Causal Inference for Event Pairs in Multivariate Point Processes
Authors: Tian Gao
Dharmashankar Subramanian
Debarun Bhattacharjya
Xiao Shou
Nicholas Mattei
Kristin Bennett
Venue: NeurIPS 2021 Poster (2021)
Keywords: causal inference
point process

Abstract: Causal inference and discovery from observational data has been extensively studied across multiple fields. However, most prior work has focused on independent and identically distributed (i.i.d.) data. In this paper, we propose a formalization for causal inference between pairs of event variables in multivariate recurrent event streams by extending Rubin's framework for the average treatment effect (ATE) and propensity scores to multivariate point processes. Analogous to a joint probability distribution representing i.i.d. data, a multivariate point process represents data involving asynchronous and irregularly spaced occurrences of various types of events over a common timeline. We theoretically justify our point process causal framework and show how to obtain unbiased estimates of the proposed measure. We conduct an experimental investigation using synthetic and real-world event datasets, where our proposed causal inference framework is shown to exhibit superior performance against a set of baseline pairwise causal association scores.

Causal Inference for Event Pairs in Multivariate Point Processes
Tian Gao IBM Research tgaouus. ibm. com
Dharmashankar Subramanian IBM Research dharmash@us, ibm. com
Debarun Bhattacharjya
IBM Research debarunb@us .iaiass.bbme imm. com
Xíao Shou RPI shoux@rpi.ede edu
Nicholas Mattei Tulane University nsmattei@tul ane. edu
Kristin Bennett
RPI bennek@rpi.c edu
Abstract
Causal inference and discovery from observational data has been extensively stud- ied across multiple fields. However, most prior work has focused independent and identically distributed (i.i.d.) data. In this paper, we propose formalization for causal inference between pairs of event variables multivariate recurrent event streams by extending Rubin's 's framework for the average treatment effect (ATE) and propensity scores to multivariate point processes. Analogous joint probability distribution representing i.i.d. data, multivariate point process repre- sents data involving asynchronous and irregularly spaced occurrences of various types of events over common timeline. We theoretically justify our point process causal framework and show how to obtain unbiased estimates of the proposed measure. We conduct experimental investigation using synthetic and real-world event datasets, where our proposed causal inference framework is shown to exhibit superior performance against set of baseline pairwise causal association scores.
Introduction
is widely known that the gold standard for effective causal inference is through the use of inter- ventional data such randomized controlled trials, for measuring the impact of some treatment on an outcome of interest 330. However, intervening the system can often impractical or even impossible, in which case one must conduct causal analysis using observational data alone.
Observational data the form of multivariate event streams common many domains including health, finance, and retail. Analogous to how an observation in an i.i.d dataset can be viewed as sample from a joint distribution over a of random variables, a multivariate event stream can be viewed as a sample from nultivariate point process over set of event labels [[C, where the instantaneous rate any event label depends on the historical occurrences some subset of the event labels. Modeling, fitting, and predicting future occurrences given a multivariate event stream is an active area in statistics
40 data mining
and machine learning
44
Recent work in this area leverages advances in deep learning 45
21
Related work in survival analysis focuses only on pairs of events with continuous covariates involving short event streams with both continuous and discrete variables, and most notably 47 only single occurrence the outcome
2X]
2 While a general theory for point process counterfactual practical algorithms focus on continuous covariates and hazard
inference has been developed B77 M.
models [39]. Moreover, work in dynamic treatment continuous-time data, which aims to reduce bias or high variance in discretizing time 38
226. [8 for discrete time, motivating work for
43 In
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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contrast, here we focus on the setting where data is of the form of long event streams with multiple occurrences of various types of events, including both the treatment event and outcome event.
Causal inference entails drawing a conclusion about a causal relationship between potential causes and effects; in this work, we focus on causal inference between pairs of event labels observed in event stream data, which differs from standard Granger causal graph learning 22] 49. Specifically, we pose the following causal inference problem for multivariate point processes: how can one meaningfully measure the causal relationship between a cause event label z and an effect event label y? Such a causal measure would explain whether the cause label 2 amplifies, inhibits, or has no impact on the effect label y, while taking into account the potential effects of all other event labels x.
This problem deviates from the typical causal inference setting in several ways. First, most methods assume i.i.d. observations; in our setting, events may be correlated across time and therefore it is necessary make certain independence assumptions in order to identify the effect of interest. Second, most causal inference settings are concerned with the expected value of a single observable outcome such as mortality. Our setting has repeated occurrences of outcome over time and therefore involves event frequency (or rate), which existing pairwise causal inference models are unable to handle and therefore necessitates new methods. Note that since event label 2's historical occurrences could have a complex dynamic effect on another event y's occurrence at any time in a multivariate point process, there are a number of questions to address around the fundamental problem of even defining the notions of treatment, outcome and covariates, before one can study the causal effect of z on y.
There several real-world applications that motivate our work on causal inference with event stream data. For instance, one may be interested in understanding causal relations between events relevant to diabetic patient, such as meals, insulin intake, exercise and blood glucose level changes ( Detecting issues in microservice or other web applications requires detecting which faults or abnormal states cause other downstream faults from system logs K Example causal pairs from other domains are: food shortage and protests, which are recurring event labels in socio-economic datasets 27] and earning call events and stock price jumps, which are recurring event labels in financial datasets 42 All these examples have an observable outcome that occurs repeatedly, necessitating a new approach.
