a class is assigned the average probability 1/K. We show selection of the resulting sample distributions in Fig.
class deviation factor
{0.0, 0.2). Specifically,
H PRACTICAL APPLICATIONS OF CECAA-CAUSAL MODELS
As the main focus of our initial work on meta-causal model lies on providing a first, formal, defi- nition of meta-causal models, we tried to approach MCM from spectrum of different theoretical
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Published as a conference paper at ICLR 2025
Class Deviation
1 Mechanism 0.0349/0.0590
Mechanisms
3 Mechanisms
Mechanisms
0.0
0.0370/0.0543
0.0380 0.0592
0.0389/0.0514
0.1
0.0381 00.055
0.0381 0.0566
0.0346 0.0528
0.2
0.0414|0.0548
0.0412 0.0567
0.0402
Table 4: Mean average error for the slope and intercept of the correctly predicted mechanism for different class imbalances and number of mechanisms. Mechanisms are accepted their parameters do not differ by more than 0.2 from the ground truth parameterization. 'hee results show that converged samples are typically estimated with an error well below the threshold.
perspectives. In this Section, we provide a brief discussion on two possible practical fields of appli- cations for MCM.
Health and Medicine. First, we would like to expand on our stress-induced fatigue example as it can not be reduced to a standard SCM, and provide an actionable perspective on MCM. While we still assume the underlying neuropsychological process to be more complex, with multiple interplaying factors to influence each other, we consider the same simplified model as presented in the paper. We now assume that some drug exists that is able to influence certain health related processes within the patient, such that the underlying -previously self-reinforcing- stress relation is unconditionally changed to suppressing one. (Upon closer consideration, the previous intervention might constitute a meta-causal do-operator, as we detach the functional type from the underlying dynamics and fix its particular functional type.)
This perspective not only allows the forecast of system changes, but also yields an actionable model which can be actively steered between meta-causal states. To permanently treat a patient, one could consider the objective to reach a self-stabilizing meta-causal state. Note how this meta-causal ob- jective might be different to that of classical causal one, where stress levels would similarly be reduced, but no attention is placed on the (possibly unchanged) system dynamics, such that stress levels might rise up again after the intervention ends.
Second, recall that our MCMs are defined as finite state machines. Figuring out exact transition conditions that induce meta-causal state transitions also yield important insights on the volatilitysstabiiity of systems in terms of risk analysis and policy making. Such scenarios might commonly arise in economies, where relations markets can change due to the sudden appearance of disrupting factors (e.g. new competitor entering the market or financial crisis) while effeets might persist even with the disrupting factor having vanished.
Economics.
I ASSERTIVE META-CAUSAL MODELS
In this section we will briefly touch upon possible assertive properties of meta-causal models. this initial paper we chose to utilize very general definition of types, which intentionally held flexible to allow for the most descriptive models. As for this defimition, there might exist a gap between between the descriptive modeling of MCM and their 'assertive', data generating, properties. This gap primarily stems from the mapping of specific structural equations onto abstract types, which prevents back projection of types structural equations in the general case. With the special class of conditionally reducible SCM we, however, presented a particular class of MCM that yield "assertive' properties by being able translate types back onto the level of structural equations. Further restrictions on types and their relations to structural equations might be placed in specific applications, to shield users from misuse of the framework. Still, we where able to present several relevant applications of MCM even with this this most general formalization of MCM.
While meta-causal states might be mapped back to configurations of 'classical' causal models, we highlight that MCM are primarily concerned with the modeling of the overarching meta-causal state transitions. By modeling MCM as (possibly non-deterministic) finite-state machines, we, fact, can make predictions about systems' future course the meta-level. This includes the sampling of state trajectories and making assertions about their stability and similar properties.
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Published as a conference paper at ICLR 2025
Algorithm Recovering Mechanisms for the Bivariate Case
procedure RECOVERMECHAAIISSSS,) maxClassDev Kmax EMSteps)
2:
for all & [L../ do bestModelLoggL
3:
4:
N + requreddSSSall maxClassDev, 0.95)
Compute # of samples (c.f. Sec. RANSAC iteration.
5:
for all [1..N] do
6:
px, py <- Gample(r;, V/. k)
Initialize parameters with 2 x & points.
7:
for all k' E )0..k) do
8:
ag (py2k+1 Pyzk) / (p02k+1 p(22)
9:
3x + Py2k ag * 22k
10:
by 1.0
Assume initial avg. deviation of the Laplacion to be 1. > Assume X Y direction first.
11:
dg <- 'XY'
C₁ Praplacian
(Ka 3,,,,,
Initial class probabilities for all samples.
13:
end for
for all / [MSees do
EM Iteration.
15:
a. 3, b. + regressLines c C Pa-pppaactaa
> (Weighted) median regression.
17:
end for modelLogl-- + E; LogP Lapieiinn x, 33 b, d); if modelLogL bestModelLogL then
Obtain the joint log probs.
