Systems with Switching Causal Relations: A Meta-Causal Perspective

Dear Reviewer puHX,
Thank you for your thoughtful review and for highlighting our strengths, including the novel theoretical contributions, clear motivation and insight for practical application. We will address your questions in the following:

Q1 [Relation to Measure-Theoretic Axiomatisations and Phenomenological Causality]: Thanks for referencing both works to us. We cite and (due to space constraints) briefly discuss the relation of measure-theoretic considerations towards the induction of causal relations in mediation processes, as well as the role of phenomenological considerations towards the overall constitution of causal variables.

Q2 [Abstract Motivation / Practical Applications]: [“The introduction is very hard to follow and could be motivated better. While the example in Figure 1 is definitely helpful to understand the idea and limitations, the introduction often sounds too abstract. Maybe starting with the Figure 1 (a) example on a high-level description without further introduction of any formalisms around causal modeling could be helpful for the motivation.]

Since we are first to introduce a rather novel formal definition of meta-causality within the restricted space of a conference paper, we are forced to keep things compact. Examples of meta-causality, by their nature, presume some existing understanding of classical causal models. Here, we decided to trade the inclusion of multiple relevant perspectives and applications of meta-causality in the main paper (identifiability of meta-causal states [game of tag], relation to contextual independencies, attribution of policy, discovery of meta-causal states and the stress induced fatigue example) against a short and, admittedly, rather technical introduction.

Q3 [Lack of Theoretical Statements]: The role of contextual independencies with regard to the reduction of MCM onto SCM was discussed in section 3, but indeed lacked a rigorous theoretical foundation.

After further consideration of the problem we are happy to present the conditions under which MCM can be reduced to SCM with conditioned structural equations. Roughly speaking, a reduction to SCM is possible whenever all transitions on the meta-causal model are self-loops. Generally, this is the case when the types of structural equations are determined by their functional form and do not change dynamically with the current state of the system.. Based on your feedback and requests of the other reviewers, we provide and prove a theorem for the exact conditions for the reduction of MCM. Due to space constraints the theorem and proof has been placed in Appendix C “Reduction of Meta-Causal Models to Conditioned SCM”.

We have uploaded a revised version of the paper, with changes color coded, for which we hope to have clarified most of your stated concerns. Your color code is orange. Please let us know if there is any additional information we can provide to further clarify our contributions. Thank you once again for your time and effort in reviewing our paper!

Sincerely,
Authors

Dear Reviewer PEUy,
Thank you for your thoughtful review and for highlighting our strengths, including the elegant implementation of SCM dynamics, the framing of meta-causal models as finite state machines, and the application of meta-causal attributions for preventive mechanisms. We will address your concerns in the following:

W1: [Do Meta-Causal Model Cause Standard Causal Model?] Thank you for raising this important question, we will try to clarify this in the following. Please note that, upon the feedback of you and reviewer DQ6p we have revised the definition of meta-causal frames in Section 3 to make the relation between the mediation process, MCM and SCM more clear and provide an extended discussion in the appendix. We believe that our answer Q3 to DQ6p might also be insightful in that regard.

Meta-Causal Models do not cause SCM, but are descriptive models that model how changes in the underlying process are reflected in the causal graph and, therefore, capture the changing dynamics of SCM. To be more specific, consider that the type encoder derives the type of edges based on the functional relationship between variables $X_j$ on $X_i$ by considering the underlying mediation process. The MCM is governed by the mediation process, as is the SCM. However, both models pursue a different perspective on the process. While SCM are concerned with how values propagate through the system, MCM are concerned how system dynamics affect the causal graph.

W2: [Subtle Differences Between Standard and Meta Causal Models.] You are correct in pointing out that meta-causal models might be reduced to classical SCM under certain conditions. The role of contextual independencies with regard to the reduction of MCM onto SCM was already discussed in section 3, but indeed lacked a rigorous theoretical foundation. As also requested by other reviewers, we are happy to extend our formalism and prove conditions under which this reduction is possible in Appendix C “Reduction of Meta-Causal Models to Conditioned SCM”. Roughly speaking, a reduction to SCM is possible whenever all transitions on the meta-causal model are self-loops. Generally, this is the case when the types of structural equations are determined by their functional form and do not change dynamically with the current state of the system. Note in that regard, that the Stress-Induced Fatigue example (Sec. 4.3) provides a counter-example where a reduction to a conditioned SCM is not possible for the reasons discussed in the paper. In such scenarios meta-causal inferences might be able to identify general conditions that induce changes in the SCM, while classical causal inferences are bound to reason within the currently given causal graph.

