When Causal Dynamics Matter: Adapting Causal Strategies through Meta-Aware Interventions

Dear Reviewer,
thank you for your feedback on our paper. We will try to address your mentioned weaknesses/questions in the following:

1. 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.

   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 $\**F**\_{T\_{s,ij}}$ that have been observed earlier for a particular edge type $T_{s,ij}$, according to some previously recorded probability distribution, $F_{ij} \\sim P_{\**F**\_{T\_{s,ij}}}$. 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.

2. 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!

3. 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: 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. 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?"). 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.

The 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.

Best regards,
the authors

[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.
[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. [3] Robins, James M., Miguel Angel Hernan, and Babette Brumback. "Marginal structural models and causal inference in epidemiology." Epidemiology 11.5 (2000): 550-560.
[4] Liang, Kung-Yee, and Scott L. Zeger. "Longitudinal data analysis using generalized linear models." Biometrika 73.1 (1986): 13-22.
[5] Laird, Nan M., and James H. Ware. "Random-effects models for longitudinal data." Biometrics (1982): 963-974.

Dear reviewer,
thank you for your reply. We are not quite sure if you implied any particular assumption to be discussed within the SCM framework, but we will naturally implement all proposed changes in the paper. As our whole work is build upon the Pearlian SCM framework, we will frame all proposed changes within that. Specifically, we will make the following modifications:

• W1) The relaxation to invertible varphi will be implemented directly within the definitions. We will briefly mention implications in the main text and provide the slightly longer discussion, that we gave in the above reply, on the resulting implications and special case of bijectiveness in the appendix.
• W2) We will briefly clarify causal effect identification of ATE beyond the bivariate case, explicitly hinting to the full use of do-calculus.
• W3) We will use some of the extra space of the camera-ready version to discuss related work in longitudinal statistics. Generally, we will change the existing related work section to be more constrastive as discussed/proposed by you.
• MCA) We will reserve the majority of the additional space to add an new subsection on meta-causal analysis and the previously discussed perspectives, together with the newly proposed algorithm, before moving on to the experiments. We will provide the presented worked out examples in the appendix and discuss briefly qualitatively in the main text at the end of the corresponding experiment.

Please let us know if you would like us to go into more detail on any of the points above.

Dear Reviewer,
thank you for your feedback on our paper. We will try to answer your mentioned weaknesses and questions in the following:

• 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.

• 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?").

• 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.

  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.

Thank 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.

Best regards,
the authors

Dear Reviewer,
thank 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:

1. 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.
2. 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?"). 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.

Other points:

1. Redundant Definition 4. Agreed. We will remove the explicit equation.
2. 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.
3. 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.
4. Repetitions in Sec. 5. We will reduce the overlap in the camera-ready version.
5. 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.
6. 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.

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.

Best regards,
the authors

The 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:

1. 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 $\\mathcal{I}: \\mathbf{X} \\to T$.
2. For every sample $\\mathbf{x}^{i,t} \\in \\mathbf{x}^{t}$:
   1. Advance the system to another timestep $\\mathbf{x}^{i,t+1} := \\mathbf{F}(\\mathbf{x}^{i,t})$.
   2. Identify the (possibly coarsened [a]) MCS of the respective states $T^{i,t} := \\mathcal{I}(\\mathbf{x}^{i,t})$ and $T^{i,t+1} := Id(\\mathbf{x}^{i,t+1})$
3. Compute the (indexed) set of unique meta-causal states: $U = (\\bigcup_i T^{i,t}) \\cup (\\bigcup_i T^{i,t+1})$ with index $l: T \\to [1..|S|]$
4. 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.
5. [Optional] Compute the continuous time rate matrix via the matrix exponential $Q := e^{P^t-I}$ where $I$ is the identity matrix.

[a] Grouping different $T$ by only considering problem relevant subsets $T'^{i,t} \\subset T^{i,t}$ of the full type matrix.

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.

A (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.

Thank you for the comment. We actually completely disagree with the notion that the added content is a completely new paper or we believe that so much content is required to be padded to the original paper. All the provided answers and changes be it the algorithm or a worked out example serve as add-ons to make the paper a bit more clear. This is not to say that the original paper was not clear in our opinion but as always there is always scope for improvement in any paper. We would like to reiterate that the original paper stands on its own and the new content is helpful but does not comprise a completely new paper in our opinion. We would be happy to discuss further.

Dear Reviewer, thank 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:

1. W1/Q1: Notation. The variables (Fatique, caseComplexity and CasePool) 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.
2. 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!
3. 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).
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.

Thank 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.

Best regards,
the authors

Worked out example: Medicating Flu.

As 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).

By counting the occurrence of individual meta-causal states at timestep $t=9$ ("age 18") we get the following vectors:

$C^t_A = \\begin{bmatrix} 10 \\\\ 9 \\\\ 981 \\end{bmatrix};\\quad C^t_B = \\begin{bmatrix} 437 \\\\ 104 \\\\ 459 \\end{bmatrix}$

Advancing individual states one step further, we obtain the following absolute transition counts:

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}

The per-row normalized transition matrices then model the state changes probabilities at time $t$:

$P_A = \\begin{bmatrix} 30.00\\% & 0.00\\% & 70.00\\% \\\\ 11.11\\% & 11.11\\% & 77.78\\% \\\\ 0.00\\% & 0.61\\% & 99.39\\% \\end{bmatrix}$

$P_B = \\begin{bmatrix} 51.95\\% & 9.15\\% & 38.90\\% \\\\ 31.73\\% & 26.92\\% & 41.35\\% \\\\ 26.14\\% & 12.64\\% & 61.22\\% \\end{bmatrix}$

And similar continuous-time rate matrix: (for discrete time steps $P$ might be sufficient.)

Q_A = \\begin{bmatrix} 0.496 & 0.001 & 0.502 \\\\ 0.050 & 0.412 & 0.537 \\\\ 0.000 & 0.004 & 0.995 \\end{bmatrix}

Q_B = \\begin{bmatrix} 0.662 & 0.066 & 0.271 \\\\ 0.211 & 0.506 & 0.282 \\\\ 0.186 & 0.082 & 0.731 \\end{bmatrix}

Specific Meta-Causal ATE (effect between both medication; more commonly effect between treatment and no treatment might be considered):

$sMCATE_{disc}(P_A,P_B) = P_B - P_A = \\begin{bmatrix} 21.94\\% & 9.15\\% & -31.09\\% \\\\ 20.61\\% & 15.81\\% & -36.43\\% \\\\ 26.14\\% & 12.02\\% & -38.16\\% \\end{bmatrix}$

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.

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.

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.