Counterfactual Temporal Point Processes
Authors: Kimia Noorbakhsh
Manuel Gomez Rodriguez
Venue: NeurIPS 2022 Accept (2022)
Keywords: counterfactual explanations
temporal point processes
structural causal model
explainability

Abstract: Machine learning models based on temporal point processes are the state of the art in a wide variety of applications involving discrete events in continuous time. However, these models lack the ability to answer counterfactual questions, which are increasingly relevant as these models are being used to inform targeted interventions. In this work, our goal is to fill this gap. To this end, we first develop a causal model of thinning for temporal point processes that builds upon the Gumbel-Max structural causal model. This model satisfies a desirable counterfactual monotonicity condition, which is sufficient to identify counterfactual dynamics in the process of thinning. Then, given an observed realization of a temporal point process with a given intensity function, we develop a sampling algorithm that uses the above causal model of thinning and the superposition theorem to simulate counterfactual realizations of the