Laplacian
19:
20:
bestModelLogL + modelLogL
21:
bestParameters +- (a, B, b.d)
22:
end end for
23:
24:
alll aplaciaa <- true
D. Check if residuals are Laplacian distributed.
25:
for all I E k1..k] do
26:
(a. b, d) + bestParameters
Select points where argmax, = x., y' filter(x y', c¹, 0.4) Keep points s.t. 2 0.4(1 max #1(c))
x.
select y,c,))
28:
29:
r = (inEg(x' y a,, B, d:) yy
not mandersonDaring(( 0.95) then
31:
allLaplacian +- false break
32:
33:
end if end for
34:
35:
if allaaacaaa then return k
36:
37:
end if
38:
end for
39:
return 0
40: end procedure
21


Reviews:
[{"review_id": "77OiHD9agn", "reviewer": "Reviewer_pnp9", "summary": "The paper introduces the concept of *meta-causal states*, which attempt to represent how the underlying dynamics of a causal model can change based on context. The authors then examine how meta-causal states could be used in the attribution of responsibility, describe how the number of distinct meta-causal states can be identified empirically (a variety of EM), and provide an example analysis of stress-induced fatigue.", "questions": "What sorts of practical causal inference in medicine, economics, social science, or other domains would be improved by consideration of meta-causal states? \n\nWhat errors are made now using conventional causal modeling that would be avoided by the use of meta-causal states?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The concept is interesting and the paper is well-written and detailed.\n\nSection 4 presents some diverse and interesting applications of the basic theory.\n\nThe graphics, particularly Figure 1, are clear and easy to understand.", "weaknesses": "The paper should cite some relevant prior work in graphical models with “gates” (Minka 2008; Winn 2012). The concepts in those papers are somewhat different, but closely related to the topic of this paper.\n\nThe paper should note that some meta-causal states can actually be represented by ordinary SCMs. For example, in an SCM in which Z is caused by both X and Y, it is entirely possible for Y to control *the manner in which* X affects Z. In the simplest case, Y could be a binary variable that determines the functional form of how X affects Z. This complicates the structural equation of Z, but it’s not outside of the standard definition of SCMs. Of course, this approach has limitations. For example, it cannot be used to reverse the direction of causation between two other variables (at least not within the standard definition of SCMs). However, it can toggle the existence of an edge or change the manner in which one or more parent variables affects their child. This should be noted more clearly in the discussion.\n\nThe utility of meta-causal states is not immediately evident. That is, what sorts of practical causal inference in medicine, economics, social science, or other domains would be improved by consideration of meta-causal states? What errors are made now using conventional causal modeling that would be avoided by the use of meta-causal states? I suspect that there are good answers to these questions, but it would improve the paper if they were more clearly explained.\n\n**References**\n\nMinka, T., & Winn, J. (2008). Gates. *Advances in Neural Information Processing Systems*, *21*.\n\nWinn, John. \"Causality with gates.\" In *Artificial Intelligence and Statistics*, pp. 1314-1322. PMLR, 2012.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kYKCytvmSM", "reviewer": "Reviewer_DQ6p", "summary": "This paper defines meta-causal systems to allow us to ask what caused a structural equation to be the way it is. This allows us to attribute responsibility for some variable sometimes to another variable, and sometimes to an external force that intervened on the system, as appropriate. An algorithm using RANSAC and EM is proposed to learn such meta-causal systems from data.", "questions": "- In section 3, I assumed from the matching notation that the variables $\\mathbf{X}$ are part of an SCM. Are they?\n- In definition 1, line 141, in what way does an element of  $\\mathcal{X}_j^{S}$ represent a \"change\" of $\\mathcal{X}_j$?\n- An odd point I wanted to note: if an SCM has arrows $X \\to Y$ and $Y \\to Z$, the meta-causal state would also have an arrow for the indirect effect $X \\to Z$. So these arrows aren't structural, merely descriptive. It depend on the answer to my question about $\\mathbf{X}$ above whether this is a drawback or merely remarkable.\n- Line 171 \"As standard causal relations emerge from the underlying mediation process\": How do they emerge? Is this always true?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "The idea of looking for the causes behind an SCM is very interesting and could be relevant in certain contexts. Putting these ideas in practice required the development of a nontrivial algorithm; while generalizing it is left for future work, this does help to solidify the contribution.", "weaknesses": "- The core concept of the paper seems to be multiply defined. I understand \"meta-causal\" to refer to questions of what *caused* the structural equations to be what they are (as in line 044-045). But in other places in the paper, it seems to refer to an abstraction where we don't look at the precise structural equations but instead at some qualitative characteristics. E.g. in the dynamical system of stress-induced fatigue in section 4.3, the structural equations remain the same but it is emphasized that there are are different causal states (e.g. self-reinforcing or self-suppressing). It is presented as a major premise of the paper (final sentence of abstract) that meta-causal states do not always result from external influences. But according to the first definition, there is only one meta-causal state, because the structural equations don't change, so this is not an example of meta-causality. (As a consequence of this ambiguity of \"meta-causality\" in the paper, I find it hard to assess to what extent the emprical results support the claims of the paper.)