Contributions We make several major contributions: 1) we propose formalization of causal inference between a pair of event variables in a multivariate point process, 2) we define an average treatment effect (ATE) for multivariate point processes the effect of having observed treatment in proximal window, 3) we derive and obtain the equivalence of propensity scores and balancing scores for ATE in multivariate point processes, 4) we propose an inverse propensity score re-weighting procedure for better ATE estimation to adjust for the impact of other covariates, and 5) we conduct a detailed empirical evaluation on both synthetic and real datasets with baselines show the superior performance of our proposed methods in pairwise causal inference in multivariate point processes.
2 Background
A multivariate event stream (or event dataset) is sequence of events, D (D,) where each event D; includes an event label (or synonymously type or variable) and a time-stamp, i.e. D. (I., fz). z; belongs to the label set L, whose cardinality is M, i.e. |C] M, and t, is the time of occurrence, 4 E R+. We assume a strictly temporally ordered dataset, I2 < Lj for < j. initial time I1 0 and end time |N+1 T. Let z, y refer an arbitrary pair of event labels in L.
Multivariate event streams can be regarded as samples from multivariate point process (MPP), where each label has counting process [0]] It uses conditional intensity functions Xy (t)H,) > 0 that denote the instantancous rate of type y given the history up to t, i.c. Hi = {(x,, t.) t4 < f). In multivariate point process, the probability of observing y as the next event at time t is:
He.)) (t)7t) S. >>> Ex((((H))
(1)
ZEC
where in is the most recent event occurrence time before t. It can be shown that (t =
E[N,((t, dt 10 33), where NA( is the number of y occurrences in time interval A.
Prior work has proposed the notion of process independence among event labels' counting processes M1 The basic idea is that the intensity of one type of event does not depend on certain past events once we know about specific other past events. This an asymmetric concept, similar to Granger causality. Informally, for sets of labels x. y and z s.t. y O 0, x is process independent of y given

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Z when all labels in x have conditional intensities that do not functionally depend at any time on the history of labels in y. given the history labels in z. Process independence captured in graphical event models [ 15 where the conditional intensity for any event label at any time / depends only on historical occurrences of its event label parents in the underlying graph.
A minimal graphical representation can be used to define direct causes in multivariate point processes, analogous those in causal networks. Here we provide simplified definition of direct causes M. Definition 1. Event label z is direct cause of label if belongs to the minimal set of nodes u s.f. y is process independent of all other labels given u.
minimal set entails that none of its subsets satisfies the property. We highlight that in this work, we interested in identifying pairwise causal relationships without learning full graphical model. This is because can often be computationally intensive as well as unnecessary to learn a full model when one is merely interested in pairwise causality.
3 Causal Inference in Point Processes
The Neyman-Ruude Baiiann-Ruuinggge potential outcomes causal inference framework estimates the treatment effeets between treated variable and effect for i..dd. data BT 20 average treatment effects (ATE), to study how an event label z affects event label in a MPP. In this class of models, treatment is denoted potential outcome y" for each 2 is the outcome if treatment 2 is applied. The main difficulty in causal inference stems from the fact that only outcomes from the administered treatments are observed and never any of the other outcomes. Hence it may be viewed as a missing data problem.
We extend this framework, including
E {0, 1}. where 0 is the control and the treatment. The
We estimate the treatment effects between treatment variable associated with historical occurrences of z and an outcome variable (or response) associated with y's occurrence, under existing assumptions in the literature. Covariate variables x involve historical occurrences of labels other than i.c. x = L \ 2. We assume all variables are observed, hence strong ignorability is satisfied (which we will discuss further in Section B.2). Next we define treatment, outcome, and covariate variables, and then derive the propensity score for ATE computation.
3.1 Defining Treatment, Outcome and Covariates
There are many possible ways to define the treatment, outcome, and covariates in a multivariate point process. We begin with the following general formulation. We use Z, to represent the treatment variable at time t, distinguishing it from the event label z. Let HH be the history of occurrences of event z before f, and K? be the future of event y occurrences at
Definition 2. General Formulation for Causal Inference: for pair of event labels (2,8), the treatment variable Ze at time t is a function of z's historical occurrences, i.e. Z, = fH(H). The outcome variable is function of y future occurrences at t, i.e. Y, fy (KT). The covariates X. function of historical occurrences of event labels other than 2, i.e. X, f
Treatment and Covariate Models We need to specify the functions in the general formulation of Definition 2 for instance, we need to summarize historical occurrences H₁, Due to potential multiple ooce-occueeecce of treatment and outcome, the representation of history needs careful modeling based on specific assumptions at each time I. We take simplified yet practical view of treatment, making the proximal assumption that the most recent history sufficient for causation. We do this by assuming only recent time window [ - (.w,,, w,() for some window could directly have a causal impact on the outcome at time t; such an assumption has been used previously in point process models f and provides efficient and interpretable definition for treatment. It fits many real-world situations where the influence comes primarily from recent history and repeated occurrences have little additional impact the intensity rate. It also has a strong statistical justification: matching long treatment history could result in infinite sample requirements. We assume covariates are also defined by whether they have appeared at least once in some recent time window.
Outcome Model We wish to measure a statistical quantity fy associated with the outcome in a multivariate point process, based on future occurrences of y, KCY, from time 1. Many such quantities (XC)) can be used. It is natural to consider either the instantaneous expected number of occurrences of the outcome y. ([t, t + dt]) H.). in place of Kcy, or cumulative expected number
3

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of occurrences of outcome in a future duration wy. ((t, + wy)))) Moreover, the expected count E|N, y (it, + dt,) equalss (t)dtt by definition. Note that at each time, the history can be very different with the proximal assumption, hence the rate at each time in K., can also be different. The cumulative counts can be similarly written as Ag (1) p X (t)dtt for some future duration Wy, which can be the total time horizon or infinity, but practically it is more meaningful for a specific duration suitable for the application.
