When Causal Dynamics Matter: Adapting Causal Strategies through Meta-Aware Interventions
Authors: Moritz Willig
Tim Tobiasch
Devendra Singh Dhami
Kristian Kersting
Venue: NeurIPS 2025 poster ()
Keywords: Meta-Causality
Causal Dynamics
Decision-Making
Reflection
Primary area: probabilistic_methods

Abstract: Many causal inference frameworks rely on a staticity assumption, where repeated interventions are expected to yield consistent outcomes, often summarized by metrics like the Average Treatment Effect (ATE). This assumption, however, frequently fails in dynamic environments where interventions can alter the system's underlying causal structure, rendering traditional `static' ATE insufficient or misleading. Recent works on meta-causal models (MCM) offer a promising avenue by enabling qualitative reasoning over evolving relationships. In this work, we propose a specific class of MCM with desirable properties for explicitly modeling and predicting intervention outcomes under meta-causal dynamics, together with a first method for meta-causal analysis. Through expository examples in high-impact domains of medical treatment and judicial decision-making, we highlight the severe consequences that arise when system dynamics are neglected and demonstrate the successful application of meta-causal strategies to navigate these challenges.

When Causal Dynamics Matter: Adapting Causal Strategies through Meta-Aware Interventions
Moritz Willig
Tim Woydt
Department of Computer Science Technical University of Darmstadt
Department of Computer Science Technical University of Darmstadt
Devendra Singh Dhami
Kristian Kersting
Department of Mathematics and Computer Science Eindhoven University of Technology
Department of Computer Science
Technical University of Darmstadt Hessian Center for AI (hessian.Al) German Research Center for AI (DFKI)
Abstract
Many causal inference frameworks rely on staticity assumption, where repeated interventions are expected to yield consistent outcomes, often summarized by metrics like the Average Treatment Effect (ATE). This assumption, however, fre- quently fails in dynamic environments where interventions can alter the system's 's underlying causal structure, rendering traditional 'static' ATE insufficient or mis- leading. Recent works on meta-causal models (MCM) offer a promising avenue by enabling qualitative reasoning over evolving relationships. In this work, we propose specific class of MCM with desirable properties for explicitly modeling and predicting intervention outcomes under meta-causal dynamics, together with a first method for domains of medical treatment and judicial decision making, we highlight the severe consequences that arise when system dynamics neglected and demonstrate the successful application meta-causal strategies to navigate these challenges.
analysis. Through expository examples in high-ing high-impact
decision-making,
Introduction
Exercising agency complex, real-world scenarios such as policy making, medical treatment, or autonomous agents carries many inherent risks and responsibilities. The consequences of actions often extend beyond immediate effects and induce lasting changes in system dynamics. Classical causal inference, largely based structural causal models (SCMs) Spirtes al. [2000), Pearl 2009), typically assumes a static underlying causal graph. However, this assumption often proves inadequate in dynamic settings, where causal mechanisms evolve time and especially under the influence of active interventions. The challenge intensifies when interventions do not only influence variables within fixed causal structure but actively alter the causal relationships themselves. Relying traditional metrics like direct total or Average Treatment Effect (ATE; Pearl [2009]. Rubin 1980).
which rely the local consistency of interventional outcomes, be misleading.
To address this, recent formalizations
Meta-Causal Models (MCM; Willig et al. 2025 offer promising way to model and analyze the qualitative nature of dynamical shifts in cause-effect relations. Crucially, while traditional causal analysis focuses on measuring the quantitative outcome of intervention (e.g., average treatment effects), Meta-Causal Analysis (MCA) is also interested in capturing changes the underlying transition dynamics. This allows us to answer questions that beyond the scope of existing approaches, such "How likely is system adapt desired state?", "How stable is a desired state?" or "Which transition paths lead to a particular state?" and
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Reflect
Te 
Observe
3
Adept
Illustration of meta-causal adaptation in a judicial setting: (1) The quality of judge's ' decisions degrades toward the end of sessions. (2) The judge reflects on the reasons for this decline in performance, recognizing exhaustion and complex cases as causes of increasing bias. (3) Since the judge is (hypothetically) unable to directly intervene exhaustion levels and case complexity, the judge implements new meta- veta-strategy by prioritizing unclear cases earlier in the day. While the overall system dynamics remain the same, the biasing effects of fatigue are successfully mitigated and decision quality is maintained.
Figure 1: Nauaal Adaptation in Judicial Decision Making:
these paths admissible from safety ethical standpoint. To the best of our knowledge, these questions cannot be answered by existing approaches, as the explicit graphical modeling of qualitative (meta-causal) state transitions not covered and constitutes the main novelty of our paper.
Contributions. This paper introduces a specialized class of Direct MCMs, a specific assertive class of MCM that explicitly model meta-causal dynamics from within the SCM. Building this, we formalize the first concept Meta-CCasal Analysis (MCA), a framework for analyzing system transition between meta-causal states and proposing the Linearized Meta- Meta-Causal Dynamics (LMCD) algorithm to capture these dynamics. We demonstrate the critical utility of direct MCM and MCA through two examples: a medical treatment analysis showing how neglecting dynamies leads to suboptimal long- term outcomes, and a judicial decision-making setting where an agent uses meta-causal reflection to actively adapt its strategy and mitigate emerging bias.
2 Preliminaries
Causal Models. Causal relations are commonly formalized via Structural Causal Models (SCM;
Sppires et al., 2000 Pearl 2009)). ). An SCM is defined tuple M == (U, V, F, Pu), where U is the set of exogenous variables, V is the set of endogenous variables, F is the set of structural equations that determine the endogenous variables, and Po is the distribution of the exogenous variables U. An endogenous variable V, E V is determined by a structural equation v; := fa (Pa((.:)) that takes a set of parent values pa(x;), consisting of endogenous and exogenous variables that directly cause Vi, and outputs the value of ey. The set of all variables is denoted by X = UUV with values X and N = |]] Every SCM M is associated with (causal) graph = E). that is constructed by adding edges eij E X x X from the causal parents X; E Pa(X,) ) of each variable X, X to itself. The do-operator [Paal 2009) written as do X; = f;)) alters the causal model by replacing the structural equation fi E F of the variable X; with the constant assignment X; == 2.
Meta-Causal Models. Meta-causal models (MCM) concerned with modeling the qualitative change of cause-effect relations in causal models over time (Willig et al., 2025). Rather than considering specific structural equations, MCM model the qualitative causal type of relations between variables (e.g. 'reinforcing'. 'suppressing".... ...)) [Chockler and Halpern 2004 Wolff 2007 Sloman al. 2009 Walsh and Sloman, | Gerstenberg 2024). ). To account for the factors that lead to these qualitative changes, meta-causal models assume an underlying mediation process & = (S, defined as a Markov process [Bellman 1957] that governs the overall dynamics of the environment. The Markov process is defined over state space S with a transition function : S - S that moves the system forward in time. MCM are typically modeled by set of variables of interest X, extracted from the state E S of an underlying mediating process. This is done by some abstraction function sp : S + X, which can be freely defined as a summarization or (causal) abstraction function over the state space variables [Rubenstein et al., 2017 Beckers and Halpern] 2019 Anand et al.) 2022 Wahl ct al., 2023. Kekić et al., 2023) Willig et al. 2023]. The linkage between a Markov process and a causal model is formally captured via Meia-Causal Frames (Def. D Each meta-causal state (MCS;
Def. is defined as T NNN captures the current qualitative type of relationship between all pairs

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of causal variables at given environment state. Each element Łaj E T indicates the particular rype of relationship between any two variables Xy The special type 0 E T. indicates the complete absence of an edge. Thus, MCS are a generalization of a causal model's s adjacency matrix. Given mediation process, meta-causal frame emitting meta-causal states, MCM (Def. (3) model the change of functional dependencies for different states of the environment and thus capture the transition dynamics between different configurations of the causal graph. We briefly repeat the necessary definitions of MCM for further consideration in the following sections:
Definition (Meta-Causal Frame; MCF). For given mediation process & = () a meta-causal frame is a tuple F = (E, X, (T) Type-encoders Tzj :X; x X + assign a type 1 E T to the functional dependency of X; on X,, induced by the underlying process E. which a relation between X, (values of X;) and the abstraction of the transition franitiionfunncioon
o a E x {bb : - X3. The identification function I : 8 x X x X +T T with [(s, X4, Xx) t To (p(ss, (o0) assigns type to each pair of causal variables for each state of the environment.
Within a Meta-Causal Frame every system state 8 E S is mapped to a meta-causal state T E N that captures the current type of relations between all variables:
framee Definition 2 (Meta-Causal State; MCS). In a meta-causal frame F (E,X, (Ti)), a meta-causal
state is a matrix T E T NNN, For given environment state s E S, the actual meta-causal state T, has the entries T.,új == Iss, X)) = Ti)((((s), up o).
Finally, meta-causal model identifies the meta-causal dynamics from the current state of the system 8 E S and the state transitions of the Markov process a at every point in time. The dynamic changes in the mediation process are thus observed as qualitative changes in the meta-causal state:
Definition 3 (Meta-Causal Model; MCM). For a meta-causee frame F = ((S,o), (TT). a meta-causal model is a finite-state machine defined as tuple A = (T -,,), where the set of meta-causal states T X is the set of machine states, the set of environment states S is the input alphabet, and : TNNN x S + T NNN is transition function consistent with the environment transition and type encoders Tíg-
3 Predicting Meta-Causal Change
One of the main hurdles that has made it difficult to transfer decisions from general meta-causal models back the underlying Markov process has been the abstraction that relates the Markov process to the causal variables. In this section, we consider a particular class of meta-causal models that link the two models more tightly and, in turn, are more informative in terms of feedback. First, we introduce a class of MCM that directly considers the variables of the underlying mediation process as their causal variables. Second, we show that meta-causal state transitions become directly predictable from the causal model under certain choice of abstraction functions. In particular, we identify set of meta-cause causal variables that are responsible for these transitions. Finally, we discuss context dependencies as special case of meta-causal dynamics, relevant to the later Applications section.
Direct MCM. We consider a class of MCM where the variables of the underlying mediation process S and the causal model X are considered to model the same set of variables:
Definition 4 (Direct MCM). For a meta-causal frame F = () (Ti)), I) a meta-causal model A = NxN S. 8) is called direct meta-causal model if S = X and = ld.
Since meta-causal states are governed by the transition function of the mediating process a S S. the structural equations are now also defined in terms of the state transitions F == o a y Id o a o Id- = σ. In the context of meta-causal models (or dynamical systems in general). the mediating process a factorizes into two sets of equations governing either the causal variables within a given particular time step or meta-causal state X, + Yz, which can be denoted as --2 and those that transition the system to the next time step (or meta-causal state) X, -> Y141, --=t+i
3.1 Factors of Meta- Cautal Change
Previous work on meta-causal models either had no explicit notion of factors that caused changes in the MCM, or attributed these changes either to explicit interventions to unspecified (cxogenous)
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environmental factors [Minka and Winn 2008 Peters et al. 2016 Seitzer et al. 2021 Willig et al. 2025]. In this paper, we assume that the factors that have the ability to change the (meta-causal) type of relations are observed within the model, and thus can be identified as a subset of the variables. A key aspect in distinguishing ordinary variables from such 'meta-causal' variables is their ability to influence the qualitative type Tú of any relation within the model. The notion of what constitutes a meta-causal variable also includes the currently used identification function I : S x X x X + T, since it ultimately determines what constitutes a change in meta-causal types in the first place. Similar to how parents in the standard SCM are sometimes characterized by their ability to influence the values of their respective child variables, variables that have the potential to change the identified rype of one or more causal relations within the SCM. (We write X5 to mean the vector without the k--th entry.)
formalize meta-causal variables (MCV) as the set of
C = { Xx E X XX; X2 E X Xx,,' E X s.t.
(I)
(x x A (I # 2xk) A (I(x,X,,X,)
MCV are a subset of the standard variables (C S X). Most closely related to the notion of MCVs is the work Günther et al.
[2024). which is concerned with discovering context variables from data, but does not further model their associated dynamics. A notion of MCVs was also implicitly used in the examples of Willig al. 2025). Here we provide first explicit definition of MCV.
Predictability of Meta-Causal Dynamics. The ability to make predictions about causal dynamics from observations of a system is a core aspect of meta-causal models. Here, MCV specifically capture the relevant factors governing the dynamics of the system. While other exogenous factors may influence the course of meta-causal evolution, we are interested in systems that can be predicted from within the SCM. We formalize this form of predictability as follows:
Definition 5 (Meta-Causal Predictability). A Meta Causal Model (TNxN,S,A is called meta- causal predictable if the next meta-causal state T244 E TNN can be predicted purely from the variable values at the current time step X E X S, so that the transition function takes the form 8², x + + NNNN
State transitions may not need to be fully deterministic. The above definition can be reformulated into a probabilistic variant where ,² : X - P(TNXX) is probability measure over the meta-causa states TNNN Since transition probabilities can shift depending on the current meta cautal state, relaxed notion of meta- meta-causal predictability might reintroduce the current MCS T,, as a parameter, 8x X x NNN -> NxT) The predictability of a system can then be measured using Shannon entropy (H(P)= == = (=P))=-EneE> EpEP (pl:g(p:); Shannon [1488). ). The predictability of system then increases with increasing entropy, H(P(T xx*)) + 1. For strict meta-causal predictability (H(P(TNXX)) = 1). P(TNXN) must then be a point-mass distribution, with all meta-causal state transitions being deterministic and fully governed by the set of meta-causal variables C.
For arbitrary abstractions o this cannot be guaranteed, since relevant information about the factors responsible for state transitions may be marginalized by the abstraction. The choice for direct MCM (Def. 4 to set o as identity provides a particularly straightforward way to guarantee meta-causal predictability. In general, any abstraction function that preserves full information about the underlying transition factors (e.g.. yp being bijective w.r.t. C) is suitable for preserving meta-causal predictability.
Theorem 3.1. Every MCM (T XN ,8,6) with a bijective abstraction &p is meta-causal predictable.
Proof. Since yp is bijective, its inverse - exists and invertible with s = x -xx. The assignment F := yp a s- implies (P)(s(s), (porr) and +1+,, == Tıy (s)), poor) = TTy((pp Fo (x)) po (5- Fo k)) = To Tiy (1 o F(x)), Fo vp), which is function 8x,, X NNNN that is completely governed by the SCM's x and F.
--1 Fo s. By Def. 2 types are determined by Ta,ij ==
The bijectiveness condition of yp might be relaxed further. Implications are discussed in App. Meta-
causal predictability of direct MCM follows directly from Thm. 3.1 and the identity being bijective. Assuming bijective abstraction functions, the state transition S + S can be defined in terms of
causal variables x X X with x : &p o -1 (in particular x := id o o id = for
direct MCM). As result, the overall MCM transitions d T xN S + T can also be written

