As neuroscientists we want to understand how causal interactions or mechanisms within the brain give rise to perception, cognition, and behavior. It is typical to estimate interaction effects from measured activity using statistical techniques such as functional connectivity, Granger Causality, or information flow, whose outcomes are often falsely treated as revealing mechanistic insight. Since these statistical techniques fit models to low-dimensional measurements from brains, they ignore the fact that brain activity is high-dimensional. Here we focus on the obvious confound of common inputs: the countless unobserved variables likely have more influence than the few observed ones. Any given observed correlation can be explained by an infinite set of causal models that take into account the unobserved variables. Therefore, correlations within massively undersampled measurements tell us little about mechanisms. We argue that these mis-inferences of causality from correlation are augmented by an implicit redefinition of words that suggest mechanisms, such as connectivity, causality, and flow. The state of the estimated connectivity field A fundamental goal in neuroscience is to understand brain mechanisms that underlie perception, cognition, and behavior, which, arguably, requires understanding the causal interactions between neurons and neuronal populations. Whenever we talk about causal interactions in this opinion paper, we use the counterfactual definition. A variable causally influences another variable if a perturbation of the first variable would induce a change in the activity of another variable (Pearl, 2009a) . This nicely approximates what we mean as neuroscientists: if we say a neuron influences another neuron we mean that perturbing the first (say electrically) would affect the second and if we say that a region influences another we mean that perturbing the first region (say magnetically) would affect the second. Causality has a perfectly clean definition (counterfactuals or perturbation) and we should demand our statistical approaches to be measured against it (although see also Gomez-Marin, 2017 ). Because we cannot directly measure these interactions, statistical techniques are used that aim to infer interactions from simultaneously recorded brain signals. This often boils down to asking the question of how does neural population A mechanistically affect population B? The approaches that we call estimated connectivity (eC) in this paper convert measured signals into a statistical estimate of "connectivity." The results are typically (implicitly or explicitly) thought of as a measure or at least approximation of the causal strength of interactions. A rich body of literature has described eC techniques: some techniques utilize granger causality (Bressler and Seth, 2011) , other techniques are called functional connectivity and look at delayed correlations (Friston et al., 1997 ). Yet other techniques talk about information flow (Babiloni et al., 2005; Honey et al., 2007) . Another class more directly talks about Dynamic Causal Modeling . Within the imaging community the term effective connectivity (EC) is often used when causality is explicitly claimed but within our definition they fall into our more expansive definition of eC. We will argue that the used observational approaches share the same logical weaknessstatistical confounding. Yet, the lure of extracting causality from observational data, is so powerful, that we cannot avoid feeling the pull of it and have effectively referred to correlations in connectivity terms (Stevenson et al., 2008) . The problem of unobserved confounding is the existence of unobserved variables that affect the observed variables in a way that often makes the estimation of connectivity impossible. It is easy to see why it is impossible to use any statistical techniques to estimate causal interactions in the presence of unobserved confounding. Let us say there is a causal influence from A to B. In this case, it is always possible to construct a common input C which will produce the same effect on B that A would have (e.g. by replicating A and feeding into C). Similarly, if 3 there is no influence from A to B it is always possible to use a confounder to change A and B so that they now look like they do interact, according to any algorithm used. These issues would maybe be a minor problem if we had reasons to believe that confounding is weak, i.e. if there were not many orders of magnitude more confounders than measured variables. It is thus logically impossible that an algorithm could identify the network of interactions. Techniques used for estimating connectivity typically come from fields that focus on forecasting or describing time-series. For example, Granger Causality (Bressler and Seth, 2011) and Coherence analysis (Bastos and Schoffelen, 2016; Sun et al., 2004) have been developed to describe the relationship between signals. They were then translated to address biological questions such as "How does neural population A interact with neural population B?" (Bressler and Seth, 2011; Friston et al., 2003 Friston et al., , 1997 . In many cases, scientists simply analyze the correlations and take high correlations to be a sign of neuronal communication. Alternatively, delayed or nonlinear correlations are used to estimate the direction of interactions. However, in biological questions we usually seek causal mechanisms and not just good predictions. Hence, researchers increasingly use, without much discussion of threats from confounding, techniques that were used to describe the relationships between signals to make claims about causality. Indeed, many scientists have been developing techniques for estimating the strength of connections between neurons based on simultaneous spike recordings. The underlying idea is that we want to predict each neuron's spiking probability based on the activity of other observed neurons (Pillow et al., 2008) . And indeed, if we record all neurons, they are noisy, and we know that from their transfer function we should be able to estimate the strength and nature of causal interactions (Karbasi et al., 2018) . There has been ample speculation about the meaning of the results of such a study, but it is frequently interpreted in causal terms (Pillow et al., 2008; Stevenson et al., 2008) . 