Causality in cognitive neuroscience: concepts, challenges, and distributional robustness

We consider the following simplified examples. Assume that the consumption of alcohol affects reaction times in a cognitive task. In a randomised controlled trial we find that drinking alcoholic (versus non-alcoholic) beer results in slowed reaction times hereinafter. Therefore, we may write ‘$\text{alcohol}\to\text{reaction time}$’ and call alcohol a cause of reaction time and reaction time an effect of alcohol. Intervening on the cause results in a change in the distribution of the effect. In our example, prohibiting the consumption of any alcoholic beers results in faster reaction times.

In cognitive neuroscience one may wish to describe how the neuronal activity is altered upon beer consumption and how this change in turn affects the reaction time. For this, we additionally require a measurement of neuronal activity, say a functional magnetic resonance imaging (fMRI) scan and voxel-wise blood-oxygen-level dependent (BOLD) signals, that can serve as explanans in a description of the phenomenon ‘$\text{alcohol}\to\text{neuronal activity}\to\text{reaction time}$’. We distinguish the following two scenarios:

Several methods such as Granger causality or constraint-based methods have been applied to the problem of inferring causality from cognitive neuroscience data. We describe these methods in Section 2.3. In addition, there are ongoing conceptual debates that revolve around the principle of causality in cognitive neuroscience, some of which we now mention. [65] raise concerns about the “lure of causal statements” and expound the problem of confounding when interpreting functional connectivity. Confounders are similarly problematic for multi-voxel pattern analyses [106, 122]. The causal interpretation of encoding and decoding (forward and backward, univariate and multivariate) models has received much attention as they are common in the analysis of neuroimaging data: [25] examine the differences between the model types, [38] point out that the weights of linear backward models may be misleading, and [119] extend the latter argument to non-linear models and clarify which causal interpretations are warranted from either model type. Feature relevance in mass-univariate and multivariate models can be linked to marginal and conditional dependence statements that yield an enriched causal interpretation when both are combined [119]; this consideration yields refined results in neuroimaging analyses [45, 3, 112] and explains improved functional connectivity results when combining bivariate and partial linear dependence measures [89]. Problems such as indirect measurements and varying temporal delays complicate causal Bayesian network approaches for fMRI [83, 68]. [99] present a simulation study evaluating several methods for estimating brain networks from fMRI data and demonstrate that identifying the direction of network links is difficult. The discourse on how to leverage connectivity analyses to understand mechanisms in brain networks is ongoing [110, 115, 98, 66]. Many of the above problems and findings are related to the two key challenges that we discuss in Section 3.

We begin Section 2 by formally introducing causal concepts. In Section 2.1, we outline why we believe there is a need for causal models in cognitive neuroscience by considering what types of questions could be answered by an OraCle Modelling (OCM) approach. We discuss the problem of models that are observationally equivalent yet make conflicting predictions about the effects of interventions in Section 2.2. In Section 2.3, we review different causal discovery methods and their underlying assumptions. We focus on two challenges for causality in cognitive neuroscience that are expounded in Section 3: (1) the scarcity of interventional data and (2) the challenge of finding the right variables. In Section 4, we argue that one should seek distributionally robust variable representations and models to tackle these challenges. Most of our arguments in this work are presented in an i.i.d. setting and we briefly discuss the implications for time-dependent data in Section 4.5. We conclude in Section 5 and outline ideas that we regard as promising for future research.

We do not always need causal models to answer our research question. For some scientific questions it suffices to consider probabilistic, that is, observational models. For example, if we wish to develop an algorithm for early diagnosis of Alzheimer’s disease from brain scans, we need to model the conditional distribution of Alzheimer’s disease given brain activity. Since this can be computed from the joint distribution, a probabilistic model suffices. If, however, we wish to obtain an understanding that allows us to optimally prevent progression of Alzheimer’s disease by, for example, cognitive training or brain stimulation, we are in fact interested in a causal understanding of the Alzheimer’s disease and require a causal model.

Distinguishing between these types of questions is important as it informs us about the methods we need to employ in order to answer the question at hand. To elaborate upon this distinction, we now discuss scenarios related to our running examples and the relationship between alcohol consumption and reaction time (cf. Section 1.1). Assume we have access to a powerful OraCle Modelling (OCM) machinery that is unaffected by statistical problems such as model misspecification, multiple-testing, or small sample sizes. By asking ourselves, what queries must be answered by OCM for us to ‘understand’ the cognitive function, the difference between causal and non-causal questions becomes apparent.

Assume, firstly, we ran the reaction task experiment with multiple subjects, fed all observations to our OCM machinery, and have Kim visiting our lab today. Since OCM yields us the exact conditional distribution of reaction times $\mathbf{P}_{Y\mid T=t}$ for Kim having consumed $T=t$ units of alcoholic beer, we may be willing to bet against our colleagues on how Kim will perform in the reaction task experiment they are just about to participate in. No causal model for brain activity is necessary.