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in terms of causal variables 8² X - T NxN with 8 (x) == [(ox(x) Xx), = Tis(o (x),
(xx, F)
similar the original definition with a swapped.
Contextual Independencies and Discovery of MCM. As a final part of this section, we briefly discuss the discovery of MCVs of MCM for the special class of contextually switching MCM. From a meta-causal perspective, an important class of switching causal mechanisms is that of contextual independencies, where relations either exert unique type fiy while being active or are (contextually) independent otherwise, T.j E Uijz. 0}. Identifying particular context variables that lead to switching causal graphs has been the subject of several approaches in general causal discovery [Pensar et al. 2015 Hyttinen et al., 2018 Günther et al. 2024], reinforcement learning, and general prediction of system dynamics [Seitzer et al.) 2021 Liu et al. [023]. While some works dealing with switching
dynamics do not have an explicit notion of such context variables [Seitzer al. 2021 Liu et al.
2023]. several others deal with their discovery from data (but without modeling dynamics) under the
name of 'Labeled Directed Acyclic Graphs' (LDAGs: Pensar et al.] (2015), Hyttinen et al. (2018),
Günther et al. (2024). In particular, [Günther al. 2024] relaxes the previous assumption to discover contextual graphs from shared data. Under the previous assumption of meta-causal predictability and the discovery of observed endogenous context variables (or more general MCVs), the general dynamics of MCM state transitions can be recovered. In this work we are not concerned with discovery, but use the notion of contextual independency meta-causal phenomena in examples in Sec. 6 This concludes our formal considerations of the predictability of MCM dynamics.
4 Intervention Effects on Meta-Causal Stability
Causal modeling is commonly used to support domain understanding and subsequent decision making. With meta-causality heing the main focus of this paper, we consider scenarios where decisions induce lasting changes to the system that have the ability permanently alter the system dynamies. While the previous section discussed conditions for meta-causal predictability, we will now draw attention to the insights that can be gained from such models. Drawing qualitative inferences not only for individual decisions, but also for strategies that continuously affect system, is a key capability for making reliable and robust decisions [Bareinboim et al. 2021 Zhang and Bareinboim 2022 Aalaila et al. 2025). Discussions of the role of long- versus short-term outcomes [Lear and Zhang [2025] and the adaptation of reflective strategies have been discussed in previous work [Lee and Bareinboim 2018 Dasgupta et al. [2019] Bocken et al. 2024]. In this section, we focus on describing those qualitative changes in system dynamics with the help of meta-causal considerations.
To study the influence of one variable on another, the (direct) causal effect -as for example measured by the Average Treatment Effect (ATE; Pearl [2009]. |Rbbi|| (1980) > is usually approximated from a set of individual samples (æ,, (:) Within the potential outcomes framework (under the assumptions of exogeneity and ignorability), the ATE is defined as the expected difference in outcome that would result from treating an individual i (y) (f)) compared to not treating them (yo(1)): (i)):
ATE = Ely1 yol Rz Σ, y1 (1) (i)
(2)
In the following discussion we make use of the do-operator do -where a E {0, 13 indicates the presence absence of a treatment, respectively- to distinguish the interventional treatment setting from simple conditioning, which may be susceptible to spurious confounding:
ATE = EEy do(x=1) do(X=0) 00
do(X.=1) (o((x=-)
y
y/4
(3)
Since we base our formalism in the Pearlian causal framework, the Pearlian do-calculus states a set of exact and complete conditions under which causal effects can be identified for arbitrary graphical structures and beyond the bivariate case [Pearl 1994, Tian and Pearl, 2002 Shpitser and Pearl 2006 Huang and Valtorta 2006). For ease of notation, and since variable adjustment is not the focus of this paper, we only consider cases where the above Eq. 3 directly captures unbiased estimates of the causal effects to be estimated from X on Y without further consideration of possible adjustment sets.
Stability of Meta-Causal Dynamics. These considerations of causal effects are made under various assumptions, such as the Stable Unit Treatment Value Assumption [Rubin| 1974 1980). which assumes that the treatment of one participant generally has no effect on the intervention outcome of others. Furthermore, most systems are assumed to be well-behaved in the sense that their overall local
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dynamics remain stable regardless of the intervention being studied [Kevorkian 1966 Kevorkian and Cole, 1968]. Assumptions on the stationarity of causal dynamics can be conveniently formalized from meta-causal perspective. One way to express the local stability of the system under perturbation is to assume that the system always remains in its current meta-causal state, regardless of the type of intervention applied. Assuming meta-causal predictability (A (Assm. 5). we take advantage of the fact that the MCS can be inferred from the causal variables x E X. and write Tx and Tx|do(y) to denote the meta- states arising from x and from x under the interventions do(y). respectively. In its most conservative form, strict meta- veta--taability is formalized as follows:
Vx, x' E X My S x. Txjdooy) = Tx
(4)
Under the criterion of Eq. 4 the meta-causal state must not be altered at any point in time, including points of intervention. However, this strict notion may preclude the common use of hard interventions. As hard interventions cut the edges from the intervened variable X, to its parents, it in turn changes the meta-causal types of these edges from their previous values to zero-type (0). This, alters the meta-causal state Txjdoly) * Tx so that these actions are not allowed. Under the strict notion of Eq 4 either soft interventions or interventions on root nodes (e.g., instrumental variables) are required, which either do not alter the meta-causal type or do not cut any edges. Depending on the use case, a weaker criterion may be sufficient, requiring only that the system converges back to its initial state some time after the intervention. Superscripts indicate the meta-cause state at particular points in time, as well as the time of the intervention:
Vx,x x' E My C X',31, A E N+ (1<( A -T> =T +()
(5)
The relaxed notation of Eq. 5 makes no statement about the MCS during the time of intervention, but only requires that the system converges back to its initial state after some time At. This allows in particular the use of hard interventions, as long as their effects are reversed after some time.
Stability of Stationary SCM. Stationary SCM without time-dependent system dynamics (a" 0) trivially satisfy Eq. 5. Since each x' depends only on its current (independently sampled) noise time E N+, so does T x- Any effects of an intervention (y) vanish at the end of the intervention, since information is carried over from o g-tt+1 As a result, At can be set to At == 1, (assuming no subsequent interventions change the MCS T at the following timestep). The unsatisfiability of Eqs. [5 serve as indicators that meta-causal dynamics might have been permanently altered following some interventions. Qualitative changes in mechanisms do not always come sudden, abrupt changes, but can be the consequence of a series of repeated interventions that steadily affect the system's 's dynamics until some tipping point is reached (recognized as change in T via the meta- causal identification function Z). To avoid unwanted outcomes, it is of primary importance to consider not only the short-term effects of some actions but also the qualitative long-term consequences of shifted dynamics. In the following section we propose a way of analyzing such meta-causal dynamics.
5 Meta-Causal Analyses
Meta-Causal Analysis (MCA) focuses on capturing changes in the underlying meta-causal state transition dynamics, offering a perspective distinct from existing analysis methods, such as Dynamic Treatment Regimes (DTR) or longitudinal statisties, which primarily measure the eventually emitted outcome effects of interventions. Questions amenable to MCA include determining the likelihood of a system adapting a desired meta-causal state, assessing the stability of a state, or identifying the transition pathways (sequences of meta-causal states) available to reach particular state. As transition dynamics eventually affect actual long term outcomes, MCA is intended to supplement existing methods by explicitly modeling the flow between system configurations with the help of graphical causal models, rather than only observing resulting outcomes.
Linearized Meta-Causal Dynamics Algorithm. To facilitate capturing meta-causal effects, we propose the Linearized Meta-Causal Dynamics algorithm (LMCD; Alg. []. which captures the linearized dynamics -meaning, that no second-order effects, e.g., shifts in time, on the observed transition probabilities are assumed to be present- by approximating transition probabilities between meta-causal states from a sample population _x...., , x= E x N E N and a meta-causal pre- dictable model. Under the assumption of meta-causal predictability (Assm. the structural equations are sufficient to advance the system onto the next state (Thm. B.Tp. To capture the true transition probabilities, we furthermore assume that the presented data is sampled identically to the underlying

