A phenomenological account for causality in terms of elementary actions

1.1 Notation and terminology

Random variables will be denoted by capital letters like X , Y and their values by lower case letters x , y . Further, calligraphy letters like X , Y will denote the range of variables X , Y . Causal Bayesian networks (CBNs) or functional causal models (FCMs) [2,3] will be our key framework for further discussions. Both concepts describe causal relations between random variables X 1 , … , X n via directed acyclic graphs (DAGs). According to the causal Markov condition, the respective causal DAG G is compatible with any joint density that factorizes^[3] according to

where p a j denotes the values of the parents P A j of X j in G . Then, the CBN is given by a DAG G together with the Markov kernels^[4] p ( x j ∣ p a j ) .

FCMs, in contrast, provide a deterministic model of the joint distribution, in which every node is a function of its parents and an unobserved noise variable N j :

where all N 1 , … , N n are statistically independent. Both frameworks, CBNs and FCMs, admit the derivation of interventional probabilities, e.g. the change of the joint distribution after setting variables to fixed values. While CBNs only provide statements on how probabilities change by an intervention, FCMs, in addition, tell us how the intervention affected each individual statistical unit and counterfactual causal statements [3].

1.2 Interventions

To briefly review different types of interventions, note that the point intervention d o ( X j = x j ) [3], also called “hard intervention,” adjusts the variable to x j , while generalized (also called “soft”) interventions on X j replace p ( x j ∣ p a j ) with a different conditional p ˜ ( x j ∣ p a j ) or the FCM (2) with a modification X j = f ˜ j ( P A j , N ˜ j ) , see the study by Peters et al. [4], page 89, and references therein. Structure-preserving interventions [11] preserve all the dependences on the parents by either operating on the noise N j or adjusting X j to the parent-dependent value f j ( p a j , N j ′ ) , where N j ′ is an independent copy of N j that is generated by the experimenter. The simple observation that any of these interventions on X j affect only X j and a subset of its descendants defines a consistency condition between a hypothetical G and the hypothesis that an action is an intervention on X j .

While the aforementioned framework has provided a powerful language for a wide range of causal problems, it implicitly requires the following two questions to be clarified:

The second question reads:

Note that the word “only” in Question 2 is meant in the sense that the action intervenes on none of the other variables under consideration directly, it only affects them in their role of descendants of X j , if they are.

Question 1 captures what is often called “causal representation learning” [12,13], while this article will mainly focus on Question 2 only although we believe that future research may not consider them as separate questions for several reasons. First, because some definitions of variables may seem more meaningful than others because they admit more natural definitions of interventions. Second, because different parameterizations of the space results in different consistency conditions between variables which may or may not be interpreted as causal interactions between them.

Following the study by Woodward [14], page 98, one may define interventions on X j as actions that affect only X j and its descendants, but this way one runs into the circularity of referring to the causal structure for defining interventions although one would like to define causality via interventions, see the study by Baumgartner [15] for a discussion. The idea of this article is to define causal directions between a set of variables by first defining a set of elementary transformations acting on the system, which are later thought of being interventions on one of the variables only. While they can be concatenated to more complex transformations, defining which transformations are elementary, defines the causal direction. While the notion of interventions often appears to be a primary concept in the graphical model based framework of causal inference, this article will tentatively describe a notion of intervention as a secondary concept derived from an implicit or explicit notion of complexity of actions.

1.3 Structure of the article

Section 2 argues that there are domains (mostly technical devices) in which an action can be identified as an intervention on a certain variable via analyzing the “hardware” of the respective system. In contrast, Section 3 describes a few scenarios with ill-defined causal relations to highlight the limitations of the “hardware analysis” approach, without claiming that our proposal will offer a solution to all of them. Section 4 argues that the idea that some operations are more elementary than others is already necessary to make sense of one version to read the principle of independent mechanisms. Based on this insight, Section 5, which is the main part of the article, describes how to define causal directions via declaring a set of transformations as elementary and illustrates this idea for examples of “phenomenological” causality where the causal directions are debatable and paradoxical. Section 6 shows that phenomenological causality appears more natural when the system under consideration is not considered in isolation, but in the context of further variables in the world. Then, the joint system can be described by a DAG that is consistent with phenomenological causality (Subsection 6.1). Further, the notion of “elementary” also satisfies a certain consistency condition with respect to such an extension (Subsection 6.2).