Assume, secondly, that we additionally record BOLD responses $\mathbf{X}=[X_{1},\dots,X_{d}]^{\top}$ at certain locations and times during the reaction task experiment. We can query OCM for the distribution of BOLD signals that we are about to record, that is, $\mathbf{P}_{\mathbf{X}\mid T=t}$, or the distribution of reaction times given we measure Kim’s BOLD responses $\mathbf{X}=\mathbf{x}$, that is, $\mathbf{P}_{Y\mid T=t,\mathbf{X}=\mathbf{x}}$. As before, we may bet against our colleagues on how Kim’s BOLD signals will look like in the upcoming reaction task experiment or bet on their reaction time once we observed the BOLD activity $\mathbf{X}=\mathbf{x}$ prior to a reaction cue. Again, no causal model for brain activity is required.

In both of the above situations, we have learned something useful. Given that the data were obtained in an experiment in which alcohol consumption was randomised, we have learned, in the first situation, to predict reaction times after an intervention on alcohol consumption. This may be considered an operational model for alcohol consumption and reaction time. In the second situation, we have learned how the BOLD signal responds to alcohol consumption. Yet, in none of the above situations have we gained understanding of the neuronal underpinnings of the cognitive function and the reaction times. Knowing the conditional distributions $\mathbf{P}_{Y\mid T=t}$ and $\mathbf{P}_{Y\mid T=t,\mathbf{X}=\mathbf{x}}$ for any $t$ yields no insight into any of the following questions. Which brain regions maintain fast reaction times? Where in the brain should we release drugs that excite neuronal activity in order to counterbalance the effect of alcohol? How do we need to update our prediction if we learnt that Kim just took a new drug that lowers blood pressure in the prefrontal cortex? To answer such questions, we require causal understanding.

If we had a causal model, say in form of an SCM, we could address the above questions. An SCM offers an explicit way to model the system under manipulations. Therefore, a causal model can help to answer questions about where to release an excitatory drug. It may enable us to predict whether medication that lowers blood pressure in the prefrontal cortex will affect Kim’s reaction time; in general, this is the case if the corresponding variables appear in the structural equations for $Y$ or any of $Y$’s ancestors.

Instead of identifying conditional distributions, one may formulate the problem as a regression task with the aim to learn the conditional mean functions $t\to\mathbb{E}[\mathbf{X}|T=t]$ and $(t,\mathbf{x})\to\mathbb{E}[Y|T=t,\mathbf{X}=\mathbf{x}]$. These functions are then parameterised in terms of $t$ or $t$ and $\mathbf{x}$. We argue in Section 2.2, point (2), that such parameters do not carry a causal meaning and thus do not help to answer the questions above.

Promoted by slogans such as ‘correlation does not imply causation’ careful and associational language is sometimes used in the presentation of cognitive neuroscience studies. We believe, however, that a clear language that states whether a model should be interpreted causally (that is, as an interventional model) or non-causally (that is, as an observational model) is needed. This will help to clarify both the real world processes the model can be used for and the purported scientific claims.

Furthermore, causal models may generalise better than non-causal models. We expect systematic differences between subjects and between different trials or recording days of the same subject. These different situations, or environments, are presumably not arbitrarily different. If they were, we could not hope to gain any scientific insight from such experiments. The apparent question is, which parts of the model we can expect to generalise between environments. It is well-known that causal models capture one such invariance property, which is implicit in the definition of interventions. An intervention on one variable leaves the assignments of the other variables unaffected. Therefore, the conditional distributions of these other variables, given their parents, are also unaffected by the intervention [37, 1]. Thus, causal models may enable us to formulate more clearly which mechanisms we assume to be invariant between subjects. For example, we may assume that the mechanism how alcohol intake affects brain activity differs between subjects, whereas the mechanism from signals in certain brain regions to reaction time is invariant. We discuss the connection between causality and robustness in Section 4.

Causal models entail strictly more information than observational models. We now introduce the notion of equivalence of models [71, 76, 10]. This notion allows us to discuss the falsifiability of causal models, which is important when assessing candidate models and their ability to capture cause-effect relationships that govern a cognitive process under investigation.

We call two models observationally equivalent if they induce the same observational distribution. Two models are said to be interventionally equivalent if they induce the same observational and interventional distributions. As discussed above, for some interventions there may not be a well-defined corresponding experiment in the real world. We therefore also consider interventional equivalence with respect to a restricted set of interventions.

One reason why learning causal models from observational data is difficult is the existence of models that are observationally but not interventionally equivalent. Such models agree in their predictions about the observed system yet disagree in their predictions about the effects of certain interventions. We continue the example from Section 2 and consider the following two SCMs:

where in both cases $N_{1},N_{2},N_{3}$ are mutually independent standard-normal noise variables. The two SCMs are observationally equivalent as they induce the same observational distribution, the one shown in Equation (1). The models are not interventionally equivalent, however, since $\mathbf{P}_{Z_{2}}^{\operatorname{do}(Z_{1}:=3)}=\mathcal{N}(0,1)$ and $\mathbf{P}_{Z_{2}}^{\operatorname{do}(Z_{1}:=3)}=\mathcal{N}(\nicefrac{{3}}{{2}},\nicefrac{{1}}{{2}})$ for the left and right model, respectively. The two models can be told apart when interventions on $Z_{1}$ or $Z_{2}$ are considered. They are interventionally equivalent with respect to interventions on $Z_{3}$.

