Norbert Wiener’s legacy in the study and inference of causation

Donatello Materassi

Abstract— Norbert Wiener has pioneered the study of causal-ity and his ideas are still shaping this active area of research.The present article explores the concept of “inferred causation”and demonstrates the continued impact of Wiener’s work. Itshows how the statistical notion of causation introduced by CliveGranger in the 70’s and the counterfactual notion of causationintroduced by Judea Pearl in the 90’s can be interpreted andmerged under a single unifying framework that is based onWiener filtering.

Determining whether a given process “causes” another hasbeen a central subject of philosophical debate since even beforethe time of David Hume and Immanuel Kant. Aside from purelyphilosophical or gnoseological considerations, practicing scien-tists are content to be able to infer causation in operative ways.In practical terms, if it is possible to isolate the processes of in-terest within a controlled laboratory environment, causality canbe inferred by performing a sufficient number of experiments.However, a deeper understanding of the meaning of causationis required if the processes can not be perfectly isolated, notall variables can be measured or only passive observationsare available. Traditionally, Clive Granger is credited with theintroduction of the first quantitative statistical tests to infercausation from passive observations. In Granger’s formulation,one variable is causal to another, if the ability to predictthe second variable is significantly improved by incorporatinginformation about the first. As Granger himself acknowledgesin his seminal paper, both the mathematical formalizationand the fundamental ideas of his notion of causality havebeen deeply inspired by the contributions of Wiener in thearea of estimation and prediction. More recently, a differentapproach to the study of causality has been developed by thecomputer scientist and philosopher Judea Pearl. Central toPearl’s work is the description of probability distributions andconditional independence among random variables by usingdirected acyclic graphs. Over this class of graphs, Pearl hasintroduced a semantic language and computational rules toformally describe the concept of causality in a network ofpartially observed variables. These computational rules areusually known as “do-calculus.” Typically, Pearl’s approachis static, in the sense that there is no time variable definedin the probabilistic model, and is not well-suited to considerfeedback loops. Granger’s approach, instead, is based on one-step ahead predictors exploiting the fact that the processesare dynamical systems, but is not capable of dealing withunobserved quantities.

In this article, we provide a combination of both approaches.We first show how, by using certain variations of the Wienerfilter, specific sparsity properties can be represented overdirected graphs where loops can be present as well. Theseresults are in line with the results obtained by Pearl for hisgraphical models. Under relatively mild hypotheses, we, as well,provide an interpretation for a generalized (multivariate) notionof Granger causality over these graphs. Finally, we prove thatthese Wiener filters can be used within the framework providedby do-calculus and seamlessly take into account scenarios wherefeedback loops are also present.

Donatello Materassi is with Laboratory for Information and DecisionSystems, Massachusetts Institute of Technology

I. HISTORICAL BACKGROUND AND INTRODUCTION

Determining whether a given process “causes” anotherhas always been a central subject of philosophical debate.Aristotle (384-322 BCE) distinguished categorically betweenwhat was to be considered scientific knowledge (scientia)and what was simply belief (opinio). Scientific knowledgewas the knowledge of deterministic relations of cause andeffect between natural phenomena. Given this knowledge,only logic was necessary to determine the future outcomes ofa certain situation or condition. Aristotle’s view of causationas knowledge in purely deterministic terms persisted forseveral centuries until the time of Descartes (1596-1650)and his follower Malebranche (1638-1715) [1]. However,this notion of causation fails to provide practicing scien-tists with an operative way to determine from data if acertain process affects another, since it presupposes a perfectknowledge about the state of the world. Indeed, only aperfect knowledge of the state of world, along with a perfectknowledge of its physical laws, would allow us to verify it.Furthermore, most typical causal statements tend to involvegeneral classes of events and not a single one. Statements,such as “smoking cigarettes causes lung cancer” or “higherinflation causes lower unemployment”, do not take intoaccount single events, but collection of events. Indeed, froma purely logical perspective, “smoking cigarettes” and “lungcancer” are neither sufficient nor necessary conditions toeach other. Thus, it is easy to see how it would be difficult,if not impossible, to frame statements of this kind undersuch a strict and deterministic approach. A more modernview of causation was given by David Hume (1711-1776)in his “Treatise on Human Nature” and later in his “ AnEnquire Concerning Human Understanding” when he wasdiscussing what Immanuel Kant referred to as the “Problemof Induction”. In Hume’s view, we are left simply observingthat one event follows another and they seem connected,but we do not know how or why. We can only rely uponrepeated observation (induction) to determine the causal lawsimplicitly assuming that the future is akin to the past.

