Towards Reverse Causal Inference on Panel Data: Precise Formulation and Challenges
Abstract
Seeking causal explanations in panel (or longitudinal/multivariate time-series) data is a difﬁcult problem of both academic and industrial importance.
Although there exists a large amount of literature on forward causal inference, where the treatment/outcome/covariates variables are well-deﬁned, it is unclear how to answer the reverse question: which covariates have effects on the outcome?
In this paper, we set forth our expedition on this reverse question from the ﬁrst principles.
We formulate the precise problem deﬁnition in terms of causal patterns and causal paths, and propose a linear-time greedy meta algorithm that makes use of forward causal inference estimators. We further identify a set of optimality conditions under which the proposed algorithm is able to ﬁnd the optimal causal path.
To substantiate our greedy algorithm, we propose a generalized version of the synthetic control estimator by ﬁtting both synthetic treatments and controls by conditioning on the partial causal paths.
Promising results on on synthetic datasets demonstrate the potential of our method.
Introduction
Time-series data are ubiquitous and important in various scientiﬁc and business domains spanning longitudinal data analysis, econometrics, epidemiology, cloud computing, supply chain management, labor planning, to name a few (see e g., Petropoulos et al.
[2022] and Benidis et al.
[2022] for a comprehensive overview and relevant applications).
There is a growing interest on drawing causal inference on time-series data, apart from being an interesting and important research problem on its own, a causal understanding of the underlying mechanisms will facilitate the construction of more robust and generalizable solutions.
Causal inference itself has a long history and has become the working horse in many applied research ﬁelds such as econometrics, political science, clinical trials, to name a few (see, e.g., Neyman [1923], Fisher [1958], Rubin [1974], Robins [1986], Abadie and
Gardeazabal [2003], Imbens and Rubin [2015]), and much effort has been devoted to time-series/panel data (see e.g., Robins [1986], Robins et al.
[2000], Bojinov et al.
There has been an interest in moving from reasoning about effects-of-cause to causes-of-effect, see, e.g., Gelman [2011], Gelman and Imbens [2013]. The term “reverse causal inference” coined by Gelman and Imbens, refer to the problems of this ﬂavor that can be viewed as the inverse of the conventional (forward) causal queries.
Although being of great theoretical and practical importance, literature on reverse causal inference usually focuses on the philosophical aspect of the question (see e.g., Dawid et al.
It has been an intriguing open question of how to formalize and investigate this problem in a practical manner.
In this work, we provide our answer to this question by formulating the reverse causal inference problem through the notions of causal patterns and causal paths (Section 2).
Under this formulation, reverse causal inference is equivalent to identifying and estimating the causal patterns.
As exact identiﬁcation requires examining exponentially many combinations of patterns (in both time and the number of covariates), we propose to use a greedy linear-time algorithm whose solution becomes exact under a set of “monotonicity” conditions.
Given that the core of our procedure solves a subproblem of forward causal inference, our method can be ﬂexibly combined with any good causal estimator, such as the inverse-propensity weighted (IPW) estimator and synthetic control (SC) estimator (see e.g., Abadie and Gardeazabal [2003], Abadie et al. We propose to use a generalized synthetic control estimator, and tested it on synthetic datasets.
Contributions.
(1) We give a novel precise formulation of the reversal causal inference problem through causal patterns and causal paths.
(2) We propose a linear-time meta algorithm for ﬁnding causal paths and derive optimality conditions.
(3) We propose a generalized version of the synthetic control estimator, and demonstrate its beneﬁt on synthetic experiments.
1.1 Related Work
Reverse Causal Inference and Causal Evidence.
Although huge literature exists for “forward” causal queries as discussed above, it is only recently when practitioners started to contemplate reverse causal inference in applied ﬁelds, see, e.g., Gelman [2011], Gelman and Imbens [2013], Dawid et al.
[2016], Imbens [2020], where one is interested in identifying the causes of effects instead of evaluating effects of causes.
On top of several subtle differences, this problem is also relevant to graphical causal models pioneered by Robins [1986], Pearl [1995], Pearl and Mackenzie [2018].
Nonetheless, much of the recent focus is on distinguishing the “directions of causality” as discussed in Peters et al.
[2012], see, e.g., Imbens [2020] for a thorough comparison between these two different approaches for causal inference.
Causal inference in panel data and time-varying treatments.