The existence of observationally equivalent models that are not interventionally equivalent has several implications. (1) Without assumptions, it is impossible to learn causal structure from observational data. This is not exclusive to causal inference from data and an analogous statement holds true for regression [36]. The regression problem is solvable only under certain simplicity assumptions, for example, on the smoothness of the regression function, which have been proven useful in real world applications. Similarly, there are several assumptions that can be exploited for causal discovery. We discuss some of these assumptions in Section 2.3. (2) As a consequence, without further restrictive assumptions on the data generating process, the estimated parameters do not carry any causal meaning. For example, given any finite sample from the observational distribution, both of the above SCMs yield exactly the same likelihood. Therefore, the above structures cannot be told apart by a method that employs the maximum likelihood estimation principle. Instead, which SCM and thus which parameters are selected in such a situation may depend on starting values, optimisation technique, or numerical precision. (3) Assume that we are given a probabilistic (observational) model of a data generating process. To falsify it, we may apply a goodness-of-fit test based on an observational sample from that process. An interventional model cannot be falsified based on observational data alone and one has to also take into account the outcome of interventional experiments. This requires that we are in agreement about how to perform the intervention in practice (see also Section 3.2). Interventional data may be crucial in particular for rejecting some of the observationally equivalent models (cf. the example above). The scarcity of interventional data therefore poses a challenge for causality in cognitive neuroscience (cf. Section 3.1).

The task of learning a causal model from observational (or a combination of observational and interventional) data is commonly referred to as causal discovery or causal structure learning. We have argued in the preceding section that causal discovery from purely observational data is impossible without any additional assumptions or background knowledge. In this section, we discuss several assumptions that render (parts of) the causal structure identifiable from the observational distribution. In short, assumptions concern how causal links manifest in observable statistical dependences, functional forms of the mechanisms, certain invariances under interventions, or the order of time. We briefly outline how these assumptions can be exploited in algorithms. Depending on the application at hand, one may be interested in learning the full causal structure as represented by its graph or in identifying a local structure such as the causes of a target variable $Y$. The methods described below cover either of the two cases. We keep the description brief focussing on the main ideas and intuition, while more details can be found in the respective references.

In Section 2.2 we discussed that different causal models may induce the same observational distribution while they make different predictions about the effects of interventions. That is, observationally equivalent models need not be interventionally equivalent. This implies that some models can only be refuted when we observe the system under interventions which perturb some specific variables in our model. In contrast to broad perturbations of the system, we call targeted interventions those for which the intervention target is known and for which we can list the intervened-upon variables in our model, say “$X_{1},X_{3},X_{8}$ have been intervened upon.” Even if some targeted interventions are available, there may still be multiple models that are compatible with all observations obtained under those available interventions. In the worst case, a sequence of up to $d$ targeted interventional experiments may be required to distinguish between the possible causal structures over $d$ observables $X_{1},\dots,X_{d}$ when the existence of unobserved variables cannot be excluded while assuming Markovianity, faithfulness, and acyclicity [28]. In general, the more interventional scenarios are available to us, the more causal models we can falsify and the further we can narrow down the set of causal models compatible with the data.

Therefore, the scarcity of targeted interventional data is a barrier to causal inference in cognitive neuroscience. Our ability to intervene on neural entities such as the BOLD level or oscillatory bandpower in a brain region is limited and so is our ability to either identify the right causal model from interventional data or to test causal hypotheses that are made in the literature. One promising avenue are non-invasive brain stimulation techniques such as transcranial magnetic or direct/alternating current stimulation which modulate neural activity by creating a field inside the brain [69, 40, 7, 50]. Since the stimulation acts broadly and its neurophysiological effects are not yet fully understood, transcranial stimulation cannot be understood as targeted intervention on some specific neuronal entity in our causal model [2, 114]. The inter-individual variability in response to stimulation further impedes its direct use for probing causal pathways between brain regions [58]. [6] review the obstacles to inferring causality from non-invasive brain stimulation studies and provide guidelines to attenuate the aforementioned. Invasive stimulation techniques, such as deep brain stimulation relying on electrode implants [64], may enable temporally and spatially more fine-grained perturbations of neural entities. [26] exemplify how to revise causal structures inferred form observational neuroimaging data on a larger cohort through direct stimulation of specific brain regions and concurrent fMRI on a smaller cohort of neurosurgical epilepsy patients. In non-human primates, concurrent optogenetic stimulation with whole-brain fMRI had been used to map the wiring of the medial prefrontal cortex [54, 53]. Yet, there are ethical barriers to large-scale invasive brain stimulation studies and it may not be exactly clear how an invasive stimulation corresponds to an intervention on, say, the BOLD response measured in some voxels. We thus believe that targeted interventional data will remain a scarcity due to physical and ethical limits to non-invasive and invasive brain stimulation.