“All events seem entirely loose and separate. Oneevent follows another; but we never can observeany tie between them. They seem conjoined butnever connected. And as we can have no ideaof anything, which never appears to our outwardsense or inward sentiment, the necessary conclu-sion seems to be, that we have no idea of connexionor power at all, and that these words are absolutelywithout any meaning, when employed either inphilosophical reasoning, or in private life”

U.S. Government work not protected by U.S. copyright

[David Hume]

Hume did not deny the existence of causal laws. Indeed, hefamusly claimed

“I never asserted so absurd a proposition as thatsomething could arise without a cause”

[David Hume].

As an empiricist, he was, though, mostly interested in theproblem of how it is possible to determine from observationsthe existence of “causal connections” between classes ofevents occurring one after the other. Even with Hume’sempiricism, though, we are still far from discussing aboutcausality in scientific, or better quantitative, terms. Tradi-tionally, the first scientist credited with the introduction ofa quantitative notion of causality is Clive Granger (1934-2009). In Granger’s formulation, one variable is causal toanother, if the ability to predict the second variable isimproved, in a statistically significant way, by incorporatinginformation about the first. However, as Granger has alwaysacknowledged, his formalization was suggested by a paper byNorbert Wiener (1894-1964) [2]. This is Granger’s personalaccount of how he conceived his notion of causality [3].

“In the early 1960’s I was considering a pairof related stochastic processes which were clearlyinter-related and I wanted to know if this relation-ship could be broken down into a pair of one wayrelationships. It was suggested to me to look at adefinition of causality proposed by a very famousmathematician, Norbert Weiner, so I adapted thisdefinition into a practical form and discussed it.”

[Clive Granger]

Indeed, we find in nuce the very same idea in one ofNorbert Wiener’s papers [2].

“[W]e note that [the just derived mathematicalexpression] represents that part of the normal-ized innovation of [the first stochastic process] ifwe have as information the past of [the secondstochastic process] as well as the past of the [thefirst stochastic process].[...][T]his may be considered as a measure [...] of thecausality effect of [the second stochastic processon the first one]”.

[Norbert Wiener]

More recently, a different approach to the study of causalityhas been developed by the computer scientist and philoso-pher Judea Pearl (1936-). Central to Pearl’s work is thedescription of probability distributions and conditional inde-pendence among random variables by using directed acyclicgraphs. Over this class of graphs, Pearl has introduceda semantic language and computational rules to formallydescribe the concept of causality in a network of partiallyobserved variables. These computational rules are usuallyknown as “do-calculus.” Typically, Pearl’s approach is static,in the sense that there is no time variable defined in the

probabilistic model, and is not well-suited to consider feed-back loops. Granger’s approach, instead, is based on one-step ahead predictors exploiting the fact that the processesare dynamical systems. However, Granger’s formalization isnot capable of dealing with unobserved quantities.

In this article, we provide a combination of both ap-proaches. We first show how, by using certain variationsof the Wiener filter, specific sparsity properties can berepresented over directed graphs where loops can be presentas well. These results are in line with the results obtainedby Pearl for his graphical models. Under relatively mildhypotheses, we, as well, provide an interpretation for ageneralized (multivariate) notion of Granger causality overthese graphs. Finally, we prove that these Wiener filters canbe used within the framework provided by do-calculus andseamlessly take into account scenarios where feedback loopsare also present.