We will focus our analysis with (t) as the outcome in the following, which could be directly extended to outcome Ag (t). We will compare with both definitions of outcomes in experiments.
Hence, our specific realization of the general formulation results in the following formulation:
Definition 3. Recent History Formulation for Causal Inference: for a pair of event labels (z, y), the binary treatment variable Z at time is defined by whether or not has occurred least once within a window w into the past from I. The outcome variable is the occurrence rate of the effect label y at time given the treatment, Y₁ = >y) (t). The covariates X at time are a binary vector, depending on whether other event labels have occurred at least once in [t w. f), w E (0, t).
In event streams, it is possible for the cause and effect to be the same label in history, for instance, consider a system with negative felebeee (eating too much sugar for example). Hence and X, could be identical to or contain but 2 and X, must be disjoint sets of events.
We summarize the assumptions in this formulation of causal inference: 1) events before & w have no impact on y's occurrence rate at time t. This enables memory in time yet provides a compact representation of history, 2) only the occurrence of in the window impacts y's rate at time t (regardless of the number of occurrences, hence we do not differentiate between multiple occurrences within w), and 3) the specific times of z's occurrences do not further affect y's rate at Such model can be robust to outliers or noisy historical observations.
time [6 29 15
3.2
Defining Average Treatment Effect
To measure how label y responds to historical occurrences z, average treatment effect (ATE) can be extended to our formulation in MPPs. We define the mean potential outcome Y, under treatment assignment Z
36
as μ == } S dt) where T is the maximal time horizon. p has an intuitive interpretation: it is the rate of y given Z = k: in its history w, averaged over time horizon to = 0 to T. Note that H measures the temporal average of instantaneous event occurrence counts under treatment.
natural way to aggregate an event's expected occurrence counts in continuous time and captures the event dynamics affected by the proposed treatment.
Definition 4. Average Treatment Effect (ATE) event pairs is defined as:
ATE = EnT [ny - pb] = ENT J. x2(0)= - (t)dtt
(2)
where (2) is Ay1z, (1) at time if occurs at least once in Hr (Z.- 1) and X5(1) is A912, (1) t if does not occur in Hr (Z," 0). The Ex, [+] is the expectation with respect to the random trajectories over [0, 7' as induced by the history dependent MPP. Note that since the rate X is the instantancous expected count of occurrences, ATE can also be viewed as a measure of expected count differences, normalized by the length of the time horizon T.
For known multivariate point process, one can find the counterfactual Sy|z, zz (t) and hence the ATE by looking up the conditional intensity function with the corresponding parental conditions after making an adjustment to the treatment. Specifically, removing all occurrences of in time period [t - w. 1) the factual is z.n = 1 or inserting & at time / w if the factual is Z
Theorem provides justification for choosing ATE to measure pairwise causal relations in a MPP:
Theorem 1. If z is not direct cause of in multivariate point process, ATE for pair (z, y) is O.
All detailed proofs are in the Appendices. The proof of Theorem | directly uses Definition I to show that the intensity rate does not change per z. The above theorem also implies that if is not a parent of y in graphical event model representation of the underlying MPP, the ATE for (z, y) is as in Equation from Definition 4

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To use ATE as defined in Equation 2. there are several assumptions that must hold in order to mimic the randomized trials to truly establish causal relationships. The ignorability condition indicates that whether y is O or 1 at each time 1 does not depend on whether " 1 or 0 at that time, i.e. (A)(1), X (t)) 1 7 [X, VI (although the y's i' intensity rate still depends on 7z("). Moreover, overlap condition states that each time & has strictly positive chance for Z = 1 to happen given its history, i... 0- < P = 1|x,)= < 1. Strong ignorability is often used when both overlap and ignorability hold true. If the assumption does not hold, possibly due to covariate differences in treatment groups, we need to adjust for these effects using propensity scores. This is particularly important in observational event data, where different times with Z = 1 and Zt 0 may not be directly comparable, because the covariates X may not be similar to each other. It can be studies further with the back-door criterion 3T.
,,
3.3 Defining Propensity Scores
The propensity score is proposed to resolve covariate differences in non-randomized experiments to mimic a randomized study. It is a balancing score: conditioned on any balancing score, the distribution of observed covariates will be similar between the treated and control groups. It has been widely used in i.i.d. settings 22
1 !
We define set of balancing scores, b(H,,) (6; for short), where the goal is make treatment conditionally independent of covariates given the scores: Z, 1 H. H1,210(H4 (b(H,, 2) Vt. Based on the spec- ification provided by Definition S we use the recent window assumption and summarize the history H with a set of covariates X Hence the balancing score would achieve Z 1 X th [6(X")) Vt. The most trivial ((X)) is b(X;) = X.." However, to consider the sampling distribution of potential cause Zr, many-to-one functions of X offer better adjustment, and the coarsest of such a function is the propensity score. We define X is finer than Y (equivalently, Y is coarser than X) if y f(X). "Coarsest" here means that Y's dimension cannot be reduced further.