Sections 2 and 3 are not strictly required for understanding the main ideas of this article and can be skipped by readers familiar with the problem of ill-defined causal relations. Section 2 starts with a hypothetical technical toy system in which causal relations and interventions are tangible due to their manifestation as wires that transmit physical signals. Further, it describes a second toy system, in which causal relations get less obvious although the system is actually tangible, which can be seen as a metaphor and motivation for Section 3, which is a collection of scenarios that can render causal relations ill-defined.

3.1 Coupling induced by “coordinatization” of the world

For some person, let us define the variables W , E , S , O describing the hours he/she spends for work, exercises, sleep, and others at some day. These variables satisfy the equation

a constraint which results in statistical dependences when observing the variables over several days. According to Reichenbach’s principle of common cause [19], each dependence is due to some causal relation, either the variables influence each other or they are influenced by common causes. Since one may be tempted to explain the statistical dependence by a common cause, let us introduce a hidden variables the person’s decision influencing all four variables, as shown in Figure 3. However, this DAG suggests that independent interventions were possible. Obviously, the intervention of setting all four variables to 7 h is prohibited by (3). Likewise, it is not possible to intervene on W only without affecting the other variables. While these remarks seem obvious, they raise serious problems for defining downstream impact of the above variables, since it is impossible to isolate, for instance, the health impact of increasing W from the health impact of the implied reduction of E , S , or O . Constraints that prohibit variable combinations a priori (by definition), rather than being a result of a mechanism (which could be replaced by others), are not part of the usual causal framework. The fact that independent changes of the variables are impossible also entails the problem of interventions being an ill-defined concept: What qualifies a change of life-style which increased E and decreased W as an intervention on E ? Is it a question of the intention? If the intention to do more exercises entailed the reduction of work we tend to talk about an intervention on E , but given that people do not always understand even their own motivation behind their actions, it seems problematic that the definition of an intervention then depends on these speculations. Examples like these have motivated the development of more general causal frameworks [20] that deal with such constraints. We will elaborate on the link to our approach in Subsection 5.2.

Figure 3

The hours spent with certain “activities” like work, exercise, sleep, and others are determined by a person’s decision.

3.2 Dynamical systems

Constraints on variables like the ones mentioned earlier result naturally if we do not think of the state of the world as a priori given in terms of variables. Let us, instead, consider a model where the admissible states of the world are given as points in a topological space (in analogy to the phase space of a classical dynamical system [21]), in which variables arise from introducing coordinates, as visualized in Figure 4.

Figure 4

A model of the world where the current state is a point in a topological space, that moves from point p t to p t + Δ t .

Here, the ellipse defines the admissible states of the world, and we have introduced a coordinate system whose coordinates define the two random variables X 1 and X 2 . The states of our toy world are formally given by the set of all those pairs ( x 1 , x 2 ) that belong to the ellipse E. We assume that the dynamics of our toy world is given by a topological dynamics of the ellipse, that is, a continuous map that maps each state p t ∈ E ⊂ R 2 to its time evolved state p t + Δ t . The fact that the state cannot leave the ellipse entails a relation between the variables X 1 and X 2 of which it is unclear whether we should consider it a causal relation. We will therefore discuss how to define the notion of “interventions” on X 1 and X 2 .

It is natural to interpret the intervention set X 1 to x ˜ 1 as an operation that maps a state ( x 1 , x 2 ) to ( x ˜ 1 , x 2 ) . This is, however, only possible if ( x ˜ 1 , x 2 ) is still a point in the ellipse. Otherwise, one is forced to change x 2 to some other value x ˜ 2 to still end up in an admissible state of the world. Could we then say that X 2 changed because X 1 changed? This interpretation may be valid if someone’s intention was to change X 1 , who is then forced to also change X 2 by the existing constraints. However, without this knowledge about what has been the intention of the actor, we just observe that both variables have been changed. Interpreting the action as an intervention on X 1 becomes questionable. Our simple model of the world already suggests two different notions of causality:

1. Causality between time-like measurements. Here, we consider the variables X t 1 , X t 2 as causes of X t + Δ t 1 , X t + Δ t 2 .

2. Causality without clear time separation. Here, causal relations between the coordinates appear as a phenomenon emerging from consistency conditions that define the admissible states of the world.

where H encodes the relevant feature of the state ( X t − 1 1 , X t − 1 2 ) . However, this DAG suggests the existence of independent interventions on X t 1 and X t 2 , although the constraints given by the ellipse need to be respected by any action (similar to our remarks on Figure 3). This fact would rather be covered by a causal chain graph containing an undirected link X t 1 − X t 2 , as discussed in the study by Lauritzen and Richardson [24] for mutual interaction in equilibrium states. If we think of a force in the direction of the x 1 -axis and observe that the point moves also in x 2 -direction once it reaches the boundary, we would certainly consider X 1 the cause and X 2 the effect. However, once the boundary is reached, it is no longer visible that it’s a force in x 1 -direction that drives the curved motion. Accordingly, the causal direction becomes opaque.