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Algorithm Linearized Meta-Causal Dynamics (LMCD) Algorithm
1: Input: SCM: = (V. U, Pu), data: x = (x E X id. func.: I: X - T
2: for each x² in x do
3: xf.+1 F((x u Pu))
Advance the system.
4: (T" Tr
(I(x)), (I(x), I(x+(t])
Identify MCS transition pair.
5: U+ JU, ((T
> Determine set of unique MCS.
6: for each (u, v) in f..... 10,,,
D Approximate transition dynamics, P E ERRR [ 7: Pape (1 = u) A (I(T(++)) = v))) EiE[,.,.] (0(p++==))
8: IQ+e" 6--1 "
> Optional: Compute continuous time rate matrix. (1 is the identity matrix.)
9: return [Q]
MCS distribution. Instead of advancing single data points, pairs of consecutive data points could directly be considered, if available. This would relate the LMCD method more closely to classical time-series analysis and eliminates the meta-causal predictability assumption.
In first step, all given sample points advanced in time (indicated by 't 1'). Next, the set of actually reachable MCS is computed. For practical purposes, an indexing function : - [[] introduced that assigns unique index each MCS t E T. This function to group different by only considering a problem relevant subset T C T of the full type matrix when assigning the index. Finally, transition probabilities P E |||| are computed. (1(b) the indicator function which is 1 if is true and 0 otherwise.) A time rate matrix Q can be determined for continuous time setups.
equally be used
Meta-Causs ATE. The (specific) Meta-Causs Causal ATE (sMCATE) between two strategies A, B with transition matrices PA. PB (or optionally QA. QB) might be defined the difference in their transition probabilities SMCATE((A, P3) == P3- - PA. Note, that PA, PB already approximate the transition expectations over the respective populations. In line with the discussions in Sec. assumed that transition matrices PA, PB are able to capture the underlying system dynamics and that those do not shift due higher-order effects. The sMCATE might further be compressed into single number via the use of matrix norms for specific scenarios. In the absence of clear candidate for defining a general MCATE we abstain from deciding on a definite characterization.
6 Applications
After addressing the formal aspects of predictability and stability, we now turn to illustrative use cases of meta-causality. Sec. [6.1 compares static and dynamic ATE analyses in a flu medication example, highlighting their differing implications from meta-causal perspective. Then, Sec. 6.2 examines judicial decision-making where agent fatigue introduces bias. Here, a meta-aware agent recognize and counteract such biases through meta- meta-level intervention. Code for reproducing examples is provided at: https://githu: cww. wiloortttillgg/etttaaaaaalllllll
For the examples, the structural equations within the SCM do not switch. To identify whether parents have an actual influence on their children, we use an identification funetion that determines the presence of an edge by inspecting whether the connection structural equations have non-zero gradient. We refer to App. B for more details.
6.1 Medicating Flu
In this example, we revisit the discussion short- versus long-term effects [Lear and Zhang 2025 with particular focus the stability of meta-causal states. The example considers the effects of different drugs used to treat fever. We consider strongly simplified system with exaggerated system dynamics for clarity of the example. Similarly, deem both medications sufficient for treating all levels of fever, such that no trade offs between short- and long term-term outcomes are considered here.
At each time step, participants experience varying levels of viral exposure (F, E R+), Fever occurs when the viral load exceeds the immune system's 's current capacity F, == max F4 - T,, 0) and drugs actively given in consequence, M, := 1(F, > 0.5). Drugs Mg. k {A.B) are characterized
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Simulation: Medication
0
Medic
No Med.
Exposure
Immure Strength
-
Fever
Medication
Simulation: Medication
Medic No Med.
Temperature Average Treatment Effect
-
Medication
Medication
1
2
10
11
12
14
15
16
17
18
Figure 2: Medicating Flu. The top and center plots show the averaged simulated evolution of the system under different drugs. Shaded areas show standard deviations. Medication provides a consistently stronger direct reduction in body temperature according to the ATE shown in the bottom plot. However, it also suppresses immune development to the point where each subsequent viral exposure requires treatment. Medication B has weaker direct ATE, but does not inhibit the immunc system, resulting in a significant long-term reduction fever.
by their parameters
)) which differ in suppressing fever symptoms (O.M,) ) of rising body temperature (B, := Fg. (1-11-x0 oM, M,)), but could also impede the development of the immune system that would otherwise develop naturally over time (I2 := I + sigm(t) 3M, M,-1; where sigm is sigmoid function representing immune maturation). The corresponding causal graph is shown Fig. Full equations and starting conditions are given in App. C We consider two drugs, with arbitrary chosen parameters: Drug A has high immediate suppressive effect on fever symptoms, but also a stronger negative effect on the immune system, MA = {a = 0.95; B = 0.75). Drug B has a milder fever response, but also milder effect the immune system, MB {a 0.6; $3 0.4).
We performed simulations over 1,000 repetitions (details given in App. G. Fig. (top and center) shows the average evolution of the system under drugs A and B over time. To assess the efficacy of ei- ther drug, conventional causal analysis might measure the 'direct' ATE in terms of the immediate reduction in body temperature. We analyze how the different medications affect the qualitative system dy- namics. The result is summarized in the bottom plot Fig. 2 as the average difference in body temperature between treating and not treat- ing fever with each drug. ATE(M,) == E[B (,oo(MM)) As can be clearly seen from the figure, drug A is more effective at lowering the temperature over all time steps (with an overall average standard deviation of ATEMA = - O.34%)) than drug (ATEMs 1.26 C 0.78% C). An analysis based on this quantity alone would therefore favor drug A as the superior treatment.
Exposure
Immune Str.
Fever
8
Medication |
Body Temp.
Medicating Flu: Causal graph for viral exposure and im- mune response under the influence of medication. Solid lines indicate in- stantancous relationships; dashed lines indicate rela- tionships between succes- sive time steps.
Figure 3:
From a Meta-Causal Analysis (MCA) perspective, drug A's cumulative negative effect on the immune system is evident by analyzing the tran- sition dynamics between meta-causal states (MCS). Focusing the Fever - Temperature and Medication - Temperature edges via the indexing function of the LMCD algorithm (Alg. D transition probabil- ities between the three MCS "no fever (no treatment)", "mild fever (no treatment)", and "fever with medication" -there no treatment without fever- are approximated. Applying LMCD to the simulation data at age 18 shows that drug A leads to high probability (99.39%) remaining in the dediiatiionrequiire MCS, with all other states having a high probability (up to 77.78%) of flowing into this state, indicating drug B's dynamics are more favorable, exhibiting a lower chance for recovery-related state transitions and reduced probability of remaining in the medication-required MCS (61. .22%). The sMCATE quantifies this difference, showing that drug strongly decreases the likelihood of transitioning into (reduction up to 36. 36.43pp) remaining in the dependency- -induccnny-nniingggggge state 18pp. This analysis
induced dependency on medication. In contrast,
8

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suggests preferring drug B (if sufficient for treatment). A fully worked example, showing transition matrices and sMCATT computations is provided in App.
6.2 Judicial Decision-Making
In this example we present the case of meta-aware agent that actively reflects on the system dynamics and is therefore able to adapt its strategy to counteract emerging biases. Similar considerations of algorithmic recourse and dynamic treatment Zhang and Bareinboim (2019), Zhang
Case Pool
Schedule
[2020] , Gerstenberg 2022), ,. Von Kügelgen al. [2022] have already been considered for classical causal cases. Here, we put our focus on the meta-
Case Complexity
causal aspects which include the agent actively self-intervening on its own policy function when becoming aware of the arising bias.
Fatigue
Decision Bias
An overview of the scenario is illustrated in Fig. T with the corresponding causal graph presented in Fig. Full equations and starting conditions are given in App. D In this scenario, a judge hears N 6 cases per day, drawn from a given pool of daily cases Po = {cI..... CN }. where each case is assigned complexity, C {1. 2, 3, 4]. that is proxy for the amount of documents to read or the criticality of the case (e.g., the maximum sentence expect). For each time slot, the judge selects one of the cases according to their schedule S [R+ + [1..N] with 8t == f(re), from which the current case complexity cr; == ps, is determined and which can't be heard again afterwards, Pt == Pt-1 {se). Over the course of the day, the judge gradually becomes fatigued, Sa =- fi- + 0.5, which affects the decisions made, so that when the judge encounters complex cases while fatigued, the decisions become biased b, := mar(f. -
Figure 4: Judicial Decision-Making: The corresponding SCM for the Judicial Decision- Making setting. Solid lines indicate instanta- neous relations; dashed lines indicate relations between consequent time steps.
Reflecting on past decisions, the judge notices an emerging fatigue-induce bias when hearing complex cases. Identifying fatigue and case complexity as the direct causes of the bias, the judge considers ways to counteract its effects. Considering actual edge activations via the previously described identification function (App. B). a suitable strategy has to avoid any edge activations of fatigue- + decision bias and case complexity + decision bias. Since the emergence of decision bias not under the direct control of the judge, the only way avoid bias is steer the system dynamics into a state T where, at any point in time, neither Fr + B, nor Cr. - B, exert causal influence; T ((T F B, 0) V (Te.,... B, = 0)). Through the additive, thresholded interplay of fatigue and case complexity, both factors act as meta meta-causal variables to each other, in the sense that low fatigue prevents case complexity from exerting influence on the occurrence of bias, and vice versa. Fatigue levels are assumed to be unintervenable and steadily rising, such that during later hearings, the edges have the potential to become active and the meta meta-causal type solely dependent on case complexity. Conversely, complex cases that might lead to biased decisions might be best heard in earlier sessions when the judge's 's concentration is high (implying low fatigue). As a result of this, the judge adopts a new scheduling strategy do(S := f*) with (p) argmax, ( (p: hree)). which schedules complex cases during periods low fatigue, thus preventing them from biasing decisions in later hearings. Under the new strategy, the system dynamics remain within the desired state T*
The previous example shows how meta-causal reasoning can prevent fatigue-induced bias by reflecting on system dynamics. While similar insights may arise from earlier methods, meta- meta-causality defines explicit conditions for edge activation, enabling logical strategy adaptation. A full comparison of initial and adjusted policy dynamics is provided in App.
7 Related Work
Longitudinal Statistics. Works by Robins 1986 1997] on the g-formula aim to estimate exposure
effects in the presence of time-varying confounders. Most similar to our work, RRobin et al.] [2000) leverages marginal structural models for causal effect estimation under time-dependent confounding
Liang and Zeger 1986) and Laird and Ware [1982] utilize linearized models and random-effect
sampling to approximate the effects of long-term dynamic shifts in the resulting distributions and
corresponding ATE contrast to prior work, which primarily focuses on measuring the actual
emitted effects, e.g., in dynamic treatment regimes (Murphy 2003), MCA is designed to supplement