Propensity score-based methods and marginal models are used in various scientiﬁc domain Rubin [1974], Rosenbaum and Rubin [1983], Robins [1986], Cole and Hernan [2008], Vansteelandt and Joffe [2014].
Inverse probability weighted methods are sensitive to speciﬁcation, and even if ground truth is known, the estimator itself may suffer instability issues, and does not cope well with high-dimensional covariates Viviano and Bradic [2021], Khan and Nekipelov [2022].
Bojinov and Shephard [2019], Bojinov et al.
[2021] adopted a design-based approach by assuming the treatment assignment mechanisms are known and studied the properties of inverse probability weighted estimators when the lag parameter is known (especially when it is either zero or one).
Similar in spirit, Viviano and Bradic [2021] proposed a dynamic covariate balancing by considering a linear model.
In the econometrics communities, there is much effort in drawing causal inference on policy evaluations in panel data.
Commonly used methods include difference-in-difference (DiD), synthetic control (SC) Abadie and Gardeazabal [2003], Abadie et al.
[2010, 2015], two-way ﬁxed-effect (TWFE) modeling, to name a few.
See, e.g.,
Athey et al.
[2021] for a uniﬁed framework formulated as matrix completion problems and Imbens [2022] for a introduction.
Along these lines of research, SC methods are most relevant to our problem in spirit where the treated group is usually few and one constructs a “synthetic control group” from the untreated group that best mimics the treated unit when the intervention had not taken place.
See e.g., Doudchenko and Imbens [2017], Abadie [2021] for a recent introduction and progresses.
Inference.
Inference is usually performed to test Fisher’s sharp null of no-effect Fisher [1958] or to test Neymann’s weak null of no average effect Neyman [1923].
Exact inference under randomization can be performed for design-based experiments Bojinov et al.
[2021], Bottmer et al.
[2021] or under outcome modelling.
Inference is more subtle for SC on the effect on the single treated unit, as such usually the average effect over time horizon or over units are studied.
The recent work Athey et al. [2021] referrs to the former as the horizontal regression and the latter vertical regression and studied their properties under the matrix completion literature.
Preliminaries and Setup 2.1 Background and Notations
We use capital letters to denote potentially multivariate time-series.
For each study unit i ∈ [ N ] , we assume there exists a covariate time-series X i,t ∈ X and an outcome time-series Y i,t ∈ Y for t ∈ [ T ] . We write Z i,t for unobserved covariates. For example, each unit may correspond to a cloud computing instance, Y i,t is the network latency of the i -th instance at time t , X i,t can be one kind of telemetry of the instance such as CPU usage or memory usage, and Z i,t is some unobserved variable of the instance such as internal state of the applications that are running on this particular instance. We focus on the case where all time-series are dichotomized, i.e., binary-valued X =
Y = { 0 , 1 } , though our method and analysis can be extended to the more general case where X i,t may be multivariate and takes more values. We omit the unit index i when making general statements for all units. We use the colon in the indices to select a slice (both ends inclusive) along that index, for example, Y i, 1: t = ( Y i, 1 , . .
, Y i,t ) , and use the shorthand notation ¯ Y i,t for Y i, 1: t .
We also put a superscript on observations such as Y obs to emphasize its non-random nature.
Without loss of generality, we assume Y i,t takes place instantaneously after X i,t and Z i,t but before X i,t +1 and Z i,t +1 for all i and t . We view the potential-outcomes Y i,t as functions of both observed covariates ¯ X i,t and unobserved covariates ¯ Z i,t , as well as its past history
¯ Y i,t − 1 .
When only partial information is available, Y i,t may contain randomness arisen from other variables. For example, we view Y 1 , 2 ( ¯ X 1 , 2 ) as a ﬁxed random function dependent on Y 1 , 1 and ¯ Z 1 , 2 .
Causal inference revolves around making comparisons between potential-outcomes at different treatment levels, which is in essence a missing data problem [Neyman, 1923, Fisher, 1958].
For example, when the treatment is known and when there is only contemptuous effect (i.e., Y i,t is only affected by X i,t , a common assumption made to simplify theoretical analysis e.g., in Arkhangelsky et al. [2021]), the potential-outcomes Y i,t is a function of X i,t
[-Ti := T − 1 (cid:80) T 
 + T t-]and we write Y i,t (1) and Y i,t (0) for the potential-outcomes under X i,t = 1 and 0 , respectively.