We use the above defined treatment 2 and covariates X and derive the equivalent form of propensity score for MPPs. The goal is derive the propensity score e e(X,"), or ef for short, such that P(Z." = X et) = P(Z th = 1|cz). We know that:
p(z." 1 D(Z =:
(3)
D(X.")
where D( = 1; X is the duration that Z is observed true given observing covariates X in the dataset. We define P(Zz" = 1lez) similarly as P(Z. = lee)) = =1:e= D = e;) can be interpreted as the duration that " is observed true, i.e., the total duration for which occurs in (t-wu w, f). Vt, in the dataset with the same of in the relevant preceding windows. To ensure the required equality, we define et as:
D(eī)
Definition 5. The propensity score for all covariates in history X at time is defined as et e(X.") = P(Z," = "
Note that Z at any time follows a distribution P(Z." = 1[X,"). which depends on the values of X!" This quantity can be estimated efficiently by counting the duration of 2," 1 with different X values, which we will elaborate upon later. The major distinction between the proposed propensity score and that for the traditional i.i.d. case is that ef is time-variant and incorporates all occurrences and aonocuurreenes of events in window-viewed history.
We can show that times with the same value bū but different treatment Z can act as controls for each other, as their expected outcome difference equals the proposed ATE:
Theorem 2. Suppose the treatment event Z is strongly ignorable and bj is balancing score. Let
[ut] = + J. (d)] Then, |u||x, 2 = (u X z 0] = [bM [g]bb]].
Proof sketch. If the treatment is strongly ignorable, then it is also strongly ignorable given any balancing score. Therefore, we can show that the expected differences of both sides are equal. We further can show that treatment Z!" and covariates in history X conditionally independent, given the propensity score et, which makes ef a balancing score. Please see the Appendices for exact proofs and further discussions.
follows that the two-sampling process gives an unbiased estimate of ATE, using pair matching and sub-classification techniques (36] to adjust for the propensity score. However, in practice, pair
5

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matching is difficult to perform given the continuous nature of time which results in infinite sampling sizes. Subclassification is also difficult when the number of covariate event labels is large, leading to a high sample requirement on T.
4 Estimating ATE Scores
There are several parameters that need be provided or estimated in order to adjust for the propensity score and obtain ATE, including the window size for the treatment definition, conditional intensity rate of the treatment, and the outcome conditional or cumulative intensity rates.
Window Size w for the Treatment:
The window size w of the treatment Z definition is provided as an input to the ATE estimation procedure. Moreover, it is used as the window size for all quantities If one wants to estimate
associated with a window, namely X Zt", >y/2, (t), and P P(Z window, practitioners should exercise prior knowledge to set w since the treatment effect with different windows will produce different results.
Alternately, one could take data-driven approach to learning window for particular pair of event labels (2.y), y). One candidate is the heuristic pairwise approach in ? that chooses from a set of inter-event times in the event stream data. Note that windows that too short too long could result in treatments that are always absent or present, respectively, making the ATE computation unsuitable and unstable. There is therefore trade-off between long (which captures the historical influences as much as possible) and short windows (which captures only the important subset of history). We include more details in the Appendix.
Propensity Score ex Estimation: Let DZZ = 1;x;") be the duration that Z is observed true in the dataset and that the condition x!" is true in the relevant preceding window Formally,
N+1
D 1;x
1xy (Ze 1,t)dt
(4)
where lxx (2f = 1,t) is an indicator for whether each instantiation of XT", x!" is true and Z = 1 at time as a function of the relevant windows w. Similarly, D(x,") is the duration that the condition
x3" true in w. Hence, P(Z xx =1x"" and P(Z = 1 is a vector of such probabilities with size |X|]]
D(x)))
Outcome Conditional Intensity Rate X (f): Given the most recent history view in the treatment definition, we estimate Ay1z, (t), VO < t < T by using the same recent history formulation. This assumption has been used previously in MPPs, such as in proximal graphical event models (PGEMs) 77] where the rate of occurrence of event type depends only whether not its parents have occurred in the most recent history.
When the parents U of all nodes V are known, the log likelihood of a MPP represented by a PGEM can be simplified to a function of counts and durations in the data and the conditional intensity rates:
(D) C-Ar D(u) + N(;; u) In In().:
where u is an instantiation of the parent of each variable v, NN; u) is the number of times that X is observed in the dataset and that the condition (from 2 possible parental combinations) is true in the relevant preceding windows, and Duu) is the duration over the entire time period where the con- dition u is true. Formally, N(c;; = E. I(4 " (t,) and (uu) ΣN+1 fr I (t)dt, where I (t) an indicator for whether u is true at time as a function of the relevant windows w.- We refer the reader to 7 for more details around model learning.
We set z as the parent of y in PGEM and then can estimate x and A which are intensities when z does and does not occur in the provided proximal window w. It is efficient since we only look at two events and only need to estimate intensity rates without learning windows. Once we have these two intensities, we can then look up ÁyIz, (t) by finding the treatment Z!" state at any query time
wy Outcome Cumulative Intensity Rate 9 2 (4): The cumulative intensity is computed as the integral of the conditional intensity over a future window wy, assuming that no other events occur. Again, given a learned point process model, we take recent history view and compute the integral using

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Algorithm Inverse Probability Weighting for Events
Input:
Event data D, time horizon
Sample times from 0 to T.
Compute the propensity scores et, Vt, per Equation
Compute wy. Vt. per Equation o stabilized scores.
Train a PGEM My to estimate Ayiz, .
(1). vt.
Predict with My for given 7i, the outcome (t) with Z and (t) with 2 0.
Compute ATE per Equation i
estimates for conditional intensities Aij= and S Computing the integral from t to t + Wy is straightforward as these are piece-wise constant conditional intensities. Furthermore, it is only possible for the treatment condition change from Z = 1 to Z 0 in the future window, when the condition from the occurrence of event label = changes, but not the other way around.