Q1: [Other examples where classical examples fail. How realistic are preventive mechanisms?] We assume that preventive mechanisms might be more common than expected. Consider for example the implications of the mental jenga example in Zhou et al. [1]. Even though the initial stack of blocks is static, --in the sense that no block is falling without external perturbation, therefore preventing the actual effect of gravity--, it is clear that the removal of any of the blocks causes others to fall. While removing individual blocks (with idealized precision) from a jenga tower might initially not yield apparent changes, it is clear that not all blocks of a row can be removed. In that sense preventive mechanisms might be encountered in every-day life as humans consider them as cognitive abstractions of the structural mechanics that govern our every-day environment.

[1] Zhou, Liang, et al. "Mental jenga: A counterfactual simulation model of causal judgments about physical support." Journal of Experimental Psychology: General 152.8 (2023): 2237.

We have uploaded a revised version of the paper, with changes color coded, for which we hope to have clarified most of your stated concerns. Your color code is green. Please let us know if there is any additional information we can provide to further clarify our contributions. Thank you once again for your time and effort in reviewing our paper!

Sincerely,
Authors

I thank the authors for addressing my concerns and providing the updated paper. Appendix C, "Reduction of Meta-Causal Models to Conditioned SCM," strengthens the manuscript. I'll update my recommendation accordingly.

Dear Reviewer DQ6p,
Thank you for your thoughtful review and highlighting the interesting and relevant ideas, and the development of a nontrivial algorithm evaluation. We will address your concerns in the following:

W1 [What is “Meta-Causal”] We are sorry for the confusion regarding the use of meta-causality throughout the paper. In particular, and as identified by other reviewers, meta-causality is concerned with the underlying process dynamics that govern the structural equations of an SCM. This can indeed sometimes be reduced to the pure consideration of structural equations conditioned on some external latent factor Z. In particular, the initially provided example falls into this class of models. Note that these models however only exhibit transition functions with only self-loops and feature no transitions between states (c.f. discussion on the “Role of Contextual Independencies” at the end of Sec. 3). As MCM are also able to model systems that can transition between meta-causal states (e.g. change relations of the causal graph in relation to changes in the underlying process), MCM form a strict superset of the class of latent conditioned SCM. As you pointed out correctly, we show such a case in the stress-induced fatigue example, where the system system dynamics --the types of the edges-- also depend on the current state of the system, and therefore can not be solely controlled via some external factor $Z$.

Q1 [Variables $\mathbf{X}$] Yes, variables $\mathbf{X}$ are the variables of the SCM. We have clarified this in the paper.

Q2 [Definition of the Type Encoder] Thank you for pointing out this weakness! After reconsidering the definition of our type encoder, we decided to revise said definition to make section 3 more approachable and better defined in general. We are happy to explain further: The type encoder assigns types based on the functional relationship between variables $X_j$ on $X_i$ as induced by the underlying mediation process. In particular the type encoder considers the projection of the transition function onto $X_j$, which we now define as $\varphi_j \circ \sigma$. The type encoder, therefore, is the key binding factor between the meta-causal model and the underlying mediation process. In particular, $\tau_{ij}$ reasons about the relationship between variables $X_j$ and $X_i$ by inspecting properties of $\varphi_j \circ \sigma$.

Consider, in the stress induced fatigue example (Sec. 4.3), we employed an identification function to determine the functional relation of stress $s$ onto itself via the derivative $d(\varphi_s \circ \sigma) / ds$ to be either of reinforcing or suppressing nature. Similar to our next answer (Q3), where indirect edges may constitute a type, general classes of functional relationships, e.g. linear functions could be identified as independent types.

Q3 [Inclusion of Indirect Edges]: Whether to include or not indirect edges is indeed an interesting question and depends on the objective of the user, and ultimately on the deployed identification function. The answer eventually reduces to the question of whether indirect effects should be considered causal effects. Under the assumption that the intermediate variable $Y$ is not part of the SCM we surely want our identification function to include $X \rightarrow Z$, while whenever $Y$ is part of the SCM we rather want to include $X \rightarrow Y, Y \rightarrow Z$ and omit the indirect edge.

One could (and, in general, probably should) define the identification function to only identify direct causal effects. Given our meta-causal framework we, however, decided to leave this freedom to the practitioner. In this case one might assign particular “1-hop indirection” types to the edges - indicating the edge to be the result of an indirect relation via some (possibly unobserved) variable. Consider how this notion of edge types over unobserved variables is already in use in CPDAGs (as e.g. produced by the PC or GES algorithms) where it is used to indicate undirected or confounded edges.