Consider the following variant of our Running Example B (cf. Section 1.1). Assume that (a) the consumption of alcoholic beer $T$ slows neuronal activity in the brain regions $X_{1}$, $X_{2}$, and $Y$, (b) $X_{2}$ is a cause of $X_{1}$, and (c) $X_{2}$ is a cause of $Y$. Here, (a) could have been established by randomising $T$, whereas (b) and (c) may be background knowledge. Nothing is known, however, about the causal relationship between $X_{1}$ and $Y$ (apart from the confounding effect of $X_{2}$). The following graph summarises these causal relationships between the variables:

Assume we establish on observational data that there is a dependence between $X_{1}$ and $Y$ and that we cannot render these variables conditionally independent by conditioning on any combination of the remaining observable variables $T$ and $X_{2}$. Employing the widely accepted Markov condition, we can conclude that either $X_{1}\to Y$, $X_{1}\leftarrow Y$, $X_{1}\leftarrow H\to Y$ for some unobserved variable $H$, or some combination of the aforementioned settings. Without any further assumptions, however, these models are observationally equivalent. That is, we cannot refute any of the above possibilities based on observational data alone. Even randomising $T$ does not help: The above models are interventionally equivalent with respect to interventions on $T$. We could apply one of the causal discovery methods described in Section 2.3. All of these methods, however, employ further assumptions on the data generating process that go beyond the Markov condition. We may deem some of those assumptions implausible given prior knowledge about the system. Yet, in the absence of targeted interventions on $X_{1}$, $X_{2}$ or $Y$, we can neither falsify candidate models obtained by such methods nor can we test all of the underlying assumptions. In Section 4.2, we illustrate how we may benefit from heterogeneity in the data, that is, from interventional data where the intervention target is unknown.

Causal discovery often starts by considering observations of some variables $Z_{1},\dots,Z_{d}$ among which we wish to infer cause-effect relationships, thereby implicitly assuming that those variables are defined or constructed in a way that they can meaningfully be interpreted as causal entities in our model. This, however, is not necessarily the case in neuroscience. Without knowing how higher-level causal concepts emerge from lower levels, for example, it is hard to imagine how to make sense and use of a causal model of the $86$ billion neurons in a human brain [39]. One may hypothesise that a model of averaged neuronal activity in distinct functional brain regions may be pragmatically useful to reason about the effect of different treatments and to understand the brain. For such an approach we need to find the right transformation of the high-dimensional observed variables to obtain the right variables for a causal explanation of the system.

The problem of relating causal models with different granularity and finding the right choice of variable transformations that enable causal reasoning has received attention in the causality literature also outside of neuroscience applications. [29] fleshes out an instructive two-variable example that demonstrates that the choice of variables for causal modelling may be underdetermined even if interventions were available. For a wrong choice of variables our ability to causally reason about a system breaks. An example of this is the historic debate about whether a high cholesterol diet was beneficial or harmful with respect to heart disease. It can be partially explained by an ambiguity of how exactly total cholesterol is manipulated. Today, we know that low-density lipoproteins and high-density lipoproteins have opposing effects on heart disease risk. Merging these variables together to total cholesterol does not yield a variable with a well-defined intervention: Referring to an intervention on total cholesterol does not specify what part of the intervention is due to a change in low-density lipoproteins (LDL) versus high-density lipoproteins (HDL). As such, only including total cholesterol instead of LDL and HDL may therefore be regarded as a too coarse-grained variable representation that breaks a model’s causal semantics, that is, the ability to map every intervention to a well-defined interventional distribution [101, 102, 107].