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existing methods by explicitly considering transition dynamics and pathways between different qualitative meta-causal states. It therefore not only interested in the resulting outcome effects, but also how the resulting effects emerge within the system.
Time Series and System Dynamics. The study of dynamical systems in causality is long- nnggsnnnn
field of study [Friston et al., 2003 Mooij et al. 2013 Bongers et al. 2018 Blom et al. 2020 Peters
al. 2022 Löwe et al. al.]|]] [202]] Lear and Zhang 2025 with common application in the modeling of
climate systems [scheisccllrr et al. | Camps- Valls et al. [2023] Runge ct al., 2023 or general
time-lagged relations [Peters al. 2013 Saggioro et al. [2000] Runge et al.] 2019 Gerhardus et al.
2023 Runge et al. [023]. The general modeling and unsupervised learning of such systems under
al. 2013 Hansen and Sokol 2014 [hallppa et al. 2 Rubenstein et al. 2016)] [ongers et al. [021 Peters
causal aspects has also been widely studied [Dash] 2005 Hyttinen al. [012 Mooij |
jt alla||]222 2022] [Blom and Mooij, 2023]. In contrast to our work, most prior work either consider models
with stationary equations or does not explicitly model factors that cause changing structural equations.
Switching Causal Mechanisms. Several papers explore systems with changing dynamics, either
with an explicit notion of switching causal factors (Minka and Winn 2008 Peters et al. [0166] Willig
al. 2025] without (Chalupka et al. 2016 Seitzer et al. [202] IJ Liu al. 2023]. The direct MCMs proposed here distinguish themselves from these by allowing for the explicit and predictable control
of such qualitative changes from within the SCM, similar to the 'mechanized SCM' of Kenton et al.)
[2023) which aimed to model the influence of agentic agents over structural equations.
Algorithmic Recourse and Fairness. Our judicial decision-making example (Sec. 6.2 highlights the importance of reflecting on system dynamics in order to remedy situations that lead to unfair outcomes. This relates to work on faimess [Kusner al.]
2017 Zhann and Bareinboim 2018
Von Kügelgen et al. 2022 Plecko and Bareinboim| 2023
2024], treatment-confondder feedback
in causal reinforcement learning [Bareinboim al. 2015 Lu et al. 2018 Buesing et al. 2018
Zhang 2020 Weichwald et al. [022] and algorithmic recourse [Zhang and Bareinboim [022]] Karimi
al.] [022]] Von Kügelgen
2022).. Our contribution to this space is providing a meta-causal
framework for agents to not only optimize their actions with respect to some external metric, but be
able to reflect and reason based on the
identification of edge activations.
8 Conclusion
This work addresses the critical shortcomings of classical causal analysis, which may rely on static assumptions, particularly environments where interventions may alter the underlying causal dynamics of the system. We introduced specialized class of meta-causal models (MCMs) designed to explicitly model and predict changes in evolving causal dynamics. formalized meta-causal variables -variables that govern the system's 's meta-causal state and are observable from within the SCM- and established conditions for meta-causal predictability. We presented a meta-causal analysis method, which, in contrast to prior work, also considers the intermediate meta-causal state dynamics which cause the eventually observed outcomes.
Broader Impact. We demonstrated the value of meta-causal decision-making through examples in medical treatment (Sec.]6.) and judicial settings 6.22. Beyond these cases, meta-causal analysis helps address the risks of ignoring system dynamics, enabling more sustainable and fair outcomes. The examples show how MCMs anticipate and adapt to shifting causal relationships, moving beyond static analyses.
Limitations and Future Work. The framework for meta-causal predictability, particularly the concept of Direct MCMs, relies on the strong assumption of full observability of meta-causal variables and the preservation of all information about transition factors via the abstraction functions. While concerns about the assertiveness of causal abstractions are common to the general field of causal representation learning [Rubenstein et al., 2017, Schölkopf et al., 202 Kekić et al., 2023 Talon et al., 2024), complete observability may not be feasible in many real-world scenarios, and the factors driving meta-causal shifts might be latent or only partially captured. Further work on relaxing theoretical assumptions and empirical validation in wider range of real-world settings may be necessary to achieve robustness of the proposed framework. Although the presented applications demonstrate the usefulness of MCM-based strategies in specific examples within confined simulated environments, transferring the proposed concepts to general meta-aware agents may be a promising avenue to explore in the future.
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Acknowledgments and Disclosure of Funding
The authors acknowledge the support of the German Science Foundation (DFG) research grant "Tractable Neuro-Causal Models" (KE 1686/8-1). This work was funded by the European Union (Grant Agreement no. 101 120763 TANGO). 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. This work has benefited from the early stages of the fundings by the German Research Foundation (DFG) under Germany's 's Excellence Strategy- "Reasonable Al" (EXC-3057) and "The Adaptive Mind" (EXC-3066); funding will begin in 2026. The Eindhoven University of Technology authors received support from their Department of Mathematics and Computer Science and the Eindhoven Artificial Intelligence Systems Institute.
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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: Shortcomings of static ATE are theoretically analyzed in Section and practically demonstrated in Section 6.1. Theoretical advancements on Direct MCMs and assertive meta-causal factors are developed in Secs. and 3.1 (c.f. Def. 4, Eq (1), Def. and Thm 3.1). Discussions on meta-causal stability presented in Sec. 4 (c.f. Eqs (4) and (5)). method for performing Meta-Causal Analyses and computing a Meta-Causal ATE presented in Section 5.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize other settings.
is fine include aspirational goals as motivation long is clear that these goals are not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: Limitations are discussed within Section and concern the made assumptions on the form of the causal abstractions within direct MCM as well as the feasibility of observing full MCV information under real-world settings.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors encouraged to create separate "I "Limitations" section in their paper.
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The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't acknowledged the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
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3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
The proof of Theorem 3.1 directly follows in the main text. The assumption on the bijectiveness of the abstraction function stated within the theorem. Relaxations of the bijectiveness assumption for the required causal abstractions is furthermore given in Appendix A.
Justification:
Guidelines:
The answer NA means that the paper does not include theoretical results. All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should properly referenced.
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Answer: [Yes]
Justification: All structural equations used for the examples in sections 6.1 and 6.2 are provided in appendices C and D. Fully worked examples of a step by step meta-causal analysis on scenarios are given in appendices E and F. Information about seeding an the number of repetitions stated. Code reproduce all reported results is provided.
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the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
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(d)
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When Causal Dynamics Matter: Adapting Causal Strategies through Meta-Aware Interventions (Appendix)
The appendix is structured as follows. App. A describes the implications of bon-bijectiveness on the prediction of future meta-causal states. App. B provides description of the particular identification
function used in the examples of this paper. App. C gives details on the SCM, starting conditions and simulation of the Medicating Flu Example. App. App.[idetaill the SCM for the Judicial Decision-Makinn
example. Similarly, App. E and App. F provide fully worked examples of applying meta-causal analysis to the two example scenarios of the main paper.
A Implications of a Non-Bijective Causal Abstraction
Works on causal abstraction often require the abstraction function jo to be surjective Rubenstein et al. [217)) Beckers and Halpern (019), Willig et al.] (2023). The more information about the initial process state is marginalized by yo, the less information about the original underlying process state can be recovered again from observations of the causal variables x. A non-unique invertability of does however not restrict the expressibility of MCM, but generalizes their prediction from a unique process state for a given variable configuration x to the prediction of a distribution of possible states ((s) that all could have lead to the observed x; - X - P(s). Unobserved or exogenous variables can equally be modeled, as variables which are marginalized by the abstraction function, thus being unavailable and inducing uncertainty when inferring the underlying system state via yp
Similar, to how monndeterminism of the state transitions changes the output of the transition function to a probability distribution 8x X' + P(TNxX) instead of predicting a unique TNxN (c.f. Sec. 3.1. non-uniquely invertible yp induced uncertainty over the function's input, such that 8* changes from deterministic input 5* X P(T to 8 P(x) + P(TNXX),
Within the proof of Thm 3.1 structural equations are given as F := y 0o For the LMCD algo- rithm (Alg. D this means that observed data point x' can no longer be advanced deterministically to it next state xi++1 as the final yp component that F is composed of, now outputs a distribution of possible structural equations Fr..ss a particular MCS T, according to some previously recorded probability distribution, Fúj P(Fr,..)) ). For advancing particular data sample in line of the LMCD algorithm, either particular instantiation of structural equations F' ~ F is (possibly repeat- edly) sampled to approximate effects of the uncertain transition probabilities, or in case of a finite number of possible structural equations the sample is pushed-forward through F and the algorithm is continued with the resulting distribution.
B Choice of Identification Function in the Examples
The examples presented in this paper use an identification function that determines the presence of an edge by considering whether or not a non-zero gradient is present between any parent and child pair connected via structural equation. The function therefore only emits the types 'effect exists' (1) or 'no effect' (0), For all other variable pairs which are not connected in the causal graph the 'no effect' type is always emitted. For any two continuous variables X;, X, E X connected via direct causal edge, X, + X,, we use the 'contextual independency' function, defined as:
d.x: ClId X1, X,,x) = dx," (x) 0)
The function is true if the gradient of X, onto X; is non-zero, which is equal to X, being contextually dependent on Xj, given the subset of parent values of X; for the current variable configuration x. Otherwise, it is false, indicating that current causal influence of Xj on X; present. Furthermore. a corresponding discrete version can be defined as:
CIID disc (X, == (3a' E Xx,(ff(xxxxxx U2/) f((x))
where x sxx Xx are the values of x without zz.
23

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C Details on the Medicating Flu Example
This section provides further details on the simulations discussed Sec. 6.c. and shown in Fig. 2 All discussed results are averaged values from 1,000 independent roll-outs of the described system. Noise for every roll-out is individually seeded and sampled independently. Code to reproduce the experiments can be found in the supplementary material. For simulations the following exact structural equations where evaluated for 10 discrete time series:
Exposure, := 0.8 Binomiaa 10, - 0.5)
(8)
ImmuneStrength, == ImmuneStrengthh.-11 Normal μr 0, = ( (9)
Fever, max (Exposure, IImmuneStrennthh 0)
(10)
if Fever, > 0.5
Medication, == 0 otherwise
(11)
BodyTemperatureAA = Fever, ((1 ax) Medication,)
(12)
The starting values for all variables, except ImmuneStrengtth before advancing to the first timestep set to 0.0 (and 2.0 for ImmuneStrength). Timesteps start and are evaluated over 10 time steps, with each timestep representing a two year span and the final step t = 9 ending at age 18. All other values follow from these initial values with conjunction with randomly sampled exposure levels at every timesteps.
Compute Resources. Simulations and visualizations run in under 5 seconds on PC with AMD Ryzen Threadripper 1900X 8-Core Processor and 32GB of RAM.
D Details the Judicial Decision-Making Example
This section contains the accompanying set of structural equations for the visualization in Fig.
Sec. 6.2 and Fig.
CasePool, == CasePool \
(CasePool, (CasePool--SScheetleee---l
(13)
Schedule, initial = 0
(14)
Schedule, sdapted == aargmax(CasePool)
(15)
CaseComplexity, CasePool, [Schedulez]
(16)
Fatigue, == Fatigue, + 0.5
(17)
DecisionBias, = max(Fatigue, + CaseComplexity, 5)
(18)
Time is evaluated from [0..5]. All values, except for the case pool, are assigned value before the first timestep. The pool at timestep () is set random permutation over a fixed set of cases:
CasePoolp = permute (3, 2,
(19)
Fig. in the main paper, shows the roll-out of the scenario with unpermuted values.
E Worked example: Medicating Flu
In this section we provide fully worked example for applying MCA for the medicating flu scenario of www.MoritzWiila.
6ec.|6.1] Code for reproducing all examples is provided at: htttps://gite DetaCausaalyyaaa. of edges we utilize the Clld identification function described App. As result, we identify effects via the following influence functions:
We start off with the SCM described in App. C For identifying the activation
24

---

ExposureInf; 0
(20)
Time;: 1
ImmuneStrengthinf,
Imm.Str,,1:: 1
(21)
Medication Medication #
Exposure,: Exposure, > Itt.
FeverInf;
(22)
ImmuneStry: (Exposure, 0) (Imm.Str. 0)
MedicationInf; :=
Fevery: Fever, > 0.5
(23)
Fevery: Fever, > 0
BodyyTemperatureAAlff, Medication:; (Fever, > 0) A (Medication, 0)
(24)
The functions determine the influence of the respective parents onto the variables. The entry ImmuneStrengghlnnf, == [Imm.Str..-1 1}. for example, indicates that the edge from immune strength in the last timestep onto immune strength in the current timestep always active. All remaining edges not identified by any of the above influence functions are identified as 0.
Defining the MCM. The example models direct MCM (Def. with y Id, such that the mediation process directly becomes the SCM. With variables X = (Time, Exposure, Immune Fever, Medication, Seddyyeeperature1/ } and structural equations given in App. C the mediating process The Meta-Causal frame is then defined as
is: & = (X, Fgs
F = (E, X, Tgq. 20
(25)
and X,, X,) + 6 = (p))), x a) Túj (x, F) according to Def. 1
The identification
function identifies the pure presence or absence of edges Taks E {0, 13. such that a meta-causal state given
E T {0, 1) 10 12x12
(26)
Note that the MCS has double the entries there variables in single timestep, since effects identified between the current, but also to variables of the previous timestep. Finally, we define the Meta-Causal Model:
A = X, (40, 1J 12x12 11 X,F)
(27)
E.1 Meta-Causse Analysis.
We apply the LMCD algorithm (Alg. T under the previously defined MCM. The same 1000 roll-outs for both medications A and B as in Sec.|6.1 [.r sampled from the SCM. Environment states at the final timestep (L = then selected for further analysis, x (xf.==9)100) )
Identifying all meta-causal states -the unique sets of active edges a sample according to the above influence functions- of all samples x' E x" yields 10 unique MCS for drug A and 8 unique MCS for drug B. For this analysis, we focus on the Fever - Temperature and Medication - Temperature edges within the meta-causa states of T := I(x*) and Tr, = I(x ). This is realized via the indexing function within the LMCD algorithm.
: T + [ Fover. Temperrrmmr T Mediication. TempperaonnTmmmeuuuuue
(28)
The resulting vector can take the four distinct states [0. 0] "no fever (no treatment)", [1, 0] "mild fever (no treatment)". [1, 1] "fever with medication" and [0, 1] "treatment without fever". As there never treatment without fever in the data, the LMCD algorithm identifies the following set of unique MCS:
= { 0j, [1, [1,
(29)
Counting the occurrence of individual meta-causal states at timestep / 9 ("age 18") obtains the following MCS counts:
10
437
Ct=
:
CL=
104
(30)
[981
459
25