[-Nt := N − 1 (cid:80) Ni =
 ¯ τ i,t or-]Here one is usually interested in studying the [-population average ¯ τ POP := ( NT ) − 1 (cid:80) T 
-]treatment effect for unit i at time t deﬁned as
[Equation]
[-T 0 (cid:80) Ni =
 τ i,t .-]and perform hypothesis test on no effect across all units (i.e., Fisher’s sharp null) or estimating an average treatment effect (e.g., average over the temporal or unit axis, or both).
Speciﬁcally in the SC literature, usually one focuses on the case when the treated units are few, e.g., a single treated unit, and wishes to estimate
[Equation]
where we assume the ﬁrst unit is the single treated unit whose observed outcome is by assumption the potential-outcome under treatment.
This setup is particularly appealing for comparative case studies where potential-outcomes under treatment for other units may not always be well-deﬁned, see e.g., west Germany reuniﬁcaiton study Abadie and Gardeazabal [2003].
This canonical setup in SC literature closely mirrors our problem of causal explanation: the units that exhibit abnormal behaviors is few (compared with few treated units); estimation of τ 1 ,t is also of interest on top of inferences on average effects; units may not be comparable in the sense that exchangeability assumption may fail.
Furthermore, the most signiﬁcant complication is one does not know a priori whether there exists “multiple levels” of treatments when dynamic treatment effects are possible.
To this ends, we ﬁrst set up deﬁnitions that fasciliate our exposition.
2.2 Partial and Complete Covariate Paths
The ﬁrst complication for drawing reverse causal inference lies in the fact that compared with forward causal queries, the plausible answers may not be unique or even exist.
As such we ﬁrst deﬁne the history up to time t as H t := X t , and write H t = (cid:83) ts =1 H t as the set of (partial or complete) covariate paths .
Note that not all paths x ∈ H t :=
X t is realizable (has non-zero probability of being observed in the population) as we allow arbitrary dependency of Y i,t (cid:48) on X i, 1: t (cid:48) for any t (cid:48) ≤ t .
For example, in Figure 1, when T = 3 , (1 , 1) is a partial covariate path and (1 , 1 , 1) is a complete covariate path.
We further deﬁne for each index t , the intervention A ( s, a ; ¯ X i,t ) =
( ¯ X i,s − 1 , a, X i,s +1: t ) for any s ≤ t and use the shorthand notation A ( s, ¯ X i,t ) =
A ( s, 1 − X i,s ; ¯ X i,t ) .
In other words, these interventions alter the s -th component of the covariate series at time t , and leave everything else unchanged.
We use the notation ∪ to deﬁne path concatenation, i.e., if for some t ≤ T , ω ∈ H t then ω (cid:48) = ( ω, a )
= ω ∪ { a } ∈ H t +1 with ω (cid:48) s = ω s for s ≤ t and ω t +1 = a . With these deﬁnitions, at any time t , the potential-outcomes can be viewed as a random function taking an element in H t (or more generally, any ordered subset of X T ) as an argument where the randomness arises from the other (observed or unobserved) covariates that may affect it.
With these deﬁnitions, at any time t , the potential outcome can be viewed as a function taking an element in H t as an argument. We denote the observed outcome by Y obs i,t , which connects with the potential outcome via the consistency assumption Y obs i,t = (cid:80) x ∈H t 1 { x = X obs i, 1: t } Y i,t ( x ) , where X obs i,t is the observed outcome at time t .
We overload the notation x = y to return false whenever x and y are of different dimensions.
A concrete example in our problem for the outcome series Y is the indicator of presence of anomalies at each time step and X the metrics and other time-varying auxiliary information we collect.
We summarize our assumptions below.
Assumption 2.1 (Assumptions) .
We assume for all i ∈ [ N ] and t ∈
[ T ] the following assumptions.
Unconfoundedness: X i,t ⊥ ¯ Y i,T |{ ¯ X i,t − 1 , ¯ Z i,t − 1 } .
Non-anticipation: Y i,t ( · ) ∈
F i,t := σ ( { X i, 1: t , Z i, 1: t , Y i, 1: t − 1 } ) .
Consistency: Y obs i,t ∼ (cid:80) x ∈ H t 1 { x = ¯ X obs i,t } Y i,t ( x ) .