Yet, we may sometimes prefer to transform micro variables into macro variables. This can result in a concise summary of the causal information that abstracts away detail, is easier to communicate and operationalise, and more effectively represents the information necessary for a certain task [42, 41, 116]; for example, a causal model over $86$ billion neurons may be unwieldy for a brain surgeon aiming to identify and remove malignant brain tissue guided by the cognitive impairments observed in a patient. [88] formalise a notion of exact transformations that ensures causally consistent reasoning between two causal models where the variables in one model are transformations of the variables in the other. Roughly speaking, two models are considered causally consistent if the following two ways to reason about how the distribution of the macro-variables changes upon a macro-level intervention agree with one another: (a) find an intervention on the micro-variables that corresponds to the considered macro-level intervention, and consider the macro-level distribution implied by the micro-level intervention, and (b) obtain the interventional distribution directly within the macro-level structural causal model sidestepping any need to refer to the micro-level. If the two resulting distributions agree with one another for all (compositions of) interventions, then the two models are said to be causally consistent and we can view the macro-level as an exact transformation of the micro-level causal model that preserves its causal semantics. A formal exposition of the framework and its technical subtleties can be found in the aforementioned work. Here, we revisit a variant of the cholesterol example for an illustration of what it entails for two causal models to be causally consistent and illustrate a failure mode: Consider variables $L$ (LDL), $H$ (HDL), and $D$ (disease), where $D:=H-L+N_{D}$ for $L,H,N_{d}$ mutually independent random variables. Then a model based on the transformed variables $T=L+H$ and $D\equiv D$ is in general not causally consistent with the original model: For $(l_{1},h_{1})\neq(l_{2},h_{2})$ with $l_{1}+h_{1}=l_{2}+h_{2}$ the interventional distributions induced by the micro-level model corresponding to setting $L:=l_{1}$ and $H:=h_{1}$ or alternatively $L:=l_{2}$ and $H:=h_{2}$ do in general not coincide due to the differing effects of $L$ and $H$ on $D$. Both interventions correspond to the same level of $T$ and the intervention setting $T:=t$ with $t=l_{1}+h_{1}=l_{2}+h_{2}$ in the macro-level model. Thus, the distributions obtained from reasoning (a) and (b) above do not coincide. If, on the other hand, we had $\widetilde{D}:=H+L+N_{D}$, then we could indeed use a macro-level model where we consider $T=H+L$ to reason about the distribution of $\widetilde{D}$ under the intervention ${\operatorname{do}(T:=t)}$ without running into conflict with the interventional distributions implied by all corresponding interventions in the micro-level model. This example can analogously be considered in the context of our running examples (cf. Section 1.1): Instead of LDL, HDL, and disease one could alternatively think of some neuronal activity ($L$) that delays motor response, some neuronal activity ($H$) that increases attention levels, and the detected reaction time ($D$) assessed by subjects performing a button press; the scenario then translates into how causal reasoning about the cause of slowed reaction times is hampered once we give up on considering $H$ and $L$ as two separate neural entities and instead try to reason about the average activity $T$. [49] observe similar problems for causal reasoning when aggregating variables and show that the observational and interventional stationary distributions of a bivariate autoregressive processes cannot in general be described by a two-variable causal model. A recent line of research focuses on developing a notion of approximate transformations of causal models [5, 4]. While there exist first approaches to learn discrete causal macro-variables from data [16, 15], we are unaware of any method that is generally applicable and learns causal variables from complex high-dimensional data.

In cognitive neuroscience, we commonly treat large-scale brain networks or brain systems as causal entities and then proceed to infer interactions between those [123, 82]. [99] demonstrate that this should be done with caution: Network identification is strongly susceptible to slightly wrong or different definitions of the regions of interest (ROIs) or the so-called atlas. Analyses based on Granger causality depend on the level of spatial aggregation and were shown to reflect the intra-areal properties instead of the interactions among brain regions if an ill-suited aggregation level is considered [17]. Currently, there does not seem to be consensus as to which macroscopic entities and brain networks are the right ones to (causally) reason about cognitive processes [108]. Furthermore, the observed variables themselves are already aggregates: A single fMRI voxel or the local field potential at some cortical location reflects the activity of thousands of neurons [55, 30]; EEG recordings are commonly considered a linear superposition of cortical electromagnetic activity which has spurred the development of blind source separation algorithms that try to invert this linear transformation to recover the underlying cortical variables [70].

The concept of causality is linked to invariant models and distributional robustness. Consider again the setting with a target variable $Y$ and covariates $X_{1},\ldots,X_{d}$, as described in the running examples in Section 1.1. Suppose that the system is observed in different environments. Suppose further that the generating process can be described by an SCM, that ${\mathbf{PA}(Y)}\subseteq\{X_{1},\ldots,X_{d}\}$ are the causal parents of $Y$, and that the different environments correspond to different interventions on some of the covariates, while we neither (need to) know the interventions’ targets nor its precise form. In our reaction time example, the two environments may represent two subjects (say, a left-handed subject right after having dinner and a trained race car driver just before a race) that differ in the mechanisms for $X_{1}$, $X_{3}$, and $X_{7}$. Then the joint distribution over $Y,X_{1},\ldots,X_{d}$ may be different between the environments and also the marginal distributions may vary. Yet, if the interventions do not act directly on $Y$ the causal model is invariant in the following sense: the conditional distribution of $Y\,|\,\mathbf{X}_{{\mathbf{PA}(Y)}}$ is the same in all environments. In the reaction time examples this could translate to the neuronal causes that facilitate fast (versus slow) reaction times to be the same across subjects. This invariance can be formulated in different ways. For example, we have for all $k$ and $\ell$, where $k$ and $\ell$ denote the indices of two environments, and for almost all $x$

Equivalently,

where the variable $E$ represents the environment. In practice, we often work with model classes such as linear or logistic regression for modelling the conditional distribution $Y\,|\,\mathbf{X}_{{\mathbf{PA}(Y)}}$. For such model classes, the above statements simplify. In case of linear models, for example, Equations (2) and (3) translate to regression coefficients and error variances being equal across different environments.

For an example, consider a system that, for environment $E=1$, is governed by the following structural assignments

with $N_{1},N_{2},N_{3},N_{4},N_{Y}$ mutually independent and standard-normal, and where environment $E=-1$ corresponds to an intervention changing the weight of $X_{1}$ in the assignment for $X_{2}$ to $-1$. Here, for example, $\{X_{1},X_{2},X_{3}\}$ and $\{X_{1},X_{2},X_{4}\}$ are so-called invariant sets: the conditionals $Y|X_{1},X_{2},X_{3}$ and $Y|X_{1},X_{2},X_{4}$ are the same in both environments. The invariant models $Y|X_{1},X_{2},X_{3}$ and $Y|X_{1},X_{2},X_{4}$ generalise to a new environment $E=-2$, which changes the same weight to $-2$, in that they would still predict well. Note that $Y|X_{1},X_{2},X_{4}$ is a non-causal model. The lack of invariance of $Y|X_{2}$ is illustrated by the different regression lines in the scatter plot on the right.