---

Medication A
Medication B
SMCATE(P, P.. Ps)
9,16
Faver No Med.j
Fever Noo Med.)
Fevor
Noo Med.
9.09 0.32
0.09 0.21
6.11
042 0.13
0.52
-0.6 0.32
Med.
Med.
Med.
0.29
8.26
026
0.10
0.39
-0.31
Fever (M.d.))
Fever (Med.)
Fever (Mod.)
Figure 5: Medicating Flu Transitions. Transition probabilities between different meta-causal states, visualized as Markov processes for medication A, medication and the resulting s.MCATE(PA. PB) of the Medicating Flu example in Sec. 6.1
After advancing the individual samples in time x F((x*
U ~ Po)), and computing
its MCS Ti.+1 == -(xt++1), we obtain the following absolute transition counts:
3
0 7
-227
40
170
CA
=
1
Cn
33
28
43
(31)
0
6
975]
120
58
281
The per-row normalized transition matrices then model the transition probabilities at time
33.00%
0.00%
70.00%
PA
11.11% %
11.11%
77.78%
(32)
0.00%
0.61%
99.39%
51.95%
9.15%
38.90%
PB
31.73%
26.92%
41.35%
(33)
26.14%
12.64%
61.22%
Optionally, the continuous-timg rate matrices are computed as Q = P-1. (For this discrete time scenario P might be sufficient):
T0.496
0.001
0.502
Q
0.050
0.412
0.537
(34)
00.000
0.004
0.995
70.662
0.066
0.271
QB
0.211
0.506
0.282
(35)
0..18
0.082
0.731
Finally, the specific Meta-Causal ATE expresses the change in transition probabilities effect between both medications and B:
21.94%
9.15%
-31.09%
SMCATE(PA, PB) = PB - PA 20.61%
15.81%
-36.43%
(36)
26.14%
12.02%
-38.16%
The obtained transition probabilities PA, Pa and their difference, sM Eatsee Pn), can be visualized as Markov processes. The corresponding graphs are shown in Fig.
Interpretation. From the C matrices we see that the third state, indicating fever induced medication. is almost always active for drug A (98.1%) 1%), while it is only given in 45.9% of the cases for drug B. Considering the meta-causal transition matrices PA. PB. we do not only find that the MCS of fever induced medication is increased in A, but also that all other states eventually flow into this state (with probabilities of 70.0% and 77.78%). Drug B therefore features more favorable dynamics. Here, patients with a fever induced medication are still rather likely to to stay in that state. However, (moderately) healthy patients only transition into this state with moderate probabilities of 38.9% and 41.35% and are similarly able to leave again from it. These differences in dynamics are also visible from the sMCATE, which shows sharp decrease transition probabilities into the third state, while the healthy states observes strong increase in incoming transition probabilities.
26

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F Worked example: Judicial Decision Making
In this section provide fully worked example for applying MCA for the judicial decision making scenario of Sec. 6.2 Code for reproducing all examples is provided at: com/MoritzWill:
aaaetaaaaaaaaaamic.. We start off with the SCM described in App. [ For
identifying the activation of edges we utilize the Clld identification function, as before, described in App. B As result, identify effects via the following influence functions:
CasePoolInf; CasePoolk-1
(37)
Schedule,-1
ScheduleInff = 0
(38)
ScheduleInf, dddppedd (CasePool;: 1}
(39)
CasePool;; 1]
CaseComplexitylnf, Schedule,: 1
(40)
FatigueInf, (Fatigue;-: 1}
(41)
Fatigue,: (Fatigue, CaseComp.; 5) (Fatigue, * 0)
DecisionBiasInf; CaseComp.: (Fatigue, + CaseComp,, 2 5) A (CaseComp.r + 0)
(42)
The functions determine the influence of the respective parents onto the variables. The entry CasePoolInf; : [CasePool,-1 1}. for example, indicates that the edge from the case pool in the last timestep onto the case pool in the current timestep is always active. All remaining edges identified by any of the above influence functions are identified as 0.
Defining the MCM. The example models direct MCM (Def. with y = Id, such that the mediation process directly becomes the SCM. With variables X = {CasePool, Schedule, CaseComplexity, Fatigue, DecisionBias) and structural equations given in App. C the mediating process is: & = (X, Eq.. The PetaaCausal frame then defined as
F= (E,X, Tuq 37
(43)
and I(s, X,, + tt == (t)(pss), p Tay (x. F) according Def. |
The identification
function identifies the pure presence or absence of edges Taks E {0, 1}, such that a meta-causal state is given
TE X (0, 1) 1
(44)
Note that the MCS has double the entries as there variables in single timestep, since effects are identified between the current, but also to variables of the previous timestep. Finally, we define the Meta-Causal Model:
A A= (T X. (10, 1] 10x10 X,F)
(45)
F1 Meta-Causal Analysis.
We apply the LMCD algorithm (Alg. under the previously defined MCM. Roll-outs for all 720 initial case pool permutations are sampled from the SCM for both, the cager and reflected, policy. States transition across all timesteps and roll-out considered for further analysis, x! 2580
Identifying all meta-causal states the unique sets of active edges sample according to the above influence functions-, of all samples x E x via T : I(x*) and 7r,+11 := I(z(++1), observes 2 unique MCS for the eager policy. Either the 'unbiased" state with no influence of fatique or case
complexity on the decision bias, the "biased" state where both parents exert influence on the decision bias variable. The reflected policy always stays within the 'unbiased" MCS. We write (0] and [1] to identify the unbiased and biased MCS, instead of writing down the whole {0, 1) 1010010 type matrices in the following:
U [1]}
(46)
Counting the occurrence of individual meta-causal states, advancing the individual samples in time xi.+11 F((x' /v) U (u Pu)), and computing their MCS ri,++1 I(x)++11 obtains the following absolute transition counts:
504
936
Canger
Creftcteed =
(47)
576
864
2000
27

---

Eager
Reflected
Paggen Preftectea)
0.35
0.0
0035
Unbiased
0.05
0.40
Biased
Unbiased
0.00
Biased
Unbiased
10.05-0.40
Biased
1.0
Figure 6: Judicial Decision Making Transitions. Transition probabilities between different meta- causal states, visualized as Markov processes for the eager and reflected strategies of the Judicial Decision Making example in Sec. 6.2
The per-row normalized transition matrices then model the state changes probabilities:
35.0% 40.0%
65.0%
Penger
(48)
60.0%
0.0% 0.0%
0.0%
P Preflected
(49)
100.0%
Optionally, the continuous-time rate matrices are computed as Q = p-I, (For this discrete time scenario P might be sufficient):
0.599
0.402
Q euger
(50)
0.247
0.752
00.667
0.0
Qrefleeted
(51)
0.0
1.0
Finally, the specific Meta-Causal Causal ATE expresses the change in transition probabilities effect between the eager and reflected policy:
-3.00% AMCATE(Peagee Pretooeed) Preflected Peager 40.0%
65.0% 40.0%
(52)
The obtained transition probabilities
Penger Penger: Preflected and STCATE eugger: Preflected), can be visualized as Markov processes. graphs are shown in Fig.
their difference, The corresponding
Interpretation. The absolute MCS state counts Crager indicate a 40.0% chance of remaining with a biased decision in the eager policy case. The transition dynamics Peager and P Prefected furthermore indicate a moderate chance (40% and 65%) of transitioning between the two MCS every timestep. Conversely, the reflected policy remains solely within the unbiased state, fully eliminating all other transitions. This similarly reflected in the sMCATE, offsets in the transition probabilities eeceptt for the unbiased + unbiased self-cycle- fully counteract all other transition probabilities of Peager.
28