II. LINEAR DYNAMIC GRAPHS AS THE GENERATIVECLASS OF MODELS

The aim of this section is to introduce a relatively broadclass of network models that will provide the formal frame-work for the derivation of quantitative results about theinference of causal relations. We will refer to this class ofmathematical models as Linear Dynamic Graphs (LDGs)[4], [5]. Informally speaking, a LDG is a collection oflinear dynamical systems connected according to a directedgraph and forced by stationary additive mutually uncorrelatednoise processes. We stress that the assumption of exclusivelylinear relations and additive noise processes are not requiredto develop most of the theoretical results discussed in thearticle. However, they provide the advantage of a simplernotation making the exposition clearer.

Linear Dynamic Graphs: motivations and definition

Let us consider two discrete-time stochastic processes xiand xj . As a working assumption, let xi and xj be jointlywide-sense stationary with zero mean. Also, let Hji(z) be atransfer function such that the Laurent expansion

Hji(z) =∞∑

t=−∞hji(t)z

−t (1)

is well defined on an open domain containing the unit circle{z ∈ C : |z| = 1} for some bi-infinite sequence {hji(t)}t∈Z.We also say that Hji(z) is the (bilateral) Z-transform ofhji(t) and denote this as Hji(·) = Z {hji(·)}. Let us assume,for the moment, that the process xj is deterministicallyobtained by filtering xi via Hji(z). Namely, this means that

xj(t) =∞∑

τ=−∞hji(t− τ)xi(τ) ∀ t ∈ Z. (2)

We denote Relation (2) as xj = Hji(z)xi and represent it ina block diagram as in Figure 1(a). In the following, we willmake use of the concept of time-causality.

Definition 1 (Time-causality): A transfer function H(·) =Z {h(·)} is time-causal if t ≤ 0 implies h(t) = 0. If, in

Hji(z) xjxiHji(z)xi xj

ej

(a) (b)Fig. 1. The signal xj as a deterministic function of xi (a) and the signalxj as a superposition of its independent behavior ej and the “influence”from xi (b).

addition, h(0) = 0 the transfer function is strictly time-causal.Note that the concept of time-causality is not the same ofcausality (or influence) that we are going to introduce for anLDG.

From Equation 2 it follows that each realization of xidetermines univocally a realization of xj . Thus, in this case,if xi is taken as an autonomous and “free” process, we candefinitely say that xj is caused by xi. Now, it is possibleto extend this relation in order describe a non-deterministicform of “influence” (or “causality”) of xi on xj in thefollowing manner

xj(t) = ej(t) +

∞∑τ=−∞

hji(t− τ)xi(τ) ∀ t ∈ Z (3)

where ej is another zero-mean wide sense stationary processthat represents the behavior of xj with no “influence” fromxi. We denote this relation as xj = ej + Hji(z)xi andrepresent it in a block diagram as in Figure 1(b).

Now, let as assume that there are a finite number nof processes and that they can influence each other. Theextension to this multivariate scenario can be now describedby the relations

xj = ej +∑i6=j

Hji(z)xi ∀ j = 1, ..., n.

Since the processes e1,...,en represent the autonomousbehaviors of the processes x1,...,xn in the absence of anymutual influence, we postulate that the power spectraldensity Φe(z) of the stochastic vector e = (e1,...,en)T isdiagonal and positive definite for all |z| = 1. This motivatesthe following definition.

Definition 2 (Linear Dynamic Graph):A Linear Dynamic Graph G is defined as a pair (H(z), e)where• e = (e1, .., en)T is a vector of n rationally related

random processes such that Φe(z) is diagonal• H(z) is a n× n matrix of transfer functions defined in

an open set containing the unit circle {z ∈ C : |z| = 1}and such that Hjj(z) = 0, for j = 1, ..., n.

The output processes {xj}nj=1 of the LDG are defined bythe implicit relation

x = e+H(z)x (4)

where x = (x1, ..., xn)T , under the hypothesis thatthe operator I − H(z) is causally invertible. Let

V := {x1, ..., xn} and let E := {(xi, xj)|Hji(z) 6= 0}.The pair G = (V,E) is the associated directed graph ofthe LDG. Nodes and edges of a LDG will mean nodes andedges of the graph associated with the LDG.

Thus, given a LDG (H(z), e) with output processesx1, ..., xn, we can interpret the matrix H(z) as an adjacencymatrix of a graph where a directed edge from the node xito the node xj is present if and only if the entry Hji(z) isnot identically zero.