15% of articles that included the term "neuroscience" also mentioned "functional connectivity" ( Figure 1 ; code and data available on: https://osf.io/9cs8p/). FC is a huge area in neuroscience making it important to obtain clarity about its interpretation and usefulness for potential future translational applications. The eC measures have in common that they are functions of (sometimes generalized) correlations. However, the language used generally does not reflect that. Almost every term in popular use suggests causality. For example, we use terms like Granger Causality, functional connectivity, information flow, effective connectivity, dynamic causal models, etc. We do this despite the fact that other terms such as improvement in predictive power, correlations, conditional correlations, and model comparison can denote more precisely what we actually do. To deal with the disagreement between used correlational techniques and desired mechanistic or causal statements, we are effectively redefining the English language. Connectivity implies a connection between two places. Causality implies cause and effect. Flow implies that something moves from one place to another. Effective implies that something has an effect. This set of redefinitions gives rise to the problem that eC approaches are often misunderstood. the neuronal mechanisms that operate during and early after practice and during sleep to support motor memory consolidation." (Dimyan and Cohen, 2011) . However, all these approaches only reach level one in Pearl's hierarchy of causation (Pearl and Mackenzie, 2018) . While the impossibility of making causal statements in such situations is well known (Dawid, 2008; Holland, 2015) , the field regularly makes causal statements based on statistical analyses that cannot support such statements. Altogether, we conclude that this debate is not merely about semantics and second that "whereof one cannot speak, thereof one must be silent" (Wittgenstein, 1922). Here we ask if the eC techniques used should be expected to estimate causal interactions. We ask if the techniques measure causality, connectivity, or flow, in the standard meanings of these words. We will conclude that, due to massive confounding, they merely describe the statistics of the joint neural data without convincing causal insights. It is important to first observe the way neurostatisticians describe their results, themselves. When entering the field, we ourselves stated "unobserved common input is a potential confound" inference is complex, reflected by a progressively sophisticated field. However, we argue here that it is impossible to overcome the problem of massive confounding by using statistics. We want to dwell on the idea of confounders to functional connectivity. Let us say we are interested in the interactions between two neurons. Further, let us assume that there is a third neuron (confounder) that activates both neurons. In that case, if we see a correlation between the neurons we cannot know if it is due to the neurons interacting directly or through an induced interaction by the unobserved neuron. More generally, any joint distribution between two neurons could be induced by a single third neuron. Unobserved neurons or information can act as confounders for the inference of functional connectivity, threatening the validity of the results. This confounding problem is central to the statistical field of causal inference and it is generally acknowledged that, in typical situations, unobserved confounders render inferences about causality impossible. We want to use the Simpson's paradox to highlight the threat of confounding (Simpson, 1951) , in which unobserved interaction can either cancel out main effects, or artificially induce spurious main effects. Let us say that there are two brain states (e.g. related to two levels of attention), and that one state is associated with high activity of neuron A and low activity of neuron B while another is associated with low activity of neuron A and high activity of neuron B. But let us say that there is a positive instantaneous causal influence from one on the other. If we do not know the brain state (confounded) we may conclude that neuron A has a negative influence on the activity of neuron B. But if we do know the brain state we correctly see that A increases the activity of neuron B (Figure 2 ). This paradox shows how confounders with relevant structure can arbitrarily influence the resulting functional connectivity. Driver et al., 2016; Messaritaki et al., 2017; Murphy et al., 2013 ) that add structured noise to the signal. In response, the field has developed some effective data correction techniques (Ciric et al., 2017; Murphy et al., 2013; Parkes et al., 2017; Power et al., 2014; . However, for each such obvious external controllable confounder there are countless internal unobservable ones (DiDomenico and Eaton, 1988) . Our argument about the impossibility to obtain causality from functional connectivity does not come from data -there is too little ground truth about complex brains to make this possible (Jonas and Kording, 2017) . It rests on a simple consideration of the factors that are known to make causal 8 inference theoretically impossible. It rests on the justified assumption that we record only few of the neurons when recording spikes, or a few projections of neural activities in imaging. We have no convincing reasons to assume that the observed dimensions should be more important than the unobserved. Our argument thus boils down to a simple logical statement: if the bulk of causal interactions happen from and within unobserved dimensions, then the correlations between observed variables are simply epiphenomena. Correlation is not causation, regardless the mathematical sophistication we use when calculating it. Causal inference algorithms that work with observational data are generally built on the assumption of causal sufficiency, which boils down to there being no unobserved confounders (although see Ranganath and Perotte, 2018; Wang and Blei, 2018) . Without these assumptions we can at best produce families of potential models and if any pair of recorded variables is confounded then this family will contain all models (Spirtes et al., 2001) . Recording only few variables in a densely interacting causal system generally renders causal inference impossible (Jonas Peters et al., 2017; Pearl, 2009a) . When analyzing spike data, there are far more unobserved variables than observed variables. When we record a few hundred neurons (Stevenson and Kording, 2011) , the number of recorded neurons is a vanishingly small subset of all neurons. We have little reason to assume that the recorded neurons are much more important than the un-recorded neurons. As each neuron receives inputs from so many other un-recorded neurons, we should expect that the parts of neural activity driven by unobserved neurons are arbitrarily larger than the parts coming from observed neurons. In other words, the confounding signal should be many orders of magnitude more important than those coming from observed data. As such, we should not expect that causal inference is possible. When analyzing imaging data such as fMRI, or LFP, EEG, or MEG, there are also far more unobserved variables than observed variables. Within each signal source, we can, in some abstraction, observe the sum of neural activity. But the same measured activity can be realized by any combination of individual activities rendering a solution of the inverse problem (signals 
REF raise concerns about the "lure of causal statements" and expound the problem of confounding when interpreting functional connectivity.
The lure of causal statements: Rampant mis-inference of causality in estimated connectivity
arXiv: Neurons and Cognition
Mathematics Biology