The validity of (2) and (3) follows from the fact that the interventions do not act on $Y$ directly and can be proved using the equivalence of Markov conditions [[]Section 6.6]Lauritzen1996,Peters2016jrssb. Here, we try to argue that it also makes sense intuitively. Suppose that someone proposes to have found a complete causal model for a target variable $Y$, using certain covariates $\mathbf{X}_{S}$ (for $Y$, we may again think of the reaction time in Example A). Suppose that fitting that model for different subjects yields significantly different model fits – maybe even with different signs for the causal effects from variables in $\mathbf{X}_{S}$ to $Y$ such that $E\mbox{${}\perp\mkern-11.0mu\perp{}$}Y\mid\mathbf{X}_{S}$ is violated. In this case, we would become sceptical about whether the proposed model is indeed a complete causal model. Instead, we might suspect that the model is missing an important variable describing how reaction time depends on brain activity.

In practice, environments can represent different sources of heterogeneity. In a cognitive neuroscience setting, environments may be thought of as different subjects who react differently, yet not arbitrarily so (cf. Section 2.1), to varying levels of alcohol consumption. Likewise, different experiments that are thought to involve the same cognitive processes may be thought of as environments; for example, the relationship ‘neuronal activity $\to$ reaction time’ (cf. Example A, Section 1.1) may be expected to translate from an experiment that compares reaction times after consumption of alcoholic versus non-alcoholic beers to another experiment where subjects are exposed to Burgundy wine versus grape juice. The key assumption is that the environments do not alter the mechanism of $Y$—that is, $f_{Y}({\mathbf{PA}(Y)},N_{Y})$—directly or, more formally, there are no interventions on $Y$. To test whether a set of covariates is invariant, as described in (2) and (3), no causal background knowledge is required.

The above invariance principle is also known as ‘modularity’ or ‘autonomy’. It has been discussed not only in the field of econometrics [37, 1, 43], but also in philosophy of science. [121] discusses how the invariance idea rejects that ‘either a generalisation is a law or else is purely accidental’. In our notion, the criteria (2) and (3) depend on the environments $E$. In particular, a model may be invariant with respect to some changes, but not with respect to others. In this sense, robustness and invariance should always be thought with respect to a certain set of changes. [121] introduces the possibility to talk about various degrees of invariance, beyond the mere existence or absence of invariance, while acknowledging that mechanisms that are sensitive even to mild changes in the background conditions are usually considered as not scientifically interesting. [13] analyses the relationship between invariant and causal relations using linear deterministic systems and draws conclusions analogous to the ones discussed above. In the context of the famous Lucas critique [59], it is debated to which extent invariance can be used for predicting the effect of changes in economic policy [14]: Economy consists of many individual players who are capable of adapting their behaviour to a change in policy. In cognitive neuroscience, we believe that the situation is different. Cognitive mechanisms do change and adapt, but not necessarily arbitrarily quickly. Some cognitive mechanism of an individual at the same day can be assumed to be invariant with respect to changes in the visual input, say. Depending on the precise setup, however, we may expect moderate changes of the mechanisms, say, for example, the development of cognitive function in children or learning effects. In other settings, where mechanisms may be subject to arbitrary large changes, scientific insight seems impossible (see Section 2.1).

Recently, the principle of invariance has also received increasing attention in the statistics and machine learning community [93, 74, 60]. It can also be applied to models that do not have the form of an SCM. Examples include dynamical models that are governed by differential equations [78].

The idea of distributional robustness across changing background conditions may help us to falsify causal hypotheses, even when interventional data is difficult to obtain, and in this sense may guide us towards models that are closer to the causal ground truth. For this, suppose that the data are obtained in different environments and that we expect a causal model for $Y$ to yield robust performance across these environments (see Section 4.1). Even if we lack targeted interventional data in cognitive neuroscience and thus cannot test a causal hypothesis directly, we can test the above implication. We can test the invariance, for example, using conditional independence tests or specialised tests for linear models [19]. We can, as a surrogate, hold out one environment, train our model on the remaining environments, and evaluate how well that model performs on the held-out data (cf. Figure 2); the reasoning is that a non-invariant model may not exhibit robust predictive performance and instead yield a bad predictive performance for one or more of the folds. If a model fails the above then either (1) we included the wrong variables, (2) we have not observed important variables, or (3) the environment directly affects $Y$. Tackling (1), we can try to refine our model and search for different variable representations and variable sets that render our model invariant and robust in the post-analysis. In general, there is no way to recover from (2) and (3), however.