In the following we will make extensive use of theserelations between nodes in a graph

Definition 3: Given a graph G = (V,E) where V ={x1, ...xn} we say that• xi is a parent of xj or xj is a child of xi if (xi, xj) ∈ E.• xi and xj are spouses if xi and xj share a child.

We denote as Pa(x), Ch(x) and Sp(x) the set or parents,children and spouses of a node x ∈ V .

Problem statement

Take the class of LDGs as the generative class of modelsrepresenting, via its associated graph, the relations of directcausality among the stochastic processes. Assume that theobservable processes x1, ..., xn are the output processes of aLDG. Determine for any pair (xi, xj) if there is a relationof direct influence among them.

III. SPARSITY PROPERTIES OF WIENER FILTERS INLINEAR DYNAMIC GRAPHS

In this section we report sparsity properties that hold forcertain variations of the Wiener filter [6] when applied to theestimation of a process of a LDG using the other processes ofthe LDG. Indeed, it happens that, in a LDG, minimum leastsquare estimates of a process xj do not necessarily dependon all the other processes {xi}i6=j . Instead only a subset ofthe processes {xi}i6=j is required to estimate xj . Also, sucha subset is strictly related to the graph of the LDG. Thus, weshow how the computation from observed data (via spectralor statistical methods) of different flavors of the Wiener filtercan provide ways to reconstruct the causal structure of theLDG.

A. One step-ahead predictor in Linear Dynamic Graphs

We consider a first result that provides a connectionbetween the structure of the LDG graph and the one step-ahead predictor, the specific variant of the Wiener filterthat Clive Granger utilized in the definition of his test forcausality.

Theorem 4: Consider a well-posed LDG G = (H(z), e)where Φe(e

iω) is positive definite for ω ∈ [−π, π], andeach entry of H(z) is strictly time-causal. Let x1, ..., xn bethe output signals associated with the n nodes of its graph.Define

χ =

{n∑i=1

Wi(z)xi : Wi(z) is strictly time-causal ∀ i

}.

(5)

Consider the problem of approximating the signal xj withan element xj ∈ χ, as defined below

minxj∈χ

‖xj − xj‖2 (6)

where the norm is to be interpreted in terms of the expectedsquare error. Then the optimal solution xj exists, is uniqueand

xj =n∑i=1

Wji(z)xi (7)

where Wji(z) 6= 0 implies i = j or that xi is a parent of xj .Proof: See [5].

The interpretation of this result is straightforward. Assumethat data have been generated by a LDG where the effectof one node on another always occurs with at least one stepof delay (Hji(z) is strictly time-causal for all i and j). LetPa(xj) be the set of processes directly influencing xj . Thenthe one-step-ahead predictor to estimate xj(t+ 1) from theknowledge the processes x1, ..., xn up to time t makes useonly of the processes in Pa(xj). Since the causality testproposed by Granger [7] is based precisely on one-step-ahead predictors of this form, Theorem 4 shows that sucha test consistently determines the causality structure of theLDG. Observe that in Theorem 4 no assumption is madeon the structure of the graph. Thus, as long as there aredelays in the transfer functions modeling the influence amongnodes, the Granger-causality test consistently works, even inthe presence of cycles and feedback loops.

B. Wiener-Hopf filter in Linear Dynamic Graphs

A second result involves the so-called Wiener-Hopf filterwhen used to time-causally estimate a process xj in a LDG.

Theorem 5: Consider a well-posed LDG G = (H(z), e)where Φe(e

iω) is positive definite for ω ∈ [−π, π] and eachentry of H(z) is time-causal. Let x1, ..., xn be the outputsignals associated with the LDG. Define

χj =

n∑i6=j

Wji(z)xi : Wji(z) is time-causal ∀ i

. (8)

Consider the problem of approximating the signal xj withan element xj ∈ χj (non-causal Wiener filtering), as definedbelow

minxj∈χj

‖xj − xj‖2 , (9)

where the norm is to be interpreted in terms of the expectedsquare error. Then, the optimal solution xj exists, is uniqueand

xj =∑i6=j

Wji(z)xi (10)

where Wji(z) 6= 0 implies at least one of the followingconditions• xi is a parent of xj• xi is a child of xj

• xi and xj share a child.Proof: See [5].