Reviews:
[{"review_id": "PkccN7d14C", "reviewer": "Reviewer_CdGz", "summary": "This paper presents a strong theoretical framework for reasoning about dynamic causal systems through Meta-Causal Models (MCMs). The motivation — that static ATE-based methods fail when interventions shift causal dynamics — is clear and well stated. The paper demonstrates the ideas through two examples in healthcare and judicial decision-making. The main theoretical contribution is comprehensive, but the paper’s notation, examples, and clarity can be improved to make the work more approachable and practically usable.", "questions": "1. Can you clarify the preferred medication in the flu example? Figure 2 and line 306 seem to imply the conclusion is reversed — Medication B appears preferable because it does not suppress the immune system.\n2. How are `M_A` and `M_B` determined? More detail would help reproduce the example.\n3. Would you consider adding an explicit worked example of an SCM, MCF, MCS, and MCM in the appendix to improve approachability?\n4. Related work: Please contrast your contribution more explicitly with prior methods — what exactly is new vs. what exists.\n5. Judicial example: You might expand a bit more on how variables/relations change as the intervention (reordering cases) affects the system.\n6. How would suggest other researchers build on your paper/framework? Is there, e.g., some code they could get started with (perhaps with sample models/assumptions)?", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n- Solid theoretical framework with comprehensive definitions (MCF, MCS, MCM).\n- Strong abstract and introduction that motivate the need for meta-causal reasoning.\n- Relevant, high-impact application examples illustrate practical value.\n- Limitations are honestly acknowledged and related work is comprehensively listed.\n\n### Weaknesses\n- Notation could be improved: e.g., lines 324–336 (`f_t/F_t`, `c/c_i/C_i`), and `P_t` is undefined in Sec. 5.2.\n- Figure 2 caption and line 306 may be backwards — Medication B should be preferred since it avoids immunosuppression.\n- Related work is present but not clearly contrasted — assumes readers infer the differences rather than explicitly discussing them.\n- Examples are reasonable but could benefit from more explicit, step-by-step SCM/MCF/MCS/MCM illustrations in the appendix.\n- Some readability issues in plots (e.g., Figure 2) — it is hard to tell which lines correspond to what.\n- Minor details: consider `\\cdot` after `F_t` (line 278, see Appendix A Eq. 10); clarify how `M_A` and `M_B` are obtained (lines 283–284).", "was_revised": true, "final_justification": "Key concerns around notation, figure clarity, and related work have been addressed -- as such I have raised my score to accept. Authors have also committed to resolving remaining clarity and reproducibility issues in the final version, which I'm not sure how we can guarantee but it would be good to somehow check, if possible. While the proposed algorithm goes beyond the initial submission, it does not alter the paper’s core contribution -- perhaps it can be suggested to the authors to explicitly say their algorithm is one such instantiation of a formal approach, somehow implying that it wasn't fully vetted.", "rebuttal": "Dear Reviewer,\nthank you for your thorough review and feedback on our paper. While being limited in response length we will try to comprehensively answer your remarked points:\n\n1) **W1/Q1: Notation.** The variables (*F*atique, case*C*omplexity and Case*P*ool) of the SCM are defined in Appendix B. However, we admittedly missed to link the corresponding section. We added a link to the full definitions in the appendix and will state variable domains, abbreviations and equations more clearly.\n2) **W2: Figure 2 Caption.** We switched drug A and B last minute to improve readability. Apparently, we forgot to adjust the figure... We have adjusted the figure caption to now indicate that medication B should be preferred. Thanks for catching that!\n3) **W3/Q4: Related Work.** In the following we contrast our work against the broader fields mentioned in the paper. For the paper, we will provide more detailed discussions: Unlike traditional causal time series analysis which often assume fixed causal structures, MCM explicitly model transition dynamics between qualitative system configurations. With regard to works on switching causal mechanisms, the novel class of direct MCM strengthens the transition predictions w.r.t. the observed variables, eliminating purely stochastic predictions. Finally, for the vast field of longitudinal statistics, that typically assess (dynamic) average treatment effects over time, the meta-causal analyses specifically focus on the pathways through different meta-causal states, which might yield desirable or undesirable function properties (e.g. stability or active mechanisms).\n4) **W4/Q3: Step-by-step Illustrations.** Thank you for suggesting a more tangible and step-by-step explanations of our examples and general meta-causal analyses. We will put a full step-by-step on defining all of SCM/MCF/MCS/MCM in the appendix providing a complete start-to-end illustration for setting up meta-causal models. Additionally, we have proposed a novel algorithm (we kindly refer to our reply to reviewer xvfw) and present an actual application of it at the end of this reply.\n5) **W5: Readability Figure 2.** We will move the SCM of Figure 2 as a subfigure into the text (similar to Figure 3). This will give the remaining plots more space and we will increase label size. Similarly, we will put the medication levels on a separate axis to declutter the plots and visually separate the variables better.\n6) **W6/Q2: Clarity - cdot and M_A, M_B.** We agree with your suggestions regarding $F_t$. $M_A$ and $M_B$ are the drug specific properties that are given at the end of the first paragraph in Section 5.1. We choose arbitrary, but moderately distinct values of for the sake of giving an example that is easy to follow. Starting conditions in the Flu example are time=0 and immuneStrength=2. All other values follow from them together with random sampling exposure levels. We will note this in the appendix.\n7) **Q5: Judicial Example.** Full equations for the example are presented in Appendix A. Nonetheless, we will add a brief description within the main text. Basically, all structural equations in the system remain fixed for the whole scenario. Biased decision arise from increasing fatigue (which simply increases over time) in combination with complex cases. While equations stay the same, some of these relations might only exhibit *actual* effects (e.g. they propagate non-zero values) in certain contexts. Particularly, we are interest in avoiding decision bias. For this we need to prevent the decision bias relation to become active (-> preventing it to compute a non-zero value). As the activation function of the Decision Bias is a ReLu (max(Fatigue+CaseComplexity-5, 0)) we can identify inactive states by having no gradients for $F$ and $C$, in all cases of $F+C<5$ is less than the threshold (5). As fatigue is deterministic, but can not be controlled,  the only chance to avoid the function activation is to implement a case scheduling strategy that selects complex cases at earlier timesteps, and keeps the simple cases for the later timesteps. While classical causal analysis might come to the same conclusions w.r.t. choosing the preferred strategy, the meta-causal approach is purely motivated via actual effect propagation. No further metrics in terms of ATE or similar need to be specified by the user.\n8) **Q6. Other researchers build on your paper/framework.** Thank you for the interesting question. As already discussed earlier, and given your feedback and that of the other reviewers, we provide an explicit algorithm in our response to reviewer xvfw. We hope our newly provided approach enables meta-causal analyses (MCA) to be out-of-the box applied to other scenarios and make the whole topic of meta-causality more tangible. Similarly, in the answer W3/W4 to reviewer jRAi we elaborated further on the distinction of 'classical' dynamical analyses and meta-causal analyses (MCA), in that are concerned in the change of qualitative system dynamics. Natural question to answer with MCA would be \"How likely is a system to adapt a desired state?\", \"How stable is a desired state?\" or \"Which paths (in terms of MCS sequences) are available to reach a particular state?\" (and \"Are these paths admissible in terms of intermediate system behavior? E.g. are they safe to navigate under safety or ethics standpoints?\"). Such questions might, for example, arise in the course of planning actions of autonomous/agentic agents, or might be relevant for general societal or governmental decision making.\n\n\n\nThank you once again for pointing out several unclarities of our work. We believe that our planned changes will overall strengthen the paper in terms of contributions and clarity. We happy to answer any remaining questions or engage in further discussions.\n\nBest regards,  \nthe authors\n\n\n\n---\n\n**Worked out example: Medicating Flu.**\n\nAs already mentioned for the LMCD algorithm, we chose an identification function that measures *actual* influence between variables ($\\\\mathcal{I}(\\\\mathbf{x}, X_i,X_j) := (dX_i/dX_j(\\\\mathbf{x}) \\\\neq 0)$). The function is true whenever $X_i$ is contextually dependent on $X_j$ (<=> the gradient of $X_j$ on $X_i$ is non-zero) and false otherwise. By identifying the full meta-causal types of all recorded samples, we obtain 10 unique MCS for drug A and 8 unique MCS for drug B. For this analysis, we however only focus on the relevant activations of edges that affect body temperature. This coarsens the MCS into 3 unique states with either none of the edges active (1),  {Fever->Temperature} active (2) or {Fever->Temperature, Medication->Temperature} active (3). (Note that {Medication->Temperature} does never occur, since medication is only administered in case of fever).\n\nBy counting the occurrence of individual meta-causal states at timestep $t=9$ (\"age 18\") we get the following vectors:\n\n$C^t_A = \\\\begin{bmatrix} 10 \\\\\\\\ 9 \\\\\\\\ 981 \\\\end{bmatrix};\\\\quad C^t_B = \\\\begin{bmatrix} 437 \\\\\\\\ 104 \\\\\\\\ 459 \\\\end{bmatrix}$\n\nAdvancing individual states one step further, we obtain the following absolute transition counts:\n\n$C_A = \\\\begin{bmatrix} 3 & 0 & 7 \\\\\\\\ 1 & 1 & 7 \\\\\\\\ 0 & 6 & 975 \\\\end{bmatrix}; \\\\quad C_B = \\\\begin{bmatrix} 227 & 40 & 170 \\\\\\\\ 33 & 28 & 43 \\\\\\\\ 120 & 58 & 281 \\\\end{bmatrix}$\n\nThe per-row normalized transition matrices then model the state changes probabilities at time $t$:\n\n$P_A = \\\\begin{bmatrix} 30.00\\\\% & 0.00\\\\% & 70.00\\\\% \\\\\\\\ 11.11\\\\% & 11.11\\\\% & 77.78\\\\% \\\\\\\\ 0.00\\\\% & 0.61\\\\% & 99.39\\\\% \\\\end{bmatrix}$\n\n $P_B = \\\\begin{bmatrix} 51.95\\\\% & 9.15\\\\% & 38.90\\\\% \\\\\\\\ 31.73\\\\% & 26.92\\\\% & 41.35\\\\% \\\\\\\\ 26.14\\\\% & 12.64\\\\% & 61.22\\\\% \\\\end{bmatrix}$\n\nAnd similar continuous-time rate matrix: (for discrete time steps $P$ might be sufficient.)\n\n$$Q_A = \\\\begin{bmatrix} 0.496 & 0.001 & 0.502 \\\\\\\\ 0.050 & 0.412 & 0.537 \\\\\\\\ 0.000 & 0.004 & 0.995 \\\\end{bmatrix}$$\n\n $$Q_B = \\\\begin{bmatrix} 0.662 & 0.066 & 0.271 \\\\\\\\ 0.211 & 0.506 & 0.282 \\\\\\\\ 0.186 & 0.082 & 0.731 \\\\end{bmatrix}$$\n\nSpecific Meta-Causal ATE (effect between both medication; more commonly effect between treatment and no treatment might be considered):\n\n$sMCATE_{disc}(P_A,P_B) = P_B - P_A = \\\\begin{bmatrix} 21.94\\\\% & 9.15\\\\% & -31.09\\\\% \\\\\\\\ 20.61\\\\% & 15.81\\\\% & -36.43\\\\% \\\\\\\\ 26.14\\\\% & 12.02\\\\% & -38.16\\\\% \\\\end{bmatrix}$\n\n**Interpretation:** From the $C$ matrices we see that the third state, indicating fever induced medication, is almost always active for drug A (98.1%), while it is only given in 45,9% of the cases for drug B. Considering the meta-causal transition matrices $P_A, P_B$, we do not only find that the MCS of fever induced medication is increased in A, but also that all other states eventually flow into this state (with probabilities of 70,0% and 77.78%). Drug B features more favorable dynamics. Here, patients with a fever induced medication are still rather likely to to stay in that state. However, (moderately) healthy patients only transition into this state with moderate probabilities of 38.9% and 41.35% and are similarly able to leave leave it. These differences in dynamics are also visible from considering the sMCATE matrix, which shows a sharp decrease in transition probabilities into the third state, while the healthy state observes a strong increase in incoming transition probabilities.\n\n**Additional notes:** The computed $P$ and $Q$ matrices can be interpreted as Markov and continuous flow processes. While this years reviewing process does not allow to update materials, we will include corresponding graph visualizations in the paper.\n\n**Judicial Decision-Making.** While we had to skip the judicial example due to the character limit of the initial response, we will likewise provide the corresponding calculations in the final paper which follow the same algorithm steps. Since outcomes in this scenario are deterministic for the optimal strategy, we deemed the medicating flu example to be more worthwhile to present in our response."}, {"review_id": "IsdPnTdkwA", "reviewer": "Reviewer_xvfw", "summary": "The paper addresses the theme of interventions that change the underlying system dynamic. In order to reason about evolving relationships, they introduce the formalism of meta-causal models and discuss a specific subclass of those with desirable properties. Furthermore, the authors discuss a few examples where a meta-causal lens is desirable.", "questions": "Barring the addition of more content (results) there is not a lot to do on the paper to improve it I would say. See above for suggestions on what could be added.", "limitations": "The societal impact and limitations have been addressed satisfactorily.", "rating": 2, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 2, "significance": 2, "originality": 1, "strengths_and_weaknesses": "Clarity - The paper is written clearly and there seems to be enough attention on relevant literature.\n\nOriginality - The paper proposes a formalism that is novel as far as I know.\n\nSignificance - The topic of dynamical causal systems with feedback mechanisms is definitely a significant one.