Inverse Probability Weighting for Events: The idea of weighting samples is as follows. If the time units for data involving the treatment ~
group are far fewer than those in the control group Z 0 in event dataset, need upweight the time units in the treatment group and down-weight those in the control group for better estimation. We use inverse probability of treatment weighting (IPTW) as the weighting scheme, in order to adjust for this imbalance in the population.
We define the weight for the conditional intensity at time in the treated Z," group as a4 ej and for the control group Z," = 0, the weight becomes a, 1- -(2"--11
I-c7 Together, the weight for the outcome intensity rate at time t is defined as:
Z
2
C22
(6)
Using Equation
6 we propose an inverse probability weighting method for events, detailed in Algorithm We hirst estimate the propensity score using Equation 3 and then estimate w, for all t
Then we can choose PGEM My to predict Ay1z, (t), for both the factual outcome and counterfactual Zz (t). Computing their empirical expectation provides an ATE estimate:
using Equation 6
ATE E'T 'T L-d an x,(()
1 l" a. X²(( T 1
Equation requires one to compare occurrence rates for all t in T'. which includes some epochs with treatment and some without. Since this is continuous time setting, integration over time is not straightforward. Hence, we propose using a sampling procedure to compute the inner integral over time in Equation as Σ a (t)( Σ a 2t(t), where S is the desired number of epoch samples from to to T.
Corollary 3. Inverse probability weighting for events with balancing scores. Under strongly ignor- able treatment, the weighted average with inverse probability weighting is unbiased for ATE when weights equal those in Equation 0
Proof Sketch. The inner integral of ATE can be shown to equal to [x]]] [ug.. and further [t] =
T J.
z."h x7(() dt and [x)]()) J. (1 (1-2," (+))())))))))))))))))))))))))) dt.
<
1-47
We show in Corollary that our ATE definition is unbiased estimator. Consistency results, however, rely on the estimation results of intensities from specific point process models. For example, when the window sizes are given, PGEM estimator 7 for intensities are consistent.
Since we consider (interval) stationary point process here, we can follow the standard computational procedure estimate the sample variance of ATE estimator and sample-based confidence interval for the ATE estimate (for example, results from Theorem 6.2 in [9 can be directly extended to our setting), as the samples are from the same distribution.
Factual and Counterfactual Treatment Usages compute the ATE, we need both factual and
counterfactual treatment for time. For time / with factual Z. = 1, the counterfactual Z. 0
treatment modeled by removing occurrences of in [t - w., t). For time with factual 2/
the counterfactual Z. 1 treatment is modeled by introducing one occurrence of in [t t).
7

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ATE accuracy results different causal pair scores for PGEM synthetic datasets.
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(h) ATE accuracy results on different causal pair scores for Hawkes synthetic datasets.
For ease of computation, we introduce the counterfactual occurrence at - w for instantaneous conditional intensity computation, although its exact location would not impact the outcome due to the proximal assumption. For cumulative intensity outcomes, we introduce the counterfactual occurrences f öt, with öt which is the middle point of the window. Other choices át of introducing the counterfactual treatment are also possible.
Stabilized IPTW One common issue with IPTW is that sometimes the propensity scores for some time unit can get very close to 0, which indicates Z is extremely unlikely to occur in the window [t f- w, t); this is even more likely for continuous time data. Hence, the weights for those t become extremely large, causing unstable estimations. To combat this issue, the stabilized IPTW (35) uses the marginal probability of treatment counteract such an instability. It is formulated 04 2 =1) (1-
1-6
5 Empirical Evaluation
Evaluating causal inference algorithms is more difficult than those for prediction tasks since observa- tional datasets rarely contain the ground truth treatment effects. To this end, most experiments in the literature analyze causal models using a synthetic dataset where the ground truth is known (20) We begin by comparing the ATE estimation performance of our proposed IPTW methods on synthetic event datasets, generated using different parameters. Per standard practice causal inference litera- ture, we use root mean squared етror (RMSE) to measure the ATE accuracy of each method, along with its standard deviation. We first focus on the experiments with the instantaneous conditional intensity outcome. All experiments are done on a machine with 2.9 GHz quad-core CPU.
Since outcome Xy is directly observable, simple adaptation of ATE from the i.i.d. case would work. There are not many well-established baselines that use event occurrence rate as the outcome in multivariate point processes for comparison. We adapt two bascline scores from fitting parametric models to the intensity rates from cause-effect associate scores 1 8 C'1 (conditional intensity) and C'M., which consider single parent event and set, respectively. For the first bascline, we consider the association between a pair of events, (z, y) and assume that the intensity of only depends on whether or has occurred at least once within a specified time window w. Hence we define conditional intensity score to estimate the causal effect of on CI y) = Ag]= > We extend this definition to allow for y depending the historical arrival of set of events instead of

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Table I: Hits@K K results for the diabetes dataset (test set).
Method
K= = 10
K =
15
K = 20
Itable
6
7
CI
2
3
Clu
6
just single z. This means that there are possibly 2|X conditional intensity rates, and need to aggregate the score for 2 given all other conditions of the parent set. For our purposes we will use the mean y over all settings. Formally, Clar(z,)) y) = mean >/ and otherwise the score is O.
when z is a parent of y > y/5,x
We compare CI scores with three versions of ATE estimation based on our proposed IPTW methods. First, we compute ATE with no weighting as per Equation 2 labeled as IP-NW. Second, we use the proposed IPTW with non-stable weight (IP-NS) with weights as per Equation K Last, we use IPTW with stable weight (IP-Stable) to compute ATE.