The interpretation of this result is slightly more intricate thanthe interpretation of Theorem 4. Assume that data have beengenerated by a LDG where the effect of one node on anotheralways occurs at the same time or with at least one step ofdelay (Hji(z) is time-causal for all i and j). The Wiener-Hopf filter used to estimate xj from the other nodes makesuse only of the nodes influencing xj (Pa(xj)), the nodesinfluenced by xj (Ch(xj)), and the nodes influencing thenodes influenced by xj (Sp(xj)). The Wiener-Hopf filtercan be determined from data, but we find that it does notdirectly provide the oriented structure of the original networkas in the case of Theorem 4 for the Granger causality test.Instead, the sparsity pattern of the Wiener-Hopf filter detectsthe parents and the children of the node xj along with thespouses of xj , that are not necessarily connected with it.

IV. SEPARATION IN LINEAR DYNAMIC GRAPHS

A different approach to determine causation has beendeveloped by Judea Pearl [8] for the study of interconnectedrandom variables. Pearl’s framework is static in the sensethat, having been developed in the domain of random vari-ables, there is no time variable to be taken into accountand as a consequence no concept of time-causality. Centralto Pearl’s work is a notion of separation among randomvariables.

Definition 6: Let V = {v1, ..., vn} be a set of n randomvariables. Two random variables vi and vj are separated bya set of random variables Vs ⊂ V \ {vi, vj} if vi and vj areindependent given the set Vs.Pearl also introduces a graphical representation for the jointprobability distribution of a finite set of random variables.

Definition 7: Let V = {v1, ..., vn} be a set of n randomvariables and assume that the joint probability distributioncan be factorized as

p(v1, ..., vn) =n∏j=1

p(vj |Pa(vj)) (11)

where p(vj |Pa(vj)) is the conditional probability distribu-tion of vj given the set of random variables Pa(vj) ⊆{v1, .., vj−1}. The directed acyclic graph obtained linkingeach variable in Pa(vj) to vj is the graph associated withthe factorization of the distribution.It would be tempting to say that, in the probabilistic model(11), each variable in Pa(vj) is a cause for the variablevj . However, as Pearl shows, the factorization depends onthe order of the variables. Thus, rearranging the variablesin a different order would provide a different graph and,as a consequence, a different causal structure. This, ofcourse, would not acceptable since the very concept ofcausal structure needs to be independent of the order of thevariables.

It is possible to show the following result.Theorem 8: Let the set random variables V = {v1, ..., vn}

be defined according to a structural equation model

vj = fj(gj , PaG(vj)) ∀ j = 1, ..., n (12)

where g1, ..., gn are independent random variables andPaG(vj) is the set of nodes that are parents of vj accordingto a directed acyclic graph G. If vi /∈ PaG(vj) and vj /∈PaG(vi), then there exists a set Vji ⊆ V \{vi, vj} separatingvi and vj .

Proof: It is a special instance of Verma-Pearl Theorem.See [9].Theorem 8 is a fundamental property since it provides aconnection between the structure of the generative model(12) and the notion of separation given by Definition 6.

For a LDG we can define an analogous notion of separa-tion based on Wiener filtering.

Definition 9 (Wiener-separation): Let xj , xi be two pro-cesses and let X be a set of random processes that does notcontain neither xi nor xi. We say that xj is Wiener-separatedfrom xi given X if the Wiener filter estimating xj from theprocesses in X gives the same estimate of the Wiener filterestimating xj from the processes in X ∪ {xi}.Given this notion of separation, an analogous of Theorem 8can be derived for LDGs.

Theorem 10: Consider a LDG with output processesx1, ..., xn and associated graph G that has not directedcycles. If xi /∈ PaG(xj) and xj /∈ PaG(xi), then thereexists a set Xji separating xi and xj .