The second case is more interesting for this article since it refers to a notion of causality that is less understood. At the same time, it challenges the interpretation of causal directions as a concept that is necessarily coupled to time order. Instead, this kind of phenomenological causality can also emerge from relations that are not given by the standard physical view on causality dealing with signals that propagate through space–time from a sender to a receiver^[7] (such a physically local concept of causality admits defining interventions on a variable X j as actions for which a signal from the actor reaches the location of X j ). There is no analogous approach for phenomenological causality, since it may be too abstract to be localized in space–time. For these reasons, Question 2 should be particularly raised for causality in domains outside physics when referring to sufficiently abstract variables. However, variables X 1 and X 2 referring to different coordinate of the same system can also have an abstract causal relation.^[8]

3.3 Undetectable confounding: distinction between a cause and its witness

To distinguish between the scenarios X → Y and X ← X ˜ → Y , where X ˜ is latent, is one of the most relevant and challenging problems. Methods have been proposed that address this task from passively observing P ( X , Y ) subject to strong assumptions, e.g. [26,27] or in scenarios where X , Y are embedded in a larger network of variables, e.g. the fast causal inference algorithm [2] or instrumental variable based techniques [28] and related approaches [29]. Certainly, the task gets arbitrarily hard when the effect of X ˜ on X gets less and less noisy, see Figure 5, left. In the limiting case where X is an exact copy of X ˜ , no algorithm can ever tell the difference between the two scenarios in Figure 5 from passive observations. Note that this scenario is quite common when X ˜ is some physical quantity and X , the value of X ˜ shown by a precise measurement device. In this case, we would usually not even verbally distinguish between X and X ˜ although it certainly matters whether an intervention acts on X or X ˜ . The distinction gets only relevant when we act on the system, and even then, only if actions are available that decouple X from X ˜ . Accordingly, it is the set of available actions that defines causality, which is the crucial idea presented in Section 5.

Figure 5

If X is an arbitrarily perfect copy of the confounder X ˜ (e.g. if X is a reasonably good measurement of the true quantity X ˜ ), distinction between the left and the right scenario gets arbitrarily hard.

A similar ill-definedness occurs in scene understanding in computer vision: Imagine a picture of a dog snapping for a sausage. Most people would agree that the sausage is the cause for the presence of the dog (unless the scene is taken from a location where the dog commonly stays anyway, e.g. its doghouse). However, the presence of the sausage on the image is certainly not the cause for the presence of the dog on the image – changing the respective pixels in the image will not affect the pixels of the dog. Similar to the measurement device, we can only meaningfully talk about causal relations if we do not distinguish between the cause “presence of the sausage” and its “witness,” e.g. its occurrence on the image. Ignoring actions that decouple the presence of an object in the real scene from its appearance on the image (retouching), we may also consider the presence of the sausage on the image the cause of the presence of the dog on the image and call this causal relation “phenomenological.”

3.4 Coarse-grained variables

If coarse-grained (“macrosocopic”) variables are defined, for instance, by averaging over “microscopic” variables, interventions on the former are usually no longer well-defined since different changes on the micro-level amount to the same change of the macro-variable [30]. Thus, causal relations between macro-variable may be ill-defined. Accordingly, Rubenstein et al. [31] state a consistency condition for coarse-grainings of causal structures according to which interventions on micro-variables that result on the same intervention on the macro-variables also entail the same downstream impact on other macro-variables. Beckers and Halpern [32] argue that different strengths of consistency conditions are needed for different levels of abstraction. We believe that discussions on causal relations in real life often refer to macro-variables for which the impact of interventions does highly depend on how the intervention is implemented, and it is thus hard to identify any valid consistency condition, however weak it may be. Let us consider the following toy model. Given the variables X 1 , X 2 , Y 1 , Y 2 , with the causal DAG shown in (4), where X 1 influences Y 1 and Y 2 influences X 2 :

We assume FCMs

and define the macro-variables X ¯ ≔ ( X 1 + X 2 ) ∕ 2 and Y ¯ ≔ ( Y 1 + Y 2 ) ∕ 2 , whose causal relations we want to discuss. Obviously, neither an intervention on X ¯ nor on Y ¯ is well defined. Increasing X ¯ by Δ can be done by adding any vector of the form ( Δ + c , − c ) with c ∈ R , and likewise for intervention on Y ¯ . Someone who changes X ¯ and Y ¯ by changing X 1 and Y 1 , respectively, will claim that X ¯ influences Y ¯ , while someone changing the macro-variables by acting on X 2 and Y 2 only considers Y ¯ the cause of X ¯ . This suggests that not only the quantitative effect but also the causal direction is not a property of the system alone (even after specifying the coarse-graining), but a result of which actions are available.