\n\nDespite these strengths, I have some comments and concerns. The first one is that this paper is rather thin on contributions. Beside the definitions, there is only one theorem, whose proof is rather trivial. Even though the examples are interesting, described and contextualized well, they are not linked to the theory because the theoretical formalism is not used; it is therefore not demonstrated that the formalism proposed in the earlier section is needed or better than other options. Hence, I believe this may not be enough meat for a full paper at NeurIPS. To consider this for a full paper I would want to see some more results on the theory and a full formalisation of the examples, at least. \n\nMy second concern is that, after reading the examples, I do not understand why the new formalism is needed at all. These seem to be two cases in which we have an SCM unfolded over discrete time steps, and we are interested in the effect of interventions over long-term outcomes (frequency of fever episodes in the first case, and amount of biased decisions overall in the second case). It seems to me this can be modelled satisfactorily with the current SCM formalism by taking a (simple) dynamic treatment regime (DTR) perspective. That is, for example in the flu scenario, by considering interventions not at a single time point (do I give drug A or B at time t) but interventions that change the causal mechanism of the Medication variable uniformly at all time points: in the paper’s table 2, the strategies seem to be “always give drug A” and “always give drug B”. Equivalently, one can think of this as a bundle of hard interventions on all the Medication variables, one per time point, setting each to a constant function = A or =B respectively. Of course, one can also think of DTRs that are more complex (e.g. give drug A if fever is above threshold Z, etc). We can then check end-point outcomes or long-term outcomes depending on what we are interested in, to see which DTR is better. These are classical problems in epidemiology, and there is quite a lot of literature on this. This stuff is also implemented in code packages, see e.g. the pygformula Python package by the Harvard people to calculate g formula estimates of dynamic treatment strategies. So also on the practical side I am under the impression that we do not have a problem handling these cases.\n\nI have some other points provided below:\n\nDefinition 4 seems to be redundant given the text above, I would compress the presentation\n\nwording at line 133 seems to be off: “any” seems to imply that the set of MCV can change the type of any other pair of variables, but your definition just says that a  variable A is meta-causal if there exist two other variables B, C and A can change the type of the relation from B to C. Thus the text and the definition do not seem to say the same thing.\n\nThe P in Eq 3 seems to suggest you are thinking of binary outcome, but this was not stated anywhere?\n\nThe first paragraph of Section 5 has repetition from the previous paragraph, I would shorten.\n\nLine 306, you mean drug B should be preferred?\n\nOn the desirability of outcomes in Section 5.1 and whether drug B is preferable: this seems to be a classic trade-off between short term outcomes (low fever now) and long-term ones (low recurrence of fever in the long run) which will depend on how much the agent discounts the future outcomes and some notion of utility of outcomes. Without setting these parameters, I do not think there is a clear choice of which drug is better unless you unilaterally decide that the short term outcome is less relevant (debatable).", "was_revised": true, "final_justification": "After reading the rebuttal and the other replies/reviews, I am still of the opinion that the original paper does not stand on its own, and that too much content needs to be added to make this a valuable contribution that passes the bar of NeurIPS. I see the authors have put a lot of effort in describing the revisions they intend to make, which is laudable, but I still have trouble envisioning what the paper will look like afterwards and some key additions--like the new algorithm--cannot be properly assessed during such a rebuttal. The way I see it, the bulk of the work needs to be present in the original paper, while here it looks like the main innovations (the new algorithm, the sMCATE) the motivation (what questions does this new framework address) and the convincing examples (showing how the old frameworks cannot do that they could do) have all been given in the rebuttal. In light of all this, I'll keep my score, but happy to discuss more with AC and other reviewers.", "rebuttal": "Dear Reviewer,  \nthank you for your extensive feedback on our paper, pushing us to improve and to provide more explicit contributions. As a direct response to your, but also other reviewers' comments, we have provided a new  algorithm for explicitly estimating (linearized) meta-causal dynamics from a sets of observations. Similar to 'standard' ATE, we furthermore derive a first notion of a meta-causal ATE. We believe that both contributions strengthen our paper and provide a tangible methods for performing meta-causal analyses (MCA). In the following, we will try to answer your remarked points:\n\n1) **Additional contributions.** First of all, thank you for finding our examples interesting, described and contextualized well, but also for pointing out a lack of actionable contributions and a disconnect between our initial theory and later applications. For the examples we initially aimed to give an intuition of meta-causal aspects, rather than a fully formal treatise. As this point was similarly remarked by other reviewers, we tackle this issue in a two-fold manner: first, we do provide a discussion on the aspects of meta-causal analyses (MCA) together with an explicit algorithm. Second, we provide worked out examples of our applications (utilizing the newly proposed algorithm) to highlight the interplay between classical (time-series) SCM and meta-causal aspects. To keep the discussions compact and easy to parse **we have put the proposed algorithm at the end of this reply and additionally provide a worked out example in our answer to reviewer xvfw.** We will include both in the final paper.\n2) **Need of new formalism.** We hope our new algorithm and corresponding discussion can clarify the different question being answered in contrast to classical causal (timeseries) models or longitudinal studies. Nonetheless, meta-causal analyses is not a separate analyses methods but designed to be used in combination and to supplement the existing methods. More specifically, MCA considers the transition dynamics between different meta-causal states (/different causal configurations of the system) at a particular points in time. Our newly proposed algorithm, for example, linearizes transition dynamics after the possible unrolling of a time series model to determine transition probabilities between different meta-causal states. (Modeling the applied strategy as either explicit structural equations or continuous **bundles of hard-interventions** is up to the user. However the actual realization is an implementation detail that does not affect the the final analyses. We will put a note in this in the paper). Eventually these state transitions translate to switching structural equations, that, in return, induce particular variable distributions, which can finally be captured via the suggested parametric g-formula, dynamic treatment regime methods and similar statistical analyses. (This is what we had computed so far in our paper). However, MCA is not only interested in the resulting actual effects, but also how such effects come and how states transition into each other with the help of graphical (causal) models, e.g. whether particular mechanisms are active at particular points in time, which might either pose a desired or undesired condition. To better highlight possible questions to be answered by MCA, we partially repeat our answer to reviewer CdGz on naturally arising **question to be answered with MCA**: \"How likely is a system to adapt a desired state?\", \"How stable is a desired state?\" or \"Which paths (in terms of MCS sequences) are available to reach a particular state?\" (and \"Are these paths admissible in terms of intermediate system behavior? E.g. are they safe to navigate under safety or ethics standpoints?\").\n   To the best of our knowledge, the previous questions can not be answered by existing approaches, as the explicit graphical modeling of qualitative (meta-causal) state transitions is not covered in any of the previously suggested work and constitutes the main novelty/contribution of our paper.\n\n**Other points:**\n\n1) **Redundant Definition 4.** Agreed. We will remove the explicit equation.\n2) **Targets of Meta-Causal Variables.** We agree that the presented description can be interpreted ambiguously. What we meant is that possibly every causal edge can be the target of some meta-causal variable (i.e. there is no set of edges that would be inherently protected from being modeled as targeted of a meta-causal variable). We agree that in most cases meta-causal variables influence only a confined fixed set of edge (which, of course, might include influencing the whole set of all edge relations in rare cases). We will be more precise on the nature of this relation in the paper.\n3) **Eq 3 / Binary Outcome.** You are right. However, this equation is thought to introduce the do-operator for the ATE formalism. We will simply drop the $P$'s here, as they are not required. Kindly, also see our reply to reviewer jRAi on the assumptions required for computing ATEs. We we will state them explicitly in the final version.\n4) **Repetitions in Sec. 5.** We will reduce the overlap in the camera-ready version.\n5) **Drug B should be preferred.** Thank you again for catching this mistake on our side. We where switching names of the medications to make the example more easy to follow, but forgot to adjust it in the figure caption. We have corrected it in the paper.\n6) **On the desirability of outcomes.** We agree. The described setup presents a theoretical scenario with strongly simplified and exaggerated system dynamics, treatment strategies, resulting probabilities and desirability on the purpose of presenting an easy to follow example. In our case, we deem both medications suited to treat all severeness levels of fever. In case this assumption is violated, drug A would need to be administered even in cases of an initial drug B strategy, to avoid severe harm or death. We will add a clarifying disclaimer at the start of the corresponding section to discuss the made assumptions and possible short- versus long-term considerations.\n\n**Summary:** We highlighted similarities and differences of MCM and longitudinal approaches. While we mostly focused on the formal technical details of our work, we are trying to pay attention to the extensive existing work and implications of presenting examples in the fields of general statistics, medicine and epidemiology. Finally, we would like to thank you for pushing us to be more explicit about possible similarities and differences with existing works. This has eventually helped us to better identify out the core contributions of our work. Kindly let us know in case any open points remain.\n\nBest regards,  \nthe authors\n\n---\n\nThe **Linearized-Meta-Causal-Dynamics (LMCD) Algorithm** estimates the linearized transition probabilities between different meta-causal states from a given set of observed states. For obtaining initial populations at particular timesteps standard causal timeseries SCM can be applied. While existing longitudonal approaches might be sufficient to estimate dynamic treatment effects over several timesteps, the presented algorithm captures the (linearized) state transition probabilities of the system, which reveal the underlying dynamics and paths the system might take:\n\n1) Input: an SCM $\\\\mathcal{M} = (\\\\mathbf{U}, \\\\mathbf{V}, \\\\mathbf{F}, P_{\\\\mathbf{U}})$; a record (or simulated) population of $N$ samples  $\\\\mathbf{x}^{t} \\\\in \\\\mathbf{X}^N, i \\\\in [1..N]$ at timestep $t$; and an identification function $\\\\mathcal{I}: \\\\mathbf{X} \\\\to T$.\n2) For every sample $\\\\mathbf{x}^{i,t} \\\\in \\\\mathbf{x}^{t}$:\n   1) Advance the system to another timestep $\\\\mathbf{x}^{i,t+1} := \\\\mathbf{F}(\\\\mathbf{x}^{i,t})$.\n   2) Identify the (possibly coarsened [a]) MCS of the respective states $T^{i,t} := \\\\mathcal{I}(\\\\mathbf{x}^{i,t})$ and $T^{i,t+1} := Id(\\\\mathbf{x}^{i,t+1})$\n3) Compute the (indexed) set of unique meta-causal states: $U = (\\\\bigcup_i T^{i,t}) \\\\cup (\\\\bigcup_i T^{i,t+1})$ with index $l: T \\\\to [1..|S|]$\n4) Compute the entries of the transition probability $P^t \\\\in \\\\mathbb{R}^{|S|\\\\times|S|}$ as $P^t_{u,v} := \\\\sum_{i\\\\in[1..N]} (\\\\delta(l(T^{i,t}) = u) \\\\land (l(T^{i,t+1}) = v) / \\\\sum_{i\\\\in[1..N]} (\\\\delta(l(T^{i,t}) = v)$ with $u,v \\\\in [1..|S|]$ and $\\\\delta$ being 1 if $b$ is true and 0 otherwise.\n5) [Optional] Compute the continuous time rate matrix via the matrix exponential $Q := e^{P^t-I}$ where $I$ is the identity matrix.\n\n[a] Grouping different $T$ by only considering problem relevant subsets $T'^{i,t} \\\\subset T^{i,t}$ of the full type matrix.\n\n**Choice of Identification function in the examples:** For all examples and any two continuous variables $X_i, X_j$ connected via a direct causal edge $X_j \\\\to X_i$, we use the \"contextual independency\" function $\\\\mathcal{I}(\\\\mathbf{x}, X_i, X_j) := (dX_i/dX_j(\\\\mathbf{x}) \\\\neq 0)$, which is true (meaning $X_j$ exerts influence $X_i$) if $X_i$ is contextually dependent on $X_j$ (has a non-zero influence / the gradient of $X_j$ onto $X_i$ is non-zero), given the subset of parent values of $X_i$ for current variable configuration $\\\\mathbf{x}$. Otherwise, it is false, indicating that no current causal influence of $X_j$ on $X_i$ exists.\n\nA **(specific) Meta-Causal ATE (sMCATE)** between two strategies A,B with transitions matrices $P_A, P_B$ (or optionally $Q_A, Q_B$) might be defined as the difference in their transition probabilities $sMCATE_{disc}(P_A,P_B) := P_B - P_A$. Note, that $P_A,P_B$ already represent averages over their respective populations. Contrary to our examples, $P_A$ might represent the no treatment / unintervened environment. The sMCATE might be further compressed into a single number. For particular scenarios, fitting matrix norms might be chosen for this purpose. In the absence of a clear general candidate for defining the general MCATE we abstain from deciding on a definite characterization."}, {"review_id": "3rGrrx2RDE", "reviewer": "Reviewer_D4bW", "summary": "Modelling a dynamic environment with traditional causal modelling (SCMs) assumes that the causal structure remains the same over time. In other words, causal dynamics are static and not affected by decisions under standard SCMs. Meta Causal Models (MCMs) alleviate the former concern by allowing for the causal dynamics to evolve over time, but do not resolve the latter. This paper introduces a new class of MCMs rich enough to characterize intervnentions and their long-term effects on the structure of a dynamic causal system.", "questions": "I was wondering if the dynamic environments from your paper and the effect of interventions on their underlying causal struture could be modelled by mechanised SCMs [1] where object level variables can directly affect mechanism level variables. Could you maybe share your on thoughts on that, and explain why the increased complexity of Direct MCMs is necessary?