Proof: See [10].Theorem 10 provides a quantitative method to determineif the two processes xi and xj are directly connected inthe graph G associated with the LDG: search for a set ofprocesses Xji that separates them according to the notiongiven by the Wiener-Hopf filter. If such a process can not bedetermines, then either xi causes xj or xj causes xi.

V. RELATION BETWEEN GRANGER-CAUSALITY ANDPEARL-CAUSALITY

In the case of LDGs we can compare the notion ofcausality provided by Pearl and the one provided by Granger.In Pearl’s framework (reinterpreted via LDGs) there is arelation of direct causality between the process xi and theprocess xj if it is not possible to find other processessuch that xi and xj become separated according to thenotion given by the Wiener-Hopf filter. Instead, accordingthe definition, a process xi Granger-causes a process xj ifthe information in xi improves the one-step-ahead predictionof xj . However, Granger-causality can be also interpreted interms of a notion of separation that is based on the one-step-prediction instead of the Wiener-Hopf filter. The two notionsof causality provide different results in the following sense.Granger-causality is capable of reconstructing the directedstructure of a LDG, as long as there is at least one stepof delay in the dynamics of the transfer functions. Thenotion of causality provided by Pearl, in general, is notcapable of determining the directed structure of a LDG, butonly its associated graph without link orientation. Also, thereconstruction is successful only if the associated graph ofthe LDG has no cycles.

As a fundamental fact, in the case of LDGs, we havefound that different a priori assumptions on a LDG model

require different notions of separation in order to reconstructits causal structure. If it is known that the LDG dynamicsinvolves only strictly time-causal transfer functions, then thenotion of separation given by the one-step-ahead predictorneeds to be considered. If it is known that the LDG dynamicsis only time-causal but has no loops, then the notion of sep-aration given by the Wiener-Hopf filter has to be consideredinstead.

At least in the domain of LDGs, we can interpret Grangercausality and Pearl causality as a two notions of separationdefined using two different variants of the Wiener filter(because of a-priori assumptions on the LDG).

In light of this interpretation, we now show how a novelnotion of separation that can be applied to reconstruct thecausal structure of LDGs with not necessarily strictly time-causal dynamics and in presence of loops. In Control Theoryparticular relevance is given to the concept of well-posednessin the presence of feedback loops [11]. If a feedback loop isnot well-posed, it means that it is not possible to define aninjective map from the input signals to the output signals ofthe system. Thus, the behavior of the system is not uniquelyspecified. It is well-known that a sufficient condition to havea well-posed system is the presence of a delay in every loop[12].

Theorem 11: Consider a LDG with output processesx1, ..., xn and associated graph G. Assume that for everydirected cycles of the LDG there is at least one strictly causaltransfer function. If xi /∈ PaG(xj) and xj /∈ PaG(xi), thenthere exist a subset Xji ⊆ {x1, ..., xn} \ {xj , xi} such thatthe set of processes Xji ∪ {z−1xj , z−1xi} separates xi andxj in the Wiener-Hopf sense.

Proof: The proof is omitted because of space issues,but techniques similar to the ones used to prove Theorem 5and Theorem 4 are used in the demonstration.

As we can see, this last results creates a bridge betweenthe notion of causality by Granger and by Pearl defininga notion of separation based on a set Xji of processes tobe filtered in a causal way to infer the presence of directcausality between xi and xj

VI. CONCLUSIONS

In this article, we have provided a synthesis between twoapproaches for the study of causality that are consideredquite distinct. The key point has been the interpretationof Judea Pearl’s notion of causality in terms of Wienerfiltering and Clive Granger’s notion of causality in terms ofseparation on a projective space. Pearl’s notion of causalityis shown to work consistently for networks of dynamicalsystems, as long as loops are not present. Granger’s notionof causality consistently works for network of dynamicalsystems, as long as delays are present on every link ofthe network. This interpretation has allowed to provide anovel criterion to consistently infer causality in well-posednetworks of dynamical systems in presence of feedbackloops. This criterion comes as a natural combination of thetwo approaches.

ACKNOWLEDGEMENTS

The author wants to thank Prof. Anil Seth for useful refer-ences and Prof. Maxim Raginsky for stimulating discussions.

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