[our reply is continued in the next comment]

Dear Reviewer,
thank you for being back and engaging in the discussion of our paper. Considering your further comments we reconsidered parts of our paper and fixed the mentioned inconsistencies in our formalism. We detail the changes below:

W1 [Meta-Causality]: In your quoted sentence we regard the qualitative behavior of equations to be part of the "process dynamics". You are right, however, that talking about "structural equations" in this context is an incorrect phrasing, since --as you mention correctly--, the same structural equations can exhibit different qualitative behavior. We checked our paper, and found that --while avoiding it elsewhere-- we accidentally introduced this phrasing in our added clarification in the introduction.

Just as there are different definitions of causality, the 'exact' definition of meta-causality will likely depend on the author. However, as we are (probably) first to give such a definition, we orient ourselves on previous definitions of causality. Our paper currently defines types as the "general qualitative behavior that emerges from the interplay of individual equations" and we would like to find a compatible definition for meta-causality. If we consider causality to be "the science of cause and effect", as prominently stated by the Book of Why, meta-causality might be defined as "[the science of] qualitative change in cause-effect behavior".

We have adjusted the previously added sentence in the introduction and use this new definition at the start of section 3. Thank you again for challenging us to reduce our rather vague definition of meta-causality and provide a more condensed version. We would be interested on whether you agree with us on this definition or not.

Q3 [Indirect Edges and Identification Function]: You are right in that our presented formalism, regarding the type encoder (and therefore the identification function), was somehow not properly formalized. Previously, we considered the type-encoder as a part of a Meta-Causal Frame, and therefore to be implicitly be aware the mediation process via its definition. However, in that case it is rather confusing to pass the particular mechanism $X^{\mathcal{S}}_j$ of the variable $X_j$ as an explicit parameter, as it should already be contained in the overall known set of mechanisms $X^{\mathcal{S}}$.

To resolve this inconsistency, we drop the subscript in $\mathcal{X}^S_j$ and explicitly pass all functional relations $\mathcal{X}^S$ to the identification function and the type encoder. In particular, this allows the type decoder to determine whether some a functional relation is mediated by some other (set of) intermediate mechanism(s). In situations where $X_i \rightarrow X_{\mathbf{z}} \rightarrow X_j$, the type encoder needs to identify whether the relation $X_i \rightarrow X_j$ is purely mediated via $\mathcal{X}^S_{\mathbf{z}}$ (meaning that all computational paths for $X_i \rightarrow X_j$ pass through $\mathcal{X}^S_{\mathbf{z}}$ at some point) or some additional direct effect exists that are not 'shielded' by $\mathcal{X}^S_{\mathbf{z}}$. Conversely, whenever some $X_k \in X_\mathbf{z}$ of the intermediate set is dropped or marginalized from the causal variables, the mechanism $\mathcal{X}^S_{k} \subset \mathcal{X}^S_j$ is no longer identified as a subcomponent of the mechanism within the overall $\mathcal{X}^S_j$ and (given that the remaining $\mathcal{X}^S_{\mathbf{z}\setminus \{ k \}}$ might no longer shield $X_i$ from $X_j$) a direct arrow $X_i \rightarrow X_j$ can be inferred.

We adjusted definition 1 and carried the corresponding changes to the text and notation. Additionally, we added this discussion, regarding the relation of all (and possibly intermediate) mechanisms $X^{\mathcal{S}}$ and the role of the specific $X^{\mathcal{S}}_j$, to Appendix B. (We corrected the statement regarding CPDAGs. Thanks!)

[Game of Tag Example]: Thanks for pointing this out. We corrected the right-hand side of the equation to state $t_{B \rightarrow A} = \text{``A Chasing''}$ to now indicate the correct edge.

[continued in the next comment]

[Contextual Independencies]: Yes. We, indeed, meant function composition in the mentioned case (and have adjusted the wording). Based on your feedback and that of the other reviewers, we already revised the corresponding paragraph and equation and [have added the following clarifications to the current revision (Appendix B)]:

Since we construct the conditioned SCM after the particular causal model that exists under a meta-causal state indexed by $z$, we know that a particular composition of functions $f^z_{ij}$ has to exist (and that the functions are compatible), since the full function $f^z_j$ exists for each meta-causal state. Consider that, in a naive approach, the order of composition $i \in \{1..N\}$ enforces a particular evaluation order of the functions, and in particular requires this order to be the same for every meta-causal state.

Generally, the presented function composition might not work, in the case that the individual functions get chosen badly from the start. A simple solution to overcome such problem is to assume compositions of lifted functions and assume their signatures to be compatible (which is always permitted due to the known existence of the composed $f^z_j$). Note that functions $f^z_{ij}, f^{z'}_{ij}$, [new paragraph due to openreview latex formating issues, sorry!]

whose functional type $t_{i j}$ did not change under a change of $z$ to $z'$, might be altered to make signatures compatible.

In the paper we now explicitly state compatibility of the functions and have added this extended discussion to the appendix C.

[Experiments: Causal Effect Direction]: In our experiments we exclusively consider either $X\rightarrow Y$ or $Y \rightarrow X$. However, having a cyclic relation will likewise realize a particular distribution. We assume that this case might still be identifiable as the noise distribution might no longer be Laplacian distributed from either direction, due to the interfering noise from both equations.

[Typos] Thank you for pointing out the specific typos in our paper. We have them fixed. We will perform an more thorough correction pass for the whole paper and in particular pages 8 and following for the camera-ready version.

Thank you once again for your detailed comments and suggestions regarding our paper. We have included all discussed points in the current revision of the paper. While we might not be able to upload further revisions after end-of-day AoE due to the current ICLR policy, we are happy to discuss and clarify any remaining points that might have been left unclear.

Dear Reviewer pnp9,
Thank you for your thoughtful review and for considering our work to be well-written, providing interesting applications, and to provide clear and easy to understand graphs. We will address your concerns in the following:

W1 [Prior Work] Thank you for referencing further relevant prior work that slipped our attention. We agree that the mentioned papers are closely related to the topic of modeling meta-causal function switching via context-independencies and have added a brief discussion to our paper.

W2 [Reduction of MCM to ordinary SCM] We agree with your concerns in the lack of a formal theorem. As already stated in the paper and pointed out by the other reviewers PEUy and puHX, meta-SCM might --under certain conditions-- be reduced to ordinary SCM. Note that the Stress-Induced Fatigue example (Sec. 4.3) provides a counter-example where a reduction to a conditioned SCM is not possible for the reasons discussed in the paper. We are delighted to provide and have proven a theorem stating sufficient conditions for when a reduction of MCM to standard SCM is possible. Roughly speaking, a reduction to SCM is possible whenever all transitions of the meta-causal model are self-loops. Generally, this is the case when the types of structural equations are determined by their functional form and do not change dynamically, based on the current (value) state of the system. We have revised Equation 1 and due to space constraints, provide the corresponding theorem and its proof in Appendix C.

Questions/W3 [Utility of Meta SCM / Errors of inferences of conventional causal models] As the main focus of our initial work on meta-causal models lies in providing a first formal definition of meta-causal models, we tried to approach MCM from a spectrum of different theoretical perspectives. As a response to your feedback, we will discuss possible practical applications in the following and have added this discussion in Appendix G to the paper.

Regarding possible errors of conventional causal model inference, consider the motivational example where classical and meta-causal attributions come to different conclusions; with conventional inference stating that B was the root-cause of A following it. It is not so much that one or the other perspective is per se ‘incorrect’, but that meta-causal analysis allows reasoning over structure dynamics of a system, while conventional causal analyses might be bound to the respective current graph structure.

Health and Medicine. We would like to expand on our stress-induced fatigue example as it can not be reduced to a standard SCM, and provide an actionable perspective on MCM. While we still assume the underlying neuropsychological process to be more complex, with multiple interplaying factors to influence each other, we consider the same simplified model as presented in the paper. We additionally assume that some drug exists that is able to influence certain health related processes within the patient, such that the underlying --previously self-reinforcing-- stress relation is unconditionally changed to a suppressing one. (Upon closer consideration, the previous intervention might constitute a meta-causal do-operator, as we detach the functional type from the underlying dynamics and fix its functional type.)

This perspective not only allows the forecast of system changes, but also yields an actionable model which can be actively steered between meta-causal states. To permanently treat a patient, one could consider the objective to reach a self-stabilizing meta-causal state. Note how this meta-causal objective might be different to that of a classical causal one, where stress levels would similarly be reduced, but no attention is placed on the (possibly unchanged) system dynamics, such that stress levels might rise up again after the intervention ends.

Economics. Recall that our MCMs are defined as finite state machines. Figuring out exact transition conditions that induce meta-state transitions also yield important insights on the volatility/stability of systems in terms of risk analysis and policy making. Such scenarios might commonly arise in economics, where relations in markets can change due to the sudden appearance of disrupting factors (e.g. a new competitor entering the market or a financial crisis) while effects might persist even with the disrupting factor having vanished.

While we are only able to briefly discuss possible areas of application in this paper, we believe that applications of MCM are plentiful and are keen to expand applications in follow up works.

We have uploaded a revised version of the paper, with changes color coded, for which we hope to have clarified most of your stated concerns. Your color code is blue. Please let us know if there is any additional information we can provide to further clarify our contributions. Thank you once again for your time and effort in reviewing our paper!

Sincerely,
Authors