Synthetic PGEM Datasets To test estimation methods synthetic data, we first generate event data that adheres to the proximal assumption for the intensity functions 77 We generate 3 models with different numbers of events, randomly generated graph structures among events, fixed window size of w = 30. T = 2000, and random intensities between 0.1 and 0.4. We use the data and the generated model obtain the true estimates of X (t) at chosen times and hence can compute the ground truth ATE. Since we observed that the sample size S of (103 to 10 in the ATE estimation does not impact the results much, we use sample size S = 103 for all our experiments. Shown in Figure Ta on three synthetic PGEM datasets, non-weighted IP-NW performs worse than CI scores when window size is small but better with large windows. IP-Stable's RMSE is consistently the smallest across datasets and window sizes, sometimes with a times reduction in error from CI scores. We did not include IP-NS in this figure as it has large errors in comparison to other methods. IP-NS with nta-st weight can lead to non-stable estimation. For exact numerical values, please refer to the appendix.
Synthetic Hawkes Dataset We also test our approaches on synthetic multivariate Hawkes process datasets using an existing toolbox (.. We again generate 3 datasets with 30, 40, and 50 event labels, and w 15. We use a fixed base rate 0.016 and each parental event leads to additive spike of 0.06 to the base rate, with an exponential decay rate 0.15. We generate event streams with T = 2000. For
estimating the counterfactual rate, Åz1z, (t), compute ATE in the case where there is no treatment event over [ - w. l). we introduce a counterfactual treatment event time / - Figure 1b shows the RMSE results of different algorithms. IP-NW outperforms
scores in all cases. It-Stable shows the lowest RMSE among all methods, generally achieving 3 to 10 times better accuracy. In addition, the performance of IP-NW and IP-Stable are relatively stable with respect to the window sizes, with minimal changes. For exact numerical values, please refer to the appendix.
Hybrid Dataset We also generate a synthetic hybrid dataset that combines proximal graphical event models with additive excitation similar to Hawkes processes with a constant kernel. For further details about the generating process and the results, please refer to Appendices G and H. IP-NW outperforms CI scores in all cases but one, and IP-Stable shows the lowest RMSE in all but two cases.
Diabetes Dataset We also test methods the diabetes dataset 14 real-world dataset which we process into events for meals, exercise activity, insulin dosage and changes in blood glucose measurements for 70 diabetes patients. We treat the assessments [2] as the ground truth, where an expert provided 11 pairs such that a cause label is more likely to make the effect label occur. Since the assessments are only partial and do not provide the true ATE, we use hits @ K among highest absolute estimated ATE values to measure performance in this experiment, which is popular metric for information retrieval. Specifically, we determine how many of the pairs are recovered by method's 's top K absolute scores. The dataset is split into 50%/50% training/test sets, and optimal window setting determined on the training set, which then deployed in the test set for evaluation. 10 0.3. 0.5, } days for all models were considered during training. Table [ compares the Hits @ K across all methods on the test set for K (10, 15, 20), illustrating that IP-Siable is able to recover more of the expert-assessed pairs in two out of three K. values.

---

Table 2: ATE accuracy results with cumulative intensity outcome diff. causal pairs for synthetic datasets.
Model
Future Window Size
CCI
CCIM
CIP-NW
CIP-NS
CIP-Stable
PGEM-1
13
1.,94
1.55
3.20-1.65
122.6.160.6
3.1112.42
PGEM-2
15
3.7712.96
3.9813.03
2.61 2.61.0.99 10997
538.911103.36
2.8710.99
Cumulative Intensity Outcome Experiments We also test our framework with the outcome being the cumulative intensity. The procedure be done directly by replacing (t) with its cumulative version A (t), computed from future window [t, + wy), as discussed in Section 4 Hence the overall computation becomes ATE = Exr T J 0₁ (t) T S (tty (t)) To adapt the CI baselines to cumulative setting, we compute the cumulative version of CI, CCI, with CCI(t) = CI2 wf. We also name our methods CIP as the cumulative version of the IP methods. All other computations remain the same. We are given the future window size wy = 15 for all methods. To explore the data-driven approach for learning the window, do not provide here and instead let it learn from data Z We show the results of ATE for the cumulative intensity outcome with 2 PGEMs in Table The results again confirm better performance of CIP methods over baselines, specially NW and Stable versions. CIP-NS can be unstable as indicated previously.
6 Conclusion
We have proposed a framework for pairwise event causality in a multivariate point process and formalized the problem with novel definitions of treatment, outcome, and propensity scores. Our definitions allow for efficient modeling of data where outcomes occur multiple times on the timeline. We estimated the average treatment effect using a propensity score weighting procedure that achieves the best performance against baselines. Our work bridges causal inference with multivariate point processes, showing promising performance in estimating pairwise causal relationships among events. Our framework can be extended in many ways. For example, ATE with multiple causes could be computed if we specify an outcome model given such multiple causes. Our treatment and covariate definitions focus on binary values but could be extended to marked point processes where the event has a real valued measurement, by choosing an appropriate aggregation function and outcome models. Future work could study efficient estimation approaches of ATE without sampling and different historical representations of the treatment. Another direction is a count-based estimand over the current proximal assumption, which would result in multiple treatment value problems.
Limitations And Societal Impact: Our method focuses on potentially long event sequences with multiple occurrences of all events. We assume proximal historical influence, along with strong ignorability and no confounding. The usage of causal models should be cautioned, to draw potentially harmful conclusions. In particular, the window sizes in our formulation, which can be subject to human inputs, can induce biases and provide different conclusions with different values.
Acknowledgement
We thank to anonymous reviewers who have provide helpful comments. This research was partly developed with funding from the Defense Advanced Research Projects Agency (DARPA), under Program No. FA8750-19-C 19-C-0206. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of DARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein XS is supported by supported by IBM AI Horizons Network, Nicholas Mattei was supported by an IBM Academic Award.