\n\n[1] Discovering Agents, Kenton et.al., 2022", "limitations": "Adequately discussed", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "+ The problem being addressed is well explained and adequately motivated\n+ The analysis of the proposed extension is detailed and deep\n+ The applications section provides good insights about the proposed approach\n\n- I found the paper quite heavy to read without familiarity with prior work. This is not necessarily bad, but it was difficult for me (and audience with background in more \"traditional\" causal inference) to follow the techical details and appreciate the importance of the contributions.\n- The complexity of the proposed framework together with the paper's very limited empirical evaluation raises concerns about the pracicality of this approach. (See questions for an alternative)", "was_revised": true, "final_justification": "Clarity of the paper was improved with additional discussion and examples.", "rebuttal": "Dear Reviewer,  \nthank you for your feedback on our paper. We will try to answer your mentioned weaknesses and questions in the following:\n\n* **W1: Heavy to read.** We agree that the paper is quite theoretical in its initial definitions. We tried to counter this fact, by providing two applied examples that show practical aspects and come with less formalism. Given your feedback, we will use some of the additional space of the camera ready version to contextualize definitions better and give interpretations for the particular modeling choices of direct MCM. (For example, see our reply to reviewer jRAi on the role and possible relaxation of the bijectiveness assumption of the varphi function.) Generally speaking, (direct) meta-causal models determine (or inversely infer) the functional/mechanistic relations of (or from) a system via abstracted function types. These types are then used to qualify the overall system states and transition between them, to qualitatively model their (meta-causal) dynamics. We will expand on these aspects, before starting with the formal definitions.\n\n* **W2: Practicality of the Approach.** MCMs are a rather new concept that is under current development and focuses on the qualitative state transitions. We are aware of the current rather theoretical type of papers, but hope to have presented two rather intuitive (and less theory heavy examples). In response to your and the other reviewers' remarks **we provide a Linearized-Meta-Causal-Dynamics algorithm (see our response to reviewer xvfw) and worked out examples (see our response to reviewer CdGz).** We will supplement the paper with these new methods and calculations to better highlight the actionability of our approach. Similarly to our reply to reviewer CdGz we want to elaborate further on the distinction between classical dynamical analyses and meta-causal analyses (MCA), in that are concerned in the transition of qualitative system dynamics rather than (and while intimately linked) the measurement of actual emitted effects via such changes. Natural questions to answer with MCA would be \"Is a system able to obtain a desired state?\", \"How stable is a desired state?\" or \"Which paths (in terms of MCS sequences) are available to reach a particular state?\" (and in this context \"Are these paths admissible in terms of intermediate system behavior? E.g., are they admissible to navigate under general safety or ethics standpoints?\").\n\n* **Q1: Dynamic Environments / Relation to Mechanized SCM.** While MCM eventually relate to a multitude of prior existing frameworks, we recognize that we did not consider the mentioned mechanized SCM as a rather closely related idea. As MCM initially come without explicit mechanism level variables by switching mechanism directly without the intermediate consideration of mechanistic variable, they might be more compact in representation at the expense of reduced interpretability.\n\n  While general MCM [allow for causal abstraction of the data generation process our direct MCM enforce explicit control of the endogenous variables over the structural equations and are, therefore, closer to mechanized SCM. By making the transition factors explicitly observable, we believe that both frameworks might eventually be transitioned into each other. Depending on the use case (e.g., modeling general system dynamics from observations versus modeling the influence of agentic agents on the system), one or the other representation might be better suited in certain situations. Thanks again for pointing us to this line of work. We will add the above discussion to the final paper.\n\n\nThank you once again for your helpful feedback on our paper. We hope to have answered your questions to a satisfactory degree and are happy to elaborate further, in case any open points remain.\n\nBest regards,  \nthe authors"}, {"review_id": "wAhDelo5v1", "reviewer": "Reviewer_jRAi", "summary": "This paper studies the modeling and prediction of intervened outcomes using meta-causal models (MCM).", "questions": "See weakness.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strength**\n\nThis paper provides very interesting for dynamic causal inference.\n\n**Weakness**\n\nThis paper should clearly articulate the underlying assumptions used for causal inference, as well as the limitations of the proposed approach. \n1. My main concern lies in the bijectiveness of \\varphi. How is this assumption justified within the context of meta-causal models (MCM)? Could you provide examples of cases in which the assumption is satisfied and cases in which it is violated? This is crucial for assessing the reliability of the results.\n2. The identification of the average treatment effect (ATE) typically requires an exogeneity or ignorability assumption. Are such assumptions not required in your paper?\nI believe that such assumptions are necessary, and that the meta-causal framework (MCF) exclude the problem of confounding.\n3. A substantial body of work by James Robins focuses on causal inference in time series and longitudinal settings. A discussion on the differences among his studies is necessary to clarify their respective contributions and methodological distinctions.\n4. Additionally, similar studies have been conducted under the framework of “longitudinal data” in the field of statistics.", "was_revised": true, "final_justification": "I will keep my positive score.", "rebuttal": "Dear Reviewer,  \nthank you for your feedback on our paper. We will try to address your mentioned weaknesses/questions in the following:\n\n1) **W1 Bijectiveness of \\\\varphi:** Bijectiveness of \\\\varphi does not restrict the expressibility of MCM, but rather decides which meta-causal states and future transitions can be predicted from (endogenous) observations of the SCM's variables. This observability is 1) necessary to estimate the causal type of an edges to determine the current meta-causal type (structural equation to MCS) and therefore the transition probabilities (c.f. steps 2.1 and 2.2 of the Linearized-Meta-Causal-Dynamics algorithm in our reply to reviewer xvfw). This that this direction does not exclude the presence of exogenous factors, (consider exogenous exposure levels in the Flu example), but determined the finegrainedness with which meta-causal types can be identified. 2) In the reverse direction (MCS to structural equation) consider that a particular meta-causal state $T_{s+1,ij}$ is sampled in step 2.1 of the algorithm. However, without being able to invert \\\\varphi the particular equations for the process values $s$ can not be determined, such that these values and their subsequently emitted influence on edge type can no longer be determined. \n \n   We take your comment as a suggestion to relax the requirement of \\\\varphi as to only be invertible. We therefore drop the uniqueness requirement. At the time of advancing the system state, this allows to sample an actual structural equation $F\\_{ij}$ from a set of candidate equations $\\\\textbf{F}\\_{T\\_{s,ij}}$ that have  been observed earlier for a particular edge type $T_{s,ij}$, according to some previously recorded probability distribution, $F_{ij} \\\\sim P_{\\\\textbf{F}\\_{T\\_{s,ij}}}$. This introduces uncertainty at the level of configuration of the SCM in terms of structural equations and allows for a more broader area of application. We incorporated the discussed changes in the definition and follow up texts, and will discuss their implications in an additional section in the appendix.\n\n2) **W2 Identification of the average treatment effect.** Since building meta-causal models upon the Pearlian causal framework ATE computations can be computed over the causal graph. Within the Pearlian framework the do-calculus given the sufficient and necessary conditions for such causal effect estimations under which condition unbiased (e.g. unconfounded estimators can be obtained). The mentioned exogeneity and ignorability can be directly derived from the do-calculus, and we assumed them to hold implicitly. However, we agree that these implicit assumption might be not obvious and potentially confusing. We added a corresponding note on the made assumption and potential use of do-calculus before introducing ATE calculations. Thanks for pointing this out!\n\n3) **W3/W4 Related Works by James Robins / Longitudinal data in Statistics.** Thanks for pointing us to these works and general methods on longitudinal data. We agree that these methods in longitudinal statistics are highly related to our ideas. Although a strong overlap in both areas of longitudinal and meta-causal analyses (MCA) exists, both approaches are trying to answer different questions in their respect. Nonetheless, we will supplement our related work section with a summary of the following references, and provide a more detailed discussion in a new appendix section:\n   **Robins' works and general Longitudinal Statistics** cover influential works on the g-formula [1,2] that aims to estimate exposure effects in the presence of time-varying confounders. Most similar to out work might be the paper of Robins and James [3] in which similarly leverages marginal structural models to be used for causal effect estimation under time-dependent confounding. Similarly, the works of Kung-Yee and Zegler [4] and Laird and Ware [5] utilize linearized models and random-effect sampling to approximate the effects of long term dynamical shifts in the resulting distributions and corresponding ATE.\n   In **contrast to prior works**, our paper focuses on performing Meta-Causal Analysis (MCA). MCA is not a separate set of analyses methods, but designed to be used in combination and to supplement existing methods. More specifically, MCA considers the transition dynamics between different meta-causal states (/different structural configurations of a system) at a particular points in time. Our newly proposed algorithm, linearizes transition dynamics after the possible unrolling of a time series model to determine the transition probabilities between different meta-causal states.  In this regard please note the newly proposed algorithm on performing MCA in our response to reviewer xvfw, as well as the resulting worked out examples in response to reviewer CdGz. Eventually these state transitions translate to switching structural equations, that, in return, induce particlar variable distributions, which can finally be captured via the suggested parametric g-formula, dynamic treatment regime methods and similar statistical analyses. (This is what we had computed so far in our paper). However, MCA is not only interested in the resulting actual effects, but also how such effects come to be with the help of graphical causal models, e.g. whether particular mechanisms are active at particular points in time, which might be either desired or undesired. To better highlight possible questions to be answered by MCA, we partially repeat our answer to reviewer CdGz: Natural question to answer with MCA would be \"How likely is a system to adapt a desired state?\", \"How stable is a desired state?\" or \"Which paths (in terms of MCS sequences) are available to reach a particular state?\" (and \"Are these paths admissible in terms of intermediate system behavior? E.g. are they safe to navigate under safety or ethics standpoints?\").\n   To the best of our knowledge, the previous mentioned questions have not yet been answered in a unified framework by previous approaches. The explicit graphical modeling of (meta-causal) state transitions in causal systems is not covered in the previously suggested fields of work and constitutes the main novelty of our work. Kindly note that we gave a similar answer to reviewer CdGz which pointed us in the same direction. \n\n\nThe above answers outline the role of meta-causal analyses in relation to existing methods in longitudinal statistics and dynamic treatment regimes, and separate them from the novel contributions of our paper in terms of MCA. We elaborated on the role of varphi and the relation of ATE and the graphical causal models used in our work. Please let us know in case me missed any key aspects or papers, or you want to discuss any of the given points further.\n\nBest regards,  \nthe authors\n\n\n[1] Robins, James. \"A new approach to causal inference in mortality studies with a sustained exposure period—application to control of the healthy worker survivor effect.\" Mathematical modelling 7.9-12 (1986): 1393-1512.  \n[2] Robins, James M. \"Causal inference from complex longitudinal data.\" Latent variable modeling and applications to causality. New York, NY: Springer New York, 1997. 69-117. \n[3] Robins, James M., Miguel Angel Hernan, and Babette Brumback. \"Marginal structural models and causal inference in epidemiology.\" Epidemiology 11.5 (2000): 550-560.   \n[4] Liang, Kung-Yee, and Scott L. Zeger. \"Longitudinal data analysis using generalized linear models.\" Biometrika 73.1 (1986): 13-22.  \n[5] Laird, Nan M., and James H. Ware. \"Random-effects models for longitudinal data.\" Biometrics (1982): 963-974."}]

Author rebuttal:

Meta-review: This submission builds on previous work about meta causal models (MCMs) that allow for causal structure to be dynamically updated, resulting in the same intervention having different effects at different points in time. The strengths of the paper are that it is clearly written, well motivated, and has examples of how the theoretical framework can be put into action. The main weakness of the paper identified by Reviewer xvfw is that there is only one theorem with a short proof and the examples don't use the new formalism, and so the contributions of this paper are light. There was a vigorous discussion between reviewers and authors, and several pieces of new material were proposed by the authors, which the reviewers agreed are high quality, but Reviewer xvfw felt that the resulting changes would be too great. 


Ultimately, my decision to suggest this paper be accepted is because the new material proposed during discussion with reviewers was given at a high level of detail with mathematical rigor. While we can't have a perfect picture of how the author will incorporate the material, the reviewers seem confident that the new material is of high quality and improves the submission. My inclination is to reward such a lively and productive discussion!
Decision: Accept (poster)