While a model that is not invariant across environments cannot be the complete causal model (assuming the environments do not act directly on the target variable), it may still have non-trivial prediction performance and predict better than a simple baseline method in a new, unseen environment. The usefulness of a model is questionable, however, if its predictive performance on held-out environments is not significantly better than a simple baseline. Conversely, if our model shows robust performance on the held-out data and is invariant across environments, it has the potential of being a causal model (while it need not be; see Section 4.1 for an example). Furthermore, a model that satisfies the invariance property is interesting in itself as it may enable predictions in new, unseen environments. For this line of argument, it does not suffice to employ a cross-validation scheme that ignores the environment structure and only assesses predictability of the model on data pooled across environments. Instead, we need to respect the environment structure and assess the distributional robustness of the model across these environments.

For an illustration of the interplay between invariance and predictive performance, consider a scenario in which $X_{1}\to Y\to H\to X_{2}$, where $H$ is unobserved. Here, we regard different subjects as different environments and suppose that (unknown to us) the environment acts on $H$: One may think of a variable $E$ pointing into $H$. Let us assume that our study contains two subjects, one that we use for training and another one that we use as held-out fold. We compare a model of the form $\widehat{Y}=f\left(\mathbf{X}_{{\mathbf{PA}(Y)}}\right)=f(X_{1})$ with a model of the form $\widetilde{Y}=g\left(\mathbf{X}\right)=g(X_{1},X_{2})$. On a single subject, the latter model including all observed variables has more predictive power than the former model that only includes the causes of $Y$. The reason is that $X_{2}$ carries information about $H$, which can be leveraged to predict $Y$. As a result, $g(X_{1},X_{2})$ may predict $Y$ well (and even better than $f(X_{1})$) on the held-out subject if it is similar to the training subject in that the distribution of $H$ does not change between the subjects. If, however, $H$ was considerably shifted for the held-out subject, then the performance of predicting $Y$ by $g(X_{1},X_{2})$ may be considerably impaired. Indeed, the invariance is violated and we have $E\centernot\mbox{${}\perp\mkern-11.0mu\perp{}$}Y|X_{1},X_{2}$. In contrast, the causal parent model $f(X_{1})$ may have worse accuracy on the training subject but satisfies invariance: Even if the distribution of $H$ is different for held-out subjects compared to the training subject, the predictive performance of the model $f(X_{1})$ does not change. We have $E\mbox{${}\perp\mkern-11.0mu\perp{}$}Y|X_{1}$.

In practice, we often consider more than two environments. We hence have access to several environments when training our model, even if we leave out one of the environments to test on. In principle, we can thus already during training distinguish between invariant and non-invariant models. While some methods have been proposed that explicitly make use of these different environments during training time (cf. Section 4.4), we regard this as a mainly unexplored but promising area of research. In Section 4.2.1, we present a short analysis of classifying motor imagery conditions on EEG data that demonstrates how leveraging robustness may yield models that generalise better to unseen subjects.

In summary, employing distributional robustness as guiding principle prompts us to reject models as non-causal if they are not invariant or if they do not generalise better than a simple baseline to unseen environments, such as sessions, days, neuroimaging modalities, subjects, or other slight variations to the experimental setup. Models that are distributionally robust and do generalise to unseen environments are not necessarily causal but satisfy the prerequisites for being interesting candidate models when it comes to capturing the underlying causal mechanisms.

Whether distributional robustness holds can depend on whether we consider the right variables. This is shown by the following example. Assume that the target $Y$ is caused by the two brain signals $X_{1}$ and $X_{2}$ via

for some $\alpha\neq 0$, $\beta\neq 0$, and noise variable $N_{Y}$. Assume further that the environment influences the covariates $X_{1}$ and $X_{2}$ via $X_{1}:=X_{2}+E+N_{1}$ and $X_{2}:=E+N_{2}$, but does not influence $Y$ directly. Here, $X_{1}$ and $X_{2}$ may represent neuronal activity in two brain regions that are causal for reaction times while $E$ may indicate the time of day or respiratory activity. We then have the invariance property

If, however, we were to construct or—due to limited measurement ability—only be able to observe $\widetilde{X}:=X_{1}+X_{2}$, then whenever $\alpha\neq\beta$ we would find that

This conditional dependence is due to many value pairs for $(X_{1},X_{2})$ leading to the same value of $\widetilde{X}$: Given $\widetilde{X}=\widetilde{x}$, the value of $E$ holds information about whether say $(X_{1},X_{2})=(\widetilde{x},0)$ or $(X_{1},X_{2})=(0,\widetilde{x})$ is more probable and thus–since $X_{1}$ and $X_{2}$ enter $Y$ with different weights–holds information about $Y$; $E$ and $Y$ are conditionally dependent given $\widetilde{X}$. Thus, the invariance may break down when aggregating variables in an ill-suited way. This example is generic in that the same conclusions hold for all assignments $X_{1}:=f_{1}(X_{2},E,N_{1})$ and $X_{2}:=f_{2}(E,N_{2})$, as long as causal minimality, a weak form of faithfulness, is satisfied [100].