3.5 Diversity of non-equivalent interventions

In the framework of FCMs and graphical causal models, the impact of interventions (e.g. point interventions do ( X j = x j ) ) does not depend on how this intervention has been performed. Here, the word “how” is meant in the sense of which mechanism has been used to change X j . The description of the mechanism implementing the intervention is not part of the description, which is because the framework implies that different ways of setting X j to x j have the same impact. In real-world problems, however, we talk about impact of one variable on another one without specifying the model we refer to, which renders impact of interventions ill defined. Let us elaborate on this in the context of stochastic processes. Let ( X t ) t be a time series of the electricity consumption of a household, where each value is the integral over 1 h. A simple action to reduce X t at some t could be to convince the resident not to start his/her dish washer at this point in time. Whether this action causes a change of X s at some later time s > t depends on whether the resident decides to clean the dishes by hand or to just delay the start of the machine. Likewise, the impact of changing the traffic of some road depends on how it is performed. In a scenario where the road is made less attractive by a strong speed limit, drivers may take a different route and thus increase the traffic of other roads. Reducing the traffic by offering additional public transport would not have the same effect. Again, one option to disambiguate is to specify the action that defines the causal impact and challenge the belief in absolute causal truth that holds without this additional specification.^[9]

3.6 Abstract causal mechanisms

While the previous subsection described difficulties with the concept of causality that already arise in clearly defined physical systems, we now discuss an example where the causal mechanisms lie in a more abstract domain. Examples of causal relations for which we believe the simple mechanistic view of causality to be problematic are widespread in the literature. The study by Schölkopf et al. [13], for instance, describes an scenario of online shopping where a laptop is recommended to a customer who orders a laptop rucksack. The authors argue that this would be odd because the customer probably has a laptop already. Further, they add the causal interpretation that buying the laptop is the cause of buying the laptop rucksack. We do agree to the causal order but do not believe the time order of the purchases to be the right argument. For someone who buys laptop and laptop rucksack in different shops, it can be reasonable to buy the rucksack first to safely carry the laptop home (given that he/she already decided on the size of the laptop). We believe, instead, that the recommendation is odd because the decision to purchase the laptop is the cause of the decision to buy the rucksack. Whether this necessarily implies that the decision has been made earlier is a difficult question of brain research. If they were made in two well-localized, slightly different positions in the brain, one could, again, argue that causal influence can only be propagated via a physical signal (of finite speed). We do not want to further elaborate on this question. The remarks were only intended to show that causal problems in everyday business processes refer to rather abstract notions of causality – for instance, to causality between mental states. Particularly in these domains, causality seems highly context dependent.

Economic variables often show several of the aforementioned aspects of ill-definedness coming from aggregation or psychological factors or both. For example, the price of a particular good is not only understood as the price at which one firm sells that good (unless we are in a monopoly market) but instead as an aggregation of prices. Likewise, consumer confidence indices are an aggregation of the beliefs of individual agents. Economic indices tend to be more abstract than their name might suggest.

5.1 The bivariate case

To avoid the above circularity of defining interventions in a way that relies on the concept of causality and the other way round, we suggest the following approach:

Since Idea 1 is quite informal, it leaves some room for different interpretations. We will work with two different ways of spelling it out:

Here and throughout the article, we assume actions to be implemented by external actors and not influenced by the state of the system. Further, we assume acyclic causal structures and exclude confounding.