Individualistic/No-interference: F i,t ⊥ F j,s for all i (cid:54) =
j and t, s ∈ [ T ] .
[Figure]
[Figure]
Remark.
Note that the set of assumptions are similar in spirit to those usually made in the literature Robins [1986], Bojinov and Shephard [2019], Viviano and Bradic [2021].
Comparing with the work studying effect estimation in time-varying/dynamic treatments, our formulation have the several most distinctive features: (1) we do not explicitly specify which component of X i,t ( · ) are the “treatment” and which are “covariates;” instead, there might be a non-trivial pattern/dependency among them that jointly serve as the most causally relevant explanation for the anomaly.
(2) we treat the outcome as a function of (partial and complete) covariate paths, which allows us to specify covariates after the intervention time or averaging over all paths therein (which requires an exponentially many paths and manually selected weights), as are usually done in e.g., Bojinov and Shephard [2019].
The Reverse Causal Inference Problem 3.1 Problem Formulation via Causal Paths
We have used “causal” several times in the colloquial sense so far, yet it avoids confusion and is more illuminating when we formally deﬁne what do we mean by “causation:” we say X “causes”
Y or X “has an effect on” Y or X “provides causally relevant explanations” for Y if (1) with every pre-treatment (covariates that occur prior to X ) controlled, X and Y are correlated; and (2)
X precedes Y in time.
This perspective dates back to the very ﬁrst works on causation Neyman [1923], Fisher [1958] and was inherited by the potential-outcomes framework and adopted by causal practitioners in various ﬁelds.
We formulate our problem of reverse causal query as below.
Problem 3.1 (Reverse causal inference) .
Given covariate time-series X t and (observed) outcome series Y t , obtain the set Ω ⊂ [ T ] such that X Ω := { X t : t ∈ Ω } provides causal explanations for Y T in the sense that the change of any covariate X Ω indexed by Ω from 1 to 0 leads to the change in the potential-outcome.
The nature of this problem that multiple causes are plausible precludes putting assumption on the necessity between the change in X Ω and the change in the potential outcome.
We use the following example to illustrate the rationale behind.
Example 3.2.
Suppose T = 3 , Y 1 =
Y 2 =
and Y 3 = 1 { X 1:3 ∈{ (1 , 1 , 1) , (1 , 0 , 1) }} , then Ω = { 1 , 3 } , and changing either X 1 or X 3 results a change in Y 3 .
However, when there is a change in Y 3 , it is not necessary that the change was due to X 1 (or X 3 ).
Note that in this deﬁnition, we are essentially concerned with ﬁnding combinations of 1 ’s in the covariate series that jointly have an effect on the outcome while treating 0 ’s as “reference levels.”
The condition for the set Ω in the problem statement is made precise in the following deﬁnition of causal patterns.
Deﬁnition 3.3 (Causal patterns) .
Given a set Ω ⊂ [ T ] , we write X Ω ∈ X | Ω | to be a subset of ¯ X T such that X s = 1 if s ∈ Ω and X s is left undetermined otherwise. We say Ω is a set of causal patterns if the potential outcomes
[Equation]
differ when applying intervention A s on any index s ≤ t in the set.
In Example 3.2, the set Ω = { 1 , 3 } (which selects the ﬁrst and the third covariates, X 1 and X 3 ) satisﬁes the above deﬁnition.
Note that depending on different modelling choices, the expression Y T ( X Ω ) can be viewed as either a ﬁxed quantity or a random variable.
In latter case the non-equality should be interpreted in the sense of random variables (e.g., in the almost sure sense).
Note that in Deﬁnition 3.3, X Ω is a subset of X 1: T that may contain “holes” in the middle.
It is more convenient to work with an alternative deﬁnition in terms of covariate paths.
Deﬁnition 3.4 (Causal paths) .
Fix s ≤ T , we say ω ∈ H s = X s is an s -causal path for Y T if
[Equation]
In other words, ω deﬁnes a causal path from t = 0 to t =
s if the potential outcome along this path has a higher probability (over the randomness in the future covariates and unobserved covariates) of being positive and intervening the end point of this path reduces this probability.
For example, in Example 3.2, with s = 2 , neither (1 , 0) nor (1 , 1) is a causal path but when s = 3 , both (1 , 0 , 1) and (1 , 1 , 1) are causal paths.