\n- I found the start of section 3 to be rather hard to follow, due to its density and it being unclear what intuition is being formalized. There is no overall description of how SCM and mediation process make up a more complex new process. The transition function seems to imply a dynamical view, with discrete time steps. But it is not made explicit anywhere that variables should be thought of as having a time index (should they?). In definition 1, the presence of the transition function does suggest that transitions between time steps are always happening.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dG0HGYKmKb", "reviewer": "Reviewer_PEUy", "summary": "The manuscript introduces a framework to analyze dynamic causal relationships that shift based on environmental or internal changes, called meta-causal models (MCM). The framework extends traditional causal models by defining “meta-causal states,” which group standard causal graphs into clusters of qualitative behavior. Then, MCM can dictate the detection of underlying changes in causal structures.", "questions": "* Are there other examples or mechanisms where classical models will fail to capture via observed changes in variable values and meta-causal models won’t? How often or realistic are the preventive mechanisms, like the ones described in Section 4.1?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 2, "strengths": "* The manuscript provides a formal framework for handling the dynamics of an SCM. Definitions 1 to 3 are elegant implementations of these ideas. In particular, framing meta-causal models as finite state machines is intuitive and well-suited for applications.\n* The framework allows for a systematic attributing responsibility in dynamic scenarios. In Section 4.1, the manuscript shows how counterfactual scenarios can be better explained in a meta-causal model. Furthermore, the paper provides a clear example of when attributing effects via model parameters might fail to explain preventative mechanisms.", "weaknesses": "* Meta-causality scope: the paper isn’t clear on two philosophical aspects of the framework regarding its scope and interpretation, namely (i) the causal-effect relationship between the meta-causal model and the causal model instance it is working with and (ii) the potential sutil difference between meta causal edges and classical ones\n\t- (i) From the initial example in Figure 1, the reader could question if the meta-causal model is causing the underlying classical causal model. This discussion isn’t handled in the manuscript, but there is an acknowledgment note in Section 3 under “Causal and Anti-Causal Meta-Causal States.”\n\t- (ii) How do we interpret and handle the subtle difference between meta-causal and classical relationships? For instance, in Fig 1, a classical causal discovery algorithm could attribute Api and Bx as causes of Ax.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1fHwlCDfwY", "reviewer": "Reviewer_puHX", "summary": "The paper introduces the concept of meta-causal models to formalize the emergence of causal relationships, which, for instance, allows one to formalize changes in the causal structure over time. While more classical approaches based on graphical causal models assume fixed causal relationships, the proposed meta-causal models can represent and reason about situations where causal mechanisms emerge, change, or disappear. The authors formalize meta-causal states that group similar causal behaviors and demonstrate how to infer these states from data in some toy scenarios.", "questions": "Overall, the paper provides a great first step towards an alternative formalism to describe and formalize the emergence of causal relationships. Since such types of works typically are more focused on the theoretical work and introduction of new formalisms, the concerns about the practical and experimental aspects are rather minor.\n\nGenerally, I would have expected a few more discussions related to the distinction between meta-causal states and traditional latent variable approaches. Although I am not that familiar with related work in this domain, I would have expected some more discussions with work introducing alternative ways to formalize causality, such as:\n- \"A Measure-Theoretic Axiomatisation of Causality\" by Park et al.\n- \"Phenomenological Causality\" by Janzing et al.\n\nThese works go in a different direction but seem, on a high level, related.\n\nThe introduction is very hard to follow and could be motivated better. While the example in Figure 1 is definitely helpful to understand the idea and limitations, the introduction often sounds too abstract. Maybe starting with the Figure 1 (a) example on a high-level description without further introduction of any formalisms around causal modeling could be helpful for the motivation.\n\nWhile the practical application in a general setting seems to be limited in the current form, the authors fairly discuss the current limitations and the work still provides a valuable foundation for follow-up work.\n\nWhen the authors can comment on the lack of theoretical statements (e.g., theorems ensuring that, under certain conditions, one can transform a MCM to a classical SCM), I am willing to further increase the score.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "+ Novel theoretical contribution towards a more generalized way to formalize causality\n+ Mostly clear motivation, although the introduction could be improved\n+ Although limited, the authors aimed to not only provide a theoretical discussion, but also insights towards a practical application\n+ Fair discussion of the limitation of the work", "weaknesses": "- While certain aspects of the theory are well introduced, theoretical (proven) statements and formalization of axioms are somewhat lacking, considering that the goal of the work is to establish a new formalism\n- The experimental evaluation is somewhat limited, but this is a minor point as the theoretical contribution is the main focus\n- The introduction is too complex and abstract\n- The practical applicability to larger, more complex systems remains unclear", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Spotlight)