Note that m in Definition 2 holds for all actions in A 1 , but different functions m hold for different statistical units. If we think of X and Y as related by the FCM Y = f ( X , N ) , we should think of m as the map f ( . , n ) with fixed noise value n . The statement that actions in A 1 do not change the mapping from X and Y thus refers to the counterfactual knowledge encoded by the FCM, which we assume to be given from domain knowledge about the system^[10]. Further note that changing the map m can be done by either changing f or n . Although the condition for A 1 is asymmetric with respect to swapping X and Y since m maps from X to Y , this asymmetry does not necessarily imply the mapping m to be causal. Assume, for instance, X and Y are related by the FCM Y = X + N . Then, an observed pair ( x , y ) for which N = 3 will obey the rule y = x + 3 and y ′ = x ′ + 3 for all pairs generated by actions in A 1 . However, all these pairs will also obey the rule x ′ = y ′ − 3 , and thus, there exists also a map m ˜ from Y to X . In our following examples, the crucial asymmetry between cause and effect will often not be induced by the existence of m (since a function mapping y to x may also exist), but by the existence of actions A 2 , which only act on the effect. In other words, interventions on the cause do not reveal the asymmetry because they change cause and effect, while actions on the effect only change the effect. The idea that actions change equations is just rephrasing the usual interpretation of FCMs as modular components of a system [3]. This view is even more explicit in the study by Spirtes and Scheines [20], which builds upon the study by Simon et al. [38], where interventions are not a priori thought of acting on variables. Instead, interventions act on equations, which are not even written as assignments (with the effect on the left-hand side), and the causal order then emerges from the direction in which the system of equation is solved. We will comment more on this relation at the end of Section 5.

As an aside, note that for the scenario where X and Y are only connected by a confounder, we would postulate actions that affect only X and those that affect only Y .

The idea of identifying causal structure by observing which conditionals in (1) change independently across datasets can already be found in the literature, e.g. the study by Zhang et al. [39] and references therein. In the same spirit, Definitions 1 and 2 raise the question whether they define the causal direction uniquely. This is easier to discuss for Definition 1. Generically, changes of P ( X ) results in simultaneous changes of both P ( Y ) and P ( X ∣ Y ) , which thus ensures that the available actions of class A 1 neither fall into the category corresponding to A 1 nor A 2 for the backwards direction from Y → X . The following simple result shows a genericity assumption for which this can be proven:

Abstract toy example. We now describe an example for unit level phenomenological causality according to Definition 2 where the causal direction is a priori undefined, but may be defined after specifying the set of actions, if one is willing to follow our approach. We have purposefully chosen an example whose causal interpretation seems a bit artificial.

Figure 7

Urn model in Example 2 with four different operations.

The example may be considered as representing a chemical process with two isomers, that is, molecules that are structurally different, although they have the same molecular formula. Isomer “red” can be converted into isomer “blue” and vice versa, catalyzed by appropriate experimental conditions [40]. Then, the “red” ones are the resource for a reaction that converts “red” into “blue.” Therefore, one may be surprised that not the amount of the resource K r , but the amount of the product K b of the reaction, is the cause.

To come up with another real-life example that resembles that mentioned earlier, let each red ball represent 1 Euro in a transactions account and a red ball be 1 Euro in the savings account. The actions correspond with typical actions that a bank account holder has: add/remove money from the transactions account, and move money from transactions to savings account or vice versa. We conclude that the phenomenological causal relation between the two is that the amount of money on the savings account causes the amount of money on the transactions account.^[11]

We now rephrase Example 2 into an example for the statistical version in Definition 1. To this end, we consider a random experiment for which the system is initially in the state K r = k r and K b = k b with k r , k b ≫ 0 . Then, in each round, we flip a coin for each of the four actions to decide whether they are applied. After ℓ < k r , k b many rounds, let N 1 denote the number of times A 1 + minus the number of times A 1 − has been applied. Likewise, N 2 counts the number of A 2 + minus the number of A 2 − actions. We then obtain

One easily checks that (5) and (6) are equivalent to

Since the actions are controlled by independent coin flips, we have N 1 ⊥ ⊥ N 2 . Following our interpretation that K b causes K r and A 1 ± and A 2 ± are interventions on K b and K r , respectively, we thus consider (7) and (8) as the corresponding FCM. By controlling actions A 1 ± and A 2 ± via coins with different bias, we may change the distributions P ( K r ∣ K b ) and P ( K b ) independently, and thus have an example of statistical phenomenological causality in Definition 1.

The following observation may seem paradoxical at first glance: The set A 1 = { A 1 + , A 1 − } is a priori symmetric with respect to swapping the roles of blue and red. The justification for calling it interventions on K b is derived from properties of A 2 = { A 2 + , A 2 − } . In other words, whether an action is considered an intervention on a certain variable depends on the impact of the other actions in the set of elementary actions. This context dependence of the definition of interventions may be worrisome, but in scenarios where “hardware-analysis” does not reveal (or define) whether an action is an intervention on a particular variable, we do not see a chance that circumvents this dependence on other actions. Given the abstractness of the underlying notion of causal direction, we are glad to observe that the well-known causal discovery method LiNGAM [41] would also infer K b → K r because (7) and (8) define a linear model with non-Gaussian additive noise. The crucial assumption inducing the statistical asymmetry is that actions A 1 ± are implemented independently of A 2 ± , resulting in independent noise variables N 1 , N 2 .