As causal paths need not to be unique, we study their qualities through the following notion of globally and locally maximal probable causal paths, and we will study a weaker version of the reverse causal inference that attempts to ﬁnd the globally maximal probable causal path.
Deﬁnition 3.5.
Given s ≤ T , an s -causal path ω per Deﬁnition 3.4 is maximal probable if
[Equation]
argmax
[Equation]
We say a path ω ∗ is globally maximal probable if ω ∗ satisﬁes Deﬁnition 3.5 with s =
T ; we say a path (cid:98) ω is locally maximal probable if for any s ∈ [ T − 1] , ¯ ω s +1 maximizes over the same conditional probability conditioning that ¯ X s = ¯ ω s .
3.2 Optimality Condition for Local Causal Paths Algorithm
Input: Sample i,t Y i,t )
i ∈ N t ∈ T
Output: Causal path ω . 1: ω ← ∅ 2: for t ∈ [1 ..T ] do 3: Switch then 4: ω
ω ∪
5: else 6: ω ← ω ∪ { 0 } 7: end if 8: end for [Table] Deﬁnition 3.5 provides a way of quantifying the “quality” of different causal paths, and studying global maximal probable causal path provides a reasonable way towards the reverse causal inference problem.
Nonetheless, exact identiﬁcation of the global maximal probable causal path requires examination of exponentially many paths. In practice, the local maximal probable path can be found efﬁciently (to be discussed shortly), but it could be arbitrarily bad (measured by the conditional probability
{+P ( Y T = 
+}
| ¯ X T = (cid:98) ω ) ), as shown in the following theorem.
Theorem 3.6 (Hardness of the problem) .
Fix T , suppose
[Equation]
Let ω ∗ and (cid:98) ω be a global and local maximal probable causal path, then we have
[Equation]
Y i,t
X i,t t
T
[Figure]
0 0 0 0 0 ∗ ∗ ? 1 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 ∗ ∗ ∗ 1 0 1 1 0 ∗ ∗ ∗ 0 1 0 0 1 ∗ ∗ ∗ 1 0 0 0 ∗ ∗ ∗ ∗
Y i,t
X i,t t
T
[Figure]
0 0 0 0 0 ∗ ∗ ? 1 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 ∗ ∗ ∗ 1 0 1 1 0 ∗ ∗ ∗ 0 1 0 0 1 ∗ ∗ ∗ 1 0 0 0 1 ∗ ∗ ∗
Y i,t
X i,t t
T
[Figure]
where the inequality becomes equality in the worse-case scenario.
The proof is done by analyzing the structure of the covariate paths and is deffered to the Appendix.
This theorem suggests that a greedy algorithm could result to an arbitrarily bad path if we could choose the path weights adversarial.
We now {+investigate conditions on on the assignment mechanism P ( X t +1 = 
 | ¯ X t = x ) that would make locally maximal probable causal path more desirable. To that end, we consider the scenario where all causal paths satisfy a form of+} a “monotonicity condition” under which the greedy algorithm is optimal.
Deﬁnition 3.7 (Monotonic paths) .
We say the causal paths are monotonic if
[Equation]
implies that
[Equation]
for all s ∈ [ T ] .
Theorem 3.8 (Optimality condition) .
When all causal paths are monotonic, the greedy procedure returns the globally maximally probable causal path.
Proof.
Suppose the greedy path (cid:98) ω differ from the global maximal path ω ∗ , inspecting the child node of their lowest common ancestor implies a contradiction with the monotonicity assumption.
Note that the monotonic paths condition include the important case of Bernoulli random trials (where
X t ’ s are iid Bernoulli- 12 random variables), which is itself an important model in the literature (e.g.,
Bojinov et al.
Estimation via Generalized Synthetic Control
In this section, we substantiate Algorithm 1 by providing estimators for causal estimands that can be used to implement the Switch-Condition in the meta algorithm.
4.1 The Impulse Effect Estimand
To substantiate the Switch-Condition in the meta algorithm, based on our discussion in Section 2, we consider the following estimand evaluating the impulse effect at each covariate and timestamp t on the outcome at time T conditioning on the history seen so far, while being oblivious to the covariates between t + 1 to T :
[Equation]
We deﬁne the average effect as ¯ τ t = 1 N (cid:80) Ni
=
τ i,t .