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] We described the assumptions in the main text and potential extensions in the conclusion.
(c) Did you discuss any potential negative societal impacts of your work? [No] We do not think that there are significant potential negative societal impacts beyond gross misuse of the proposed methodology.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes]
(b) Did you include complete proofs of all theoretical results? [Yes] In the appendices.
3. If you ran experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)? [No] The code will be released in Github.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]
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(c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)? [Yes]
(d) Did you include the total amount of compute and the type of resources used (c.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Only CPUs of a single machine are needed.
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) your work uses existing assets, did you cite the creators? [Yes] PGEM learner and Diabetes dataset.
(b) Did you mention the license of the assets? [N/A] No license is needed.
(c) Did you include any assets either in the supplemental material or URL? [No] They can be found the original papers cited.
(d) Did you discuss whether and how consent was obtained from people whose data you're 're using/curating? [N/A]
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
(b) Did you describe any potential participant risks, with links Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
14


Reviews:
[{"review_id": "k8lRBMdMIU-", "reviewer": "Reviewer_CcPK", "summary": "This paper proposes a model for estimating the average causal effect in point processes where the estimand of interest is the effect of an event occurring within a window on the rate parameter of a point process. The authors use the formulation commonly used for inverse propensity scores of generalized treatments (e.g., from Robins, Imai). The authors describe the properties of the propensity score as a balancing weight by leveraging known results and advocate for the use of stabilization of inverse propensity scores to reduce variance as in Robins(2000). The weights are then used within an outcome model which is a point process model. Empirical results show promising performance compared to a range of alternatives. \n", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I think this is an interesting and important problem area, and the authors provide what I feel to be a reasonable approach. While the paper is on the lighter side in terms of theoretical contributions, I think the approach is novel and provides a nice addition to the literature. \n\nMy largest concerns are as follows:\n- The authors could do more to tie the proposed approach to the larger literature. While the point process application is novel (to my eyes), there is a large literature on propensity scores and balancing weights for general treatments in the statistics literature which should be referenced in order to provide proper context.\n\n- It's unclear to me how much is being sacrificed by considering the quantity \"event happens any number of times within the window\" over \"number of times an event is observed within the window\". It would seem to me that these are different estimands, however it is not clear when we would prefer this \"event happens at all\" estimand. Can the authors provide intuition on this point?\n\n- It would be nice to have consistency / convergence results for the proposed estimator.\n\n- It's unclear how one would get confidence intervals from the proposed approach for the estimates. \n\n- The authors offer a heuristic for choosing the window size for the estimate. Intuitively, it would seem that smaller window sizes should result in greater precision. Can the authors provide intuition for the observed behavior which appears either flat or bell shaped?\n\n- It would be much more convincing if the authors ran experiments on any of the motivating domain examples. As-is it is difficult to reason over the true efficacy of the proposed approach on the synthetic domains. \n\nSome smaller points:\n- The authors consistently conflate propensity scores with balancing weights. Propensity scores are balancing weights under correct specification but the converse is not necessarily true. Given the growing literature on balancing weights this distinction is important to adhered to for the benefit of the reader.\n\n- The citation to Zubizaretta (2015) on line 298 is incorrect. Zubizarreta’s paper concerns the construction of balancing weights, not stabilized propensity scores. I believe the citation for stabilized IPTW should be Robins (2000).", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Any method for causal inference can be mis-applied an have negative societal impacts, however I don't think there is anything specific to this work that is unaddressed and should be acknowledged. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}, {"review_id": "fKrInhSwTgT", "reviewer": "Reviewer_BtNu", "summary": "This paper describes a method for the causal analysis of long streams of\nevent data that can be modeled as multivariate point processes (MPPs).\nThe paper defines the average treatment effect (ATE) between pairs of\nevents in MPPs that occur within a window, and derives how to estimate\nthe ATE as the expected difference of average outcome rates.  When\ncovariates differ or assumptions are violated, this estimate may be\nbiased, but adjusting for the propensity score yields an unbiased\nestimator, which the paper does through inverse probability of treatment\nweighting (IPTW) (including a stabilized version).\n\nExperiments on data from known, synthetic MPPs (proximal graphical event\nmodels and Hawkes processes) show that the proposed IPTW estimation\nmethod more accurately recovers the ATE than baseline conditional\nintensity scores.  Similarly, experiments on a diabetes dataset show\nthat stable IPTW recovers more true causal pairs at 2 out of 3 levels of\nrecall compared to the conditional intensity scores.  These results\nextend to using the cumulative rather than the conditional intensity\nrate.\n", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Overall\n-------\n\nOverall, I think this is an excellent paper.  In particular, it balances\nwell the presentation of high level concepts and motivations while\nproviding rigorous, supporting details.  The framework of causal\ninference in multivariate point processes (MPPs) it develops could play\na central role in establishing a comprehensive methodology for causal\ninference with event sequence data.\n\n\nOriginality\n-----------\n\nWhile all of the pieces of this paper have been well established, this\npaper combines them in a new way by deriving the standard causal\ninference measure (ATE) -- and how to estimate it well -- for a class of\nmodels (MPPs) that are very relevant to practical applications.  