Rather than taking the lack of robustness as a deficiency, we believe that this observation has the potential to help us finding the right variables and granularity to model our system of interest. If we are given several environments, the guiding principle of distributional robustness can nudge our variable definition and ROI definition towards the construction of variables that are more suitable for causally modelling some cognitive function. If some ROI activity or some EEG bandpower feature does not satisfy any invariance across environments then we may conclude that our variable representation is misaligned with the underlying causal mechanisms or that important variables have not been observed (assuming that the environments do not act on $Y$ directly).

This idea can be illustrated by a thought experiment that is a variation of the LDL-HDL example in Section 3.2: Assume we wish to aggregate multiple voxel-activities and represent them by the activity of a ROI defined by those voxels. For example, let us consider the reaction time scenario (Example A, Section 1.1) and voxels $X_{1},\dots,X_{d}$. Then we may aggregate the voxels $\overline{X}=\sum_{i=1}^{d}X_{i}$ to obtain a macro-level model in which we can still sensibly reason about the effect of an intervention on the treatment variable $T$ onto the distribution of $\overline{X}$, the ROIs average activity. Yet, the model is in general not causally consistent with the original model. First, our ROI may be chosen too coarse such that for $\mathbf{\widehat{x}}\neq\mathbf{\widetilde{x}}\in\mathbb{R}^{d}$ with $\sum_{i=1}^{d}\mathbf{\widehat{x}}_{i}=\sum_{i=1}^{d}\mathbf{\widetilde{x}}_{i}=\overline{x}$ the interventional distributions induced by the micro-level model corresponding to setting all $X_{i}:=\mathbf{\widetilde{x}}_{i}$ or alternatively to $X_{i}:=\mathbf{\widehat{x}}_{i}$ do not coincide—for example, a ROI that effectively captures the global average voxel-activity cannot resolve whether a higher activity is due to increased reaction-time-driving neuronal entities or due to some upregulation of other neuronal processes unrelated to the reaction time, such as respiratory activity. This ROI would be ill-suited for causal reasoning and non-robust as there are two micro-level interventions that imply different distributions on the reaction times while corresponding to the same intervention setting $\overline{X}:=\overline{x}$ with $\overline{x}=\sum_{i=1}^{d}\mathbf{\widehat{x}}_{i}=\sum_{i=1}^{d}\mathbf{\widetilde{x}}_{i}$ in the macro-level model. Second, our ROI may be defined too fine grained such that, for example, the variable representation does only pick up on the left-hemisphere hub of a distributed neuronal process relevant for reaction times. If the neuronal process has different laterality in different subjects, then predicting the effects of interventions on only the left-hemispherical neuronal activity cannot be expected to translate to all subjects. Here, a macro-variable that averages more voxels, say symmetric of both hemispheres, may be more robust to reason about the causes of reaction times than the more fine grained unilateral ROI. In this sense, seeking for variable constructions that enable distributionally robust models across subjects, may nudge us to meaningful causal entities. The spatially refined and finer resolved cognitive atlas obtained by [112], whose map definition procedure was geared towards an atlas that would be robustness across multiple studies and $196$ different experimental conditions, may be seen as an indicative manifestation of the above reasoning.

We now present some existing methods that explicitly consider the invariance of a model. While many of these methods are still in their infancy when considering real world applications, we believe that further development in that area could play a vital role when tackling causal questions in cognitive neuroscience.

So far, we have mainly focused on the setting of i.i.d. data. Most of the causal inference literature dealing with time dependency considers discrete-time models. This comes with additional complications for causal inference. For example, there are ongoing efforts to adapt causal inference algorithms and account for sub- or sup-sampling and temporal aggregation [23, 46]. Problems of temporal aggregation relate to Challenge 2 of finding the right variables, which is a conceptual problem in time series models that requires us to clarify our notion of intervention for time-evolving systems [88]. When we observe time series with non-stationarities we may consider these as resulting from some unknown shift interventions. That is, non-stationarities over time may be due to shifts in the background conditions and as such can be understood as shifts in environments analogous to the i.i.d. setting. This way, we may again leverage the idea of distributional robustness for inference on time-evolving systems for which targeted interventional data is scarce. Extensions of invariant causal prediction to time series data that aim to leverage such variation have been proposed by [20, 79] and the ICA procedure described in Section 4.4 also exploits non-stationarity over time. SCMs extend to continuous-time models [72], where the idea to trade off prediction and invariance has been applied to the problem of inferring chemical reaction networks [78].

A remark is in order if we wish to describe time-evolving systems by one causal summary graph where each time series component is collapsed into one node: For this to be reasonable, we need to assume a time-homogeneous causal structure. Furthermore, it requires us to carefully clarify its causal semantics: While summary graphs can capture the existence of cause-effect relationships, they do in general not correspond to a structural causal model that admits a causal semantics nor enables interventional predictions that are consistent with the underlying time-resolved structural causal model [88, 49]. That is, the wrong choice of time-agnostic variables and corresponding interventions may be ill-suited to represent the cause-effect relationships of a time-evolving system [[, cf. Challenge 2 and]]rubenstein2017causal,janzing2018structural.

The authors thank the anonymous reviewers for their constructive and helpful feedback on an earlier version of this manuscript. SW was supported by the Carlsberg Foundation.