Now given a partial causal path ω
∈ H s for s < T ,
Algorithm 1 decides whether X s +1 should be 1 or 0 in the complete causal path by examining
[Equation]
[Equation]
where, recall that we use ∪ notation to denote path concatenation.
Remark.
The rationale behind choosing this estimand is to ensure no post-treatments/mediators are adjusted Rosenbaum [1984].
Through enumerating all combinations of covariate-time indices, we are evaluating the direct effect from each covariate.
Compared with causal estimands on timeseries proposed by Bojinov and Shephard [2019], Bojinov et al.
[2021], our formulation allows for identifying speciﬁc patterns in multivariate series that serve as a cause for the outcome, and can be extended into an online algorithm.
4.2 A Generalized Synthetic Control Estimator
Outcome modeling is done by directly modelling the outcome on the treatment and pre-treatment variables.
In our setup, it amounts to modelling the outcome on the past history:
[Equation]
12) for treatment D i and covariates X i .
For any s ≤ t , given partial path ω ∈ H s , we use the shorthand notation µ i,t ( ω ) to denote E
[ Y i,t | ¯ X i,s = ω ] . Synthetic control type estimators is one good choice for this problem as the problem setups are similar: the units that exhibit abnormal behaviors is few (compared with few treated units); and estimation of τ 1 ,t is also of interest on top of inferences on average effects; units may not be comparable in the sense that the exchangeability assumption may fail.
The SC method then aims at ﬁnding the weights w = ( w 1 , . .
. , w N ) such that
[Equation]
The weights are usually selected according to pre-treatments and are constrained as
[Equation]
[Equation]
where we wrote ¯ X 2: N,t = ( ¯ X i,t ) Ni
=2 ∈
X t × ( N − 1) for the matrix with ¯ X i,t for 2 ≤ i ≤ N as its columns.
We will consider a generalization of the SC method to estimate Equation (11): given a partial path ω
∈ {+H s , we ﬁt both “synthetic treatment” (cid:98) µ s +1 ( ω ∪ { 1 } ) and “synthetic control” (cid:98) µ s +1 ( ω ∪ { 0 } ) on a “hypothetical unit” (say with unit index i = 0 ) with ¯ X 0 ,s = ω using two donor pools D 1 = { j : X obs j,s +
 = 1 } and D 
+}
= { j : X obs j,s +1 = 0 } respectively. For example, we ﬁt (cid:98) µ s +1 ( ω ∪ { 1 } ) = {+(cid:80) j ∈D 
 w j Y obs j,s +1 via+}
[Equation]
15) where ¯ X D 0 ,t similarly denotes the matrix with ¯ X i,s for s ∈ D 0 as its columns. We then form the generalized SC estimator for Equation (11) given time t ≤ T and
partial path ω ∈ H t − 1 as [Equation] The rationale is that as we reﬁne the causal path, there might be no unit that has the exact partial path.
Theoretical studies of the properties of SC estimators in the literature usually adopts the linear factor model (see e.g., Abadie et al.
Analogously, we study a generalized version of the linear factor model.
The full details and the proof are postponed to the appendix, and the proof is similar to that in Abadie et al.
Theorem 4.1.
Suppose at time T 0 , there are N = N 0 + N 1 units among which N 0 are treated (i.e., X i,T 0 = 1 ), ﬁx T 0 ≤ T , assume the potential-outcomes follows a general form of the linear factor model
[Equation]
where δ t is known, ε is the noise, and u t and λ t are known.
Under regularity conditions on λ t , u t , ε and assume SC ﬁts are perfectly well, which are given in the Appendix, the SC estimators are consistent and asymptotically normal.
p 0 .0 .0 .0 {+.
+} .ˆ p
Method SCGSC (a) T = 8 ,q = 0 .
p {+0 .0 .0 .0 .
 .ˆ p+}
Method {+SCGSC (b) T = 8 ,q = 0 . p 0 .0 .0 .0 .
 .ˆ p+}
Method SCGSC (c) T = 8 ,q = 0 . {+p 0 .0 .0 .0 .
 .ˆ p+}
SCGSC (d) T = 15 ,q = 0 .
p 0 .0 .0 .0 {+.
 .ˆ p+}
SCGSC (e) T = 15 ,q = 0 .
{+p 0 .0 .0 .0 .