The\npaper situates its contributions clearly in comparison to survival\nanalysis, hazard models, and Granger causality.  From the meticulous\nnature of the paper, I expect related work is adequately cited, but I\ndon't know the literature in this area thoroughly enough to assuredly\nmake that judgment.\n\n\nQuality\n-------\n\nThe submission is very meticulously written and is technically sound as\nfar as I have investigated.  (I did not check the supplement nor the\nproofs therein.)  Statements are well-cited, and experimental evidence\nsupports the claims of effectiveness.  I don't have any complaints about\nthe methodology or the completeness.  The authors do not include an\nexplicit evaluation of strengths and weaknesses, but they have defined\ntheir scope well enough to allow those to be inferred.\n\nMy main suggestion is to use figures to illustrate the results rather\nthan just listing them in tables.  Yes, I know it takes a lot more work,\nbut the end result is so much easier for our feeble human brains to\ninterpret.  (If you think the specific numbers are important, then put\nthem in the supplement, perhaps as a data file.)  This alone brings my\nevaluation down one point.\n\nI also suggest including proof sketches in the main body of the paper as\na general rule of thumb (if it is accepted you get an extra page), but\nthe theoretical results are as expected, so I'm less concerned compared\nto unexpected / interesting results.\n\n\nClarity\n-------\n\nThis paper is one of the clearest I have ever reviewed.  Bravo!  (I have\nno suggestions.)\n\n\nSignificance\n------------\n\nI think other researchers are quite likely to build on this work in\nhelping to establish a framework of causality for MPPs.  Since much of\nthe observational data we collect is event-based, I believe causality\nwith MPPs is quite important, and this paper could be a foundational\ncontribution to the area.\n\n\nDetails\n-------\n\nThese are just some things I noticed; they are not comprehensive.\n\n78-79 It would be nice to have a citation here, if only for the\nnew-comers.  (Where is it shown that ...?)\n\n218 Is the square brackets and pipes notation, '[...|...]', meant to be\nexpectation?  That's what the text implies, but there's no 'E'.  Please\nfix or clarify.\n\n411 \"hawkes\" -> \"Hawkes\"\n\n421 \"2010\" -> \"2020\"\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors didn't really address the limitations or potential negative\nimpacts of their work.  (However, many limitations are implicitly\naddressed in that one can deduct them from the content, but it's always\nnice to have important things explicitly addressed.)  The only thing I\ncan quickly think of other than general, routine cautions about causal\nmodels and confounding (which I don't miss here), is some discussion of\nthe window size and its consequences.  (But some experiments showed no\neffect.)  I agree with the authors (in the checklist) that there is\nunlikely to be societal impact.\n", "review_text": "", "extra_scores": {"ethical_concerns": "No ethical concerns.\n", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4.8 hours"}, "extra_text": {}}, {"review_id": "UrFsOS3bPG1", "reviewer": "Reviewer_ktAc", "summary": "The paper provides tools to estimate average treatment effect (ATE) between event pairs. The authors first define the quantity for multivariate point processes together with the necessary concepts from Neyman–Rubin causality, then they derive a propensity score in this setting. Finally, they give a method for estimating the ATE and evaluate the method on synthetic and real world datasets.\n", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The work precisely define causal assumptions of the method: ignorability and overlap. The method do not require causal sufficiency, unobserved confounders are allowed as long as the covariates form a proper adjustment set for their effect. However it will ignore indirect effects mediated by these covariate variables.\n\nWhat does it mean that $\\mathcal{K}$ is $y$'s future occurrences at $t$? Later we see that $Y_i$ is actually equal the instantaneous expected number of occurrences  OR the cumulative expected number of occurrences of the outcome. At this point the entity $\\mathcal{K}$ is somewhat mysterious.\n\nLine 131 (typo): y is the effect/outcome not a treatment\n\nLine 180: Ignorability is misinterpreted here. Ignorability means that the counterfactual outcomes are independent from the actual outcome given the covariates. It is actually telling that the current value of Z in the real world have no extra information to determine the counterfactual outcomes given the knowledge of the covariates. It is closely related to the back-door criterion, see Pearl: Causality, 2009, Chapter 3; 11.\n\nLine 198: What do you mean by \"the coarsest of such a function is the propensity score\" ? \n\nLine 204: missing : given observing covariates $\\mathbf{x}_t^w$\n\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The work precisely define causal assumptions defining the limits of applicability.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}, {"review_id": "2YeJLGk4E1B", "reviewer": "Reviewer_oC9j", "summary": "This paper proposes a causal inference framework for a multivariate point process model. The key idea is to form a series of binary processes Z_t and X_t using a moving window [t-w,t] to indicate whether processes Z and X have at least one event in the window [t-w,t].  An average treatment event (ATE) is defined and propensity scores are proposed for unbiased estimation of ATE. The authors use synthetic experiments and real-world datasets to demonstrate how the proposed method work compared to baseline approaches.", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The paper is fairly well written and the concepts proposed are clear and new. I would say that the work is quite original in terms of conceptual development.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "1. Line 154, I have some doubts on the assumption (3) \"the specific times of z’s occurrences do not further affect y’s rate at time t.\". This is at least not the case for Hawke's process. I wonder how many existing point process models satisfy this assumption. Can you give some examples?\n\n2. Line 240, if the dimension of X_t^w is high, wouldn't the propensity estimates be very unstable and very high dimensional?\n\n3. In Corollary 3. only an unbiased estimator for the ATE is given. But to do causal inference, one would need some distributional results for ATE. Can this be done? Otherwise, I do not see much usage of the proposed method in real applications. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "24"}, "extra_text": {}}]

Author rebuttal:

Meta-review: Causal inference in point processes is a novel and open area, and the reviewers agreed that this is a useful contribution both methodologically and from an applied perspective. There are a number of changes that you should definitely make to improve the manuscript.
Decision: Accept (Poster)