There is also another aspect of this example that shows the abstractness of the causal interpretation of the above scenario. The fact that actions in A 1 preserve the total number of balls has been interpreted as structural equation (8) generating K r from K b . Using the function m from Definition 2, this structural equation reads K r = m ( K b ) with m ( K b ) = − K b + k b + k r + N 2 . Since operations in A 2 change the total number of balls, they change m to m ′ by changing k b . Hence, actions in A 2 change the “mechanism” relating K b and K r . In a mechanistic interpretation of causality, one would expect changes of a mechanism a change of a kind of machine where the input–output behaviour is changed. As abstract as the “mechanism” from K b to K r is, as abstract is its change.

Context-dependent causal directions Here, we describe a system for which causal directions swap when the system moves from one regime to another one. Although the following example is hypothetical, we encourage the reader to think of similar example in realistic business processes. While context dependence of causal directions is not a novel insight [20,42,43], we describe the example to show that it is here due to the context dependent impact of elementary actions.

Figure 8

In scenario 1, the food consumption per rabbit is causing the total food consumption because changing the number of rabbits only changes the latter if there is enough food for each rabbit. In scenario 2 with food shortage, changing the number of rabbits does not affect the total food consumption, and it only changes the food consumption per rabbit. Therefore, the total food consumption is the cause. Images by Tom Paolini (carrots), michealcopley03 (single carrot), Aswathy (rabits) with unsplash license.

The aforementioned dependence of the causal direction on the regime suggests that there exists a large grey zone without clearly defined causal direction. We want to discuss this for the following business relevant example, where we will not offer a clear answer.

Example 4 shows that causal directions can also be in the grey zones because actions can be in the grey zone of being an intervention on one versus the other quantity. Acting on the factor F will in general change both variables R and Q . However, in the regime where one of it is relatively robust to changes of F , we can consider this one as the cause and consider changing F as an intervention on the effect because it affects the mechanism relating cause and effect. It is likely that many causal relations in real life show equally much room for interpretations.

Let us now revisit the example from the domain of product recommendation algorithms described in the study by Schölkopf et al. [13]:

Here, we have neglected that seeing laptop rucksacks may remind some customers that they were planning to buy a laptop already since a while, which could induce additional demand for laptops. Further, a marketing campaign changing P ( X ) and P ( Y ∣ X ) could certainly exist, e.g. one that explicitly advertises laptop and rucksack as an economically priced pair. We are thus aware of the fact that any causal statement in this vague domain of customer psychology is a good approximation at best. After all, calling the aforementioned operations “elementary” comes from the idea that they appear quite natural to us, but we admit that this view is debatable. When previously mentioning the scenario of Example 5 in the introduction, we have emphasized that the time order of customer’s purchases does not determine the causal order of the underlying decisions for the purchases. This already suggests that analyzing the causal order of the underlying materialized processes does not reveal the causal structure of their psychologic origin. The following example elaborates on this discrepancy of phenomenological causal direction and causal direction of underlying micro-processes.

5.2 The multivariate case

We first generalize Definition 1 to multiple variables. Although these generalizations are straightforward, we will see that our multivariate extensions of the urn example reveal the abstractness of phenomenological causality even more.

Note that Definition 3 may admit a large number of DAGs. However, there are cases where it uniquely determines a causal order, for instance if all actions in A j change the distributions of X i for all i ≥ j . Given the causal order, the causal DAG can be determined subject to minimality^[12]

Likewise, we generalize Definition 2:

We now generalize Example 2 to n different balls, where the causal structure suggested by our definition gets even less obvious:

Figure 9

Causal relation between the variables K j , which count the number of balls in the urn with label j (according to our definition of phenomenological causal structure).

Note that the algebraic transformations between (15) and (17) resemble the algebra in independent component analysis (ICA)-based multivariate causal discovery [45] (following the idea of LiNGAM [41] mentioned for the bivariate case mentioned earlier). This analogy is not a coincidence: ICA decomposes the vector K into independent noise variables N . Accordingly, since (16) is a linear acyclic causal model with independent non-Gaussian noise variables N j , multivariate LiNGAM would also identify the same causal structure and FCMs that we derived as phenomenological causal model. In other words, if we ensure that the choice of the actions is controlled by random generators, independently across different A j , we obtain a joint distribution P ( K 1 , … , K n ) for which the causal discovery algorithm LiNGAM infers the DAG in Figure 9.

It is instructive to discuss Example 7 from the perspective of complexity of some actions that are not elementary. Increasing K j without affecting the others requires j operations, e.g. one can first increase K 1 and propagate this increase to K j . From the causal perspective, these actions are necessary to compensate the impact of K j on its child.

A further remark on causal faithfulness [2]. The fact that an intervention only propagates to the child, but not to the grandchild shows that the structural equations are non-generic; direct and indirect influence of K j on K j − 2 compensate. Accordingly, if we control each action by independent coin flips as in the remarks after Example 2, the induced joint distribution will not be faithful to the causal DAG. The idea of “nature choosing each mechanism p ( x j ∣ p a j ) in (1) independently” seems to have its limitation here. The reason is that the actions A j ± are the building blocks of the system, rather than the Markov kernels p ( x j ∣ p a j ) , which are constructed from the former. There is also another “paradox” of our causal interpretation that becomes apparent for n > 2 , while it seems less paradoxical in Example 2: imagine what happened if we were to redefine A 0 ± as adding or removing of balls of type n instead of type 1. We would then reverse all the arrows in Figure 9. In other words, the direction of the arrows in a long chain of variables depends on the set of available actions at the end points.^[13] This idea is in stark contradiction to the spirit of modularity [46] assuming each p ( x j ∣ p a j ) is an independent mechanism of nature. The reader may see this as an indicator against interpreting the equations (18) as FCMs, but we think that causal directions on the phenomenological level may well depend on this kind of context.^[14] One option to restore modularity is to consider the equations themselves as mechanisms (for the cost of having no direction a priori), as suggested in the study by Blom et al. [20].

In Example 7, the locality of the impact of each of the actions A j ± itself (affecting only two adjacent variables) entailed long-range causal influence between the variables. Now we will describe the opposite where actions affecting a large number of variables is induced by only local causal connections (in other words: in the first example, S has only entries in the first off-diagonal, in the case following now this is true for A ).

Figure 10

Urn containing packages of n different types, where type P j contains balls will label 1 , … , j . The variable K j counts the number of balls with symbol j in the urn. The elementary operations of the system are adding one package from the stack to the urn or put it back. Changing K j thus entails the same change for K j − 1 , … , K 1 .

Figure 11

Causal relation between the variables K j , which count the number of balls in the urn with label j (according to our definition of phenomenological causal structure).

What’s the purpose of the causal interpretation? The balls in the urn show an extreme case where the causal interpretation is far away from any “mechanistic view” of causality where the functions m j from Definition 4 refers to tangible mechanisms (recall, for instance, that A j in Example 7 were symmetric with respect to swapping j and j − 1 , yet we have identified them as interventions on K j , not on K j − 1 ). To argue that our causal interpretation is not just a weird artefact of our concept, we need to show its benefit. The following section will argue that this extension of causality allows us to consistently talk about the overall causal DAG when systems with “phenomenological causality” are embedded in systems with more tangible causal structure.

Related ideas in the literature. While the idea that causal conditionals p ( x j ∣ p a j ) and structural equations define “modules” in a causal DAG which can be manipulated independently has a long tradition, we have turned it somehow around by defining these manipulations as the primary concept and the causal DAG (in case the set of elementary actions correspond to a DAG, see below) as a derived concept. The closest work to this idea seems to be the study by Blom et al. [20], where DAGs also appear as derived concepts rather than being primary. The idea is to start with a set of equations, where each can contain containing endogenous variables as well as exogenous ones. Subject to certain conditions (phrased in terms of matchings in bipartite graphs), one can uniquely solve the equations to express the endogenous variables in terms of the exogenous ones using Simon’s ordering algorithm [38]. Remarkably, general interventions in the study by Blom et al. [20] are thought to act on equations rather than variables. Note that in the usual view of causality, an action that changes the structural equation X j = f j ( P A j , N j ) to some different equation X j = f ˜ j ( P A j , N j ) is considered an intervention on X j only because the equation is read as a “structural equation” (or “assignment”) for X j , rather than for any of the parents or the noise N j . In previous studies [20,38], an equation is not a priori considered an assignment for a certain variable, but only later after analyzing the direction in which the system of equations is solved. We now argue that our view complements this view. To this end, we consider the following set of equations:

Assume N 1 and N 2 are independent, as in unconfounded structural equations. When interpreted as purely mathematical equations, they are equivalent to

when we define appropriate noise variables N ˜ 1 , N ˜ 2 . For Gaussian N 1 , N 2 , they can still be independent if α ˜ is properly chosen. The crucial questions now read: First, what is the justification for reading equations (22) and (23) as causal mechanisms as opposed to (24) and (25)? Second, what is the justification to prefer N 1 , N 2 as exogenous noise instead of N ˜ 1 , N ˜ 2 ? To start with an unsatisfactory answer, one could say that N 1 , N 2 (or, more precisely P ( N 1 ) , P ( N 2 ) ) or the equations vary independently across environments, while N ˜ 1 , N ˜ 2 would not, but this raises the question which notion of independence such a statement should refer to. If it is meant to be statistical independence, we would be concerned about building causal semantics on a somewhat arbitrary probability distribution over environments.

A different answer is given by Blom et al. [20] building on the study by Simon [38], where causal order comes from a set of equations which need to be solved in a certain order such that every variable is uniquely determined by previous ones up to a exogenous noise. Although each single equation does not show an asymmetry between cause and effect because it is a priori not written as an assignment with effect on the left-hand side (in contrast to our aforementioned equations), the asymmetry is encoded in the system of equations, where actions change only one equation (or one noise variable, see footnote 4 in the study by Blom et al. [20]) in the set.^[15] However, we could concatenate this type of intervention also in a way that they change only one equation of the derived equations (24) and (25), or one of the noise terms in the latter. Therefore, we assume that the preference for any among these equivalent set of equations in the study by Blom et al. [20] comes from an implicit notion of elementary versus concatenated actions.

To make this unified view on the study by Blom et al. [20] and ours explicit for Example 2, we can justify our choice of exogenous variables as follows: we defined K b (or its increase N 1 ) to be the first exogenous variable and K b + K r (or its increase N 2 ) to be the second exogenous noise, because then our elementary actions impact only one of the noise variables. Choosing K r as the first exogenous term and K b + K r , for instance, would have our elementary actions let influence both.

The question where this notion of complexity of actions come from goes beyond the scope of this article. We hope that examples like Ex.1 showed that in real-life scenarios there are reasons to consider some actions as obviously more elementary than others. Further, we refer to the appendix where we argue that complexity of actions can be subject of scientific research and mention some approaches from modern physics.

Technically, the present article is also related to the work on causal discovery via interventions with unknown interventions targets [48] and joint causal inference (JCI) from multiple contexts [49], which both amount to assuming Markov properties and faithfulness in an augmented graph containing a “context variable” C [49] (also called “utility F-node” [48], “regime indicator variable” [17]) representing the action, change of context, or experimental conditions. We have avoided the assumption of faithfulness on purpose. While it can be helpful for causal discovery, we would not consider it fundamental enough to build a definition of causality on it, given that its validity can be questioned [34]. In a nutshell, the JCI approach of [49] requires faithfulness because it does not have the notion of elementary actions. Let us explain this for the toy example C → X 1 → X 2 . Different contexts that change only P ( X 1 ) reveal the arrow X 1 → X 2 because of faithfulness. Otherwise, we could assume X 2 → X 1 and C influencing both such that X 2 ⊥ ⊥ C ∣ X 1 holds. This can be excluded in our setting because our notion of elementary corresponds to a context variable that is connected to only one target node. This way, we do not need to add faithfulness or restrictions of model classes as in causal discovery. Note, for instance, that Definition 3 does not imply that actions in A j influence every descendant of j in G (likewise for Definition 3). Accordingly, Definitions 3 and 3 will not specify the DAG uniquely.

6.1 Markov property of phenomenological causality

Let us first consider the mechanisms described by functions m j in Definition 4. Since they represent structural equations f j ( . , n j ) with fixed noise value n j , we will denote them with superscript and write m j n j . Whenever the noise values n j are statistically independent across different statistical units, they induce a joint distribution that is Markovian with respect to G , see the study by Pearl [3], Theorem 1.4.1. We conclude that we obtain a Markovian joint distribution of P ( X 1 , … , X n ) whenever we control actions in A j by independent random variables. The same holds true when we control the actions in Definition 3 by independent random variables and introduce formal random variables Θ j controlling the causal conditionals p θ j ( x j ∣ p a j ) (resembling the context variables in [49], for instance).

Then the joint distribution