+}
.ˆ
p SCGSC (f) T = 15 ,q = 0 . p {+0 .0 .0 .0 .
 .ˆ+}
{+p SCGSC (g) T = 100 ,q = 0 . p 0 .0 .0 .0 .
 .ˆ+}
p
{+SCGSC (h) T = 100 ,q = 0 . p 0 .0 .0 .0 .
 .ˆ p+}
SCGSC (i) T = 100 ,
[Figure]
Empirical Studies
We now test our method on a synthetic dataset.
Although the dataset is fairly simple, it helps to illustrate the challenges of the problem and fascilates the comparison of different methods.
5.1 Setup and Synthetic Dataset
[Table]
We consider a generalization of the model in Figure 1.
Given T , we consider a set of patterns of length l = 3 that starts at T − d ( d ≥ l ) with Y t = 0 for all t (cid:54)
=
T .
That is, there is an anomaly at Y T only if certain patter emerges at time T − d .
Consider the assignment probability mechanism in Table 1 and the outcome model as
Y [Equation]
) } for independent Bernoulli random variables Z 1 ∼ Ber(1 / 2) and Z 2 ∼ Ber( p ) for some parameter p .
Finally, we sample X j iid ∼ Ber( q ) for a ﬁxed parameter q , which models the background noise due of the covariate series.
We experiment with both the case where the time horizon is short ( T = 8 or T = 15 ) and is moderately long ( T = 100) , we set d = 3 in the ﬁrst case and d = 30 in the latter. In both cases, we experiment over p ∈ { 0 . {+5 , 0 . 75 , 1 } and q ∈ { 0 . 01 , 0 . 1 , 0 . 5 } . Note that by construction, the globally maximal probable causal path has the conditional probability P ( Y i,T = 
+}
| X i,T − d − l − 1: T − d ) = p .
For each parameter conﬁguration, we generate N = 500 units on which we compute the optimal causal path ω and record the mean outcome probability
E Z [ Y T | ω ] .
We repeat this procedure for m = 50 independent runs.
5.2 Methods and Evaluation
We compare the following methods.
Point causal estimand using SC (SC).
Here we directly apply synthetic control to estimate the effect of each covariate X t to Y T for t ≤ T by ﬁtting SC of a randomly chosen unit with X i,t = 1 and estimate ¯ τ t .
In this way, the baseline has the same computational complexity with the generalized SC estimator.
and 0 otherwise.
Generalized SC (GSC).
Here we apply Algorithm 1 with the generalized SC estimator introduced in Section 4.2 and append 1 to the partial path if the estimate is positive.
Evaluation.
We use the true conditional probability P ( Y T = 1 | ¯ X T = (cid:98) ω ) , which is known from dataset construction, to evaluate different methods by computing the average conditional probabilities (cid:98) p of causal paths found over independent runs.
In our synthetic dataset, the perfect algorithm should have (cid:98) p OPT = max { 0 . 5 , p } while a random algorithm will have (cid:98) p random =
5 + p ) / 8 since we are focusing on the pattern of length 3 .
5.3 Simulation Results and Discussions
Baseline method performs poorly.
We observe that in Figure 3, the baseline method, where we ﬁt
SC on point effect without adjusting for dependencies among covariates, performs poorly, though the synthetic dataset seems simple.
This suggests that unlike forward causal inference problems, reverse causal inference is harder and much care needs to be taken when performing identiﬁcation.
GSC identiﬁes the a maximal probable causal path reasonably well.
We observe that the GreedySC method is able to identify the globally probable causal path reasonably well as the GSC estimates is consistently above 0 .
5 , indicating a causal path is returned. GSC fails to identify global causal paths when the time horizon T becomes larger.
As we increase the time horizon T from 8 to 15 and to 100 , we note the it is gradually harder for GSC to ﬁnd a global causal path (while still outputting a local causal path, in contrast to the vanilla SC estimator). This issue can be alleviated through constructing a doubly-robust version of the
GSC estimator, which we discuss in a following-up work. Effects of p and q on the difﬁculty of the problem. We observe all estimators have comparable performances as we vary the background noise level q , indicating q kis not a decisive factor that governs the difﬁculty of the problem.
Concluding Remarks and Broader Impacts
In this paper we give a precise formulation of the reverse causal inference problem through causal patterns and causal paths.
We propose to use a linear-time approximate procedure that can be ﬂexibly combined with any causal estimator for this purpose and analyze its optimality conditions.
We propose to use a generalized synthetic control estimator for this problem.
There are several avenues for future work, for example, we only attempted to ﬁnd causal paths but not causal patterns, which is more difﬁcult.
As reverse causal inference is a challenging problem, we hope our expositions could cast new lights into the community and inspire the practitioners
A Miscellaneous Proofs A.1 Proof of Theorem 3.
Proof.
For ease of notation, write f ( ω 1: s ) =
P ( Y T = 1 | ¯ X 1: s = ω 1: s ) and f ( ω ) =
f ( ω 1: T ) .
First consider the case where there are only two causal paths and let T − K − 1 ≥ 0 be the time index of the lowest common ancestor of ω ∗ and (cid:98) ω .
Note that when K = 1 , (cid:98) ω cannot be worse than ω ∗ , hence it sufﬁces to consider the cases for 2 ≤ K < T .
We have
[Equation]
where the factor 2 arises as in the worst case there are (cid:12)(cid:12)
X K − 1 (cid:12)(cid:12) =
2 K − 1 choices for ω (cid:48) .
Similarly,
[Equation]
[Equation]
by leaving out every path that is not ω ∗ .
But by construction
[Equation]
) hence
[Equation]
21) for any K ∈ [ T − 1] .
In general when there are multiple causal paths, let ω ∗ be the path that has higher probability than (cid:98) ω which has the lowest common ancestor, and iteratively applying the above reasoning, we see that
[Equation]
thus completing the proof.
A.2 Proof of Theorem 4.
Proposition A.1 (Formal) .
Given N = N 0 + N 1 units among which N 0 are treated, ﬁx T 0 ≤ T , assume { u t } T 0 t =1 and { λ t } T 0 t =1 are known, writing
[Equation]
if ΛΛ (cid:62) is non-singular, ε i,t is independent with { X i,t , Z i,t } for all i .
Suppose we ﬁt SC (or synthetic treatment) for all units whence there exists weights { w ij } Ni,j such that
[Equation]
24) then
[Equation]
(25) is a consistent estimator for τ i,t , and consequently,
[Equation]
(26) is a consistent estimator for the ATE ¯ τ t .
In fact, we have
Bias( (cid:98) ¯ τ SC t ) ≤ C
[Equation]
p (cid:18)
[Equation] ξ [Equation] where C p is a universal constant that depends only on p , ρ N := max { N 0 , N 1 } /N , and ¯ m p :
=
max j T − 1 0 (cid:80) Nj =1 | ε jt | p .
Proof.
First consider the case where there is only a single unit under treatment at time T 0 ≤ T , which we assume without loss of generality to be the ﬁrst unit:
[Equation]
then the error r t at time t ≤ T 0 satisﬁes
[Equation]
Write r = ( r
[Equation]
[Equation]
[Equation]
0 , then
[Equation]
Thus r
[Equation]
(31) where we write Λ (cid:48) :
= (ΛΛ (cid:62) ) − 1 Λ .
Now taking expectation with respect to the randomness in ε j,T and those in
Y j , recall that w 1 = 1 is ﬁxed, hence the only non-zero term is
[Equation]
λ [Equation] since the condition Equation (24) may introduces correlation between ¯ ε j,t .
Write
[Equation]
we have by Cauchy-Schwarz inequality
[Equation]
since we have assumed ΛΛ (cid:62) to be non-singular, hence so is (ΛΛ (cid:62) ) − 1 .
Now assume further that
[Equation] [Equation]
and | λ sk | ≤ c for all s ∈ [ T 0 ] and k ∈ [ M X ] , then
[Equation]
By Hölder’s inequality,
[Equation]
and
[Equation]
where we assume the p -moment of ε js exists.
Applying Rosenthal’s inequality yields
[Equation]
where we write
[Equation]
and C p is an absolute constant that only depends on p .
Thus
[Equation]
where ¯ m p = max j ¯ m p,j .
In the general case, assume there are N 0 treated units and N 1 untreated such that N = N 0 + N 1 .
We then ﬁt SC for each treated units and similarly synthetic treatments for each untreated units.
Write ρ N := max { N 0 , N 1 } /N , which we assume 0 < ρ N < 1 , then we have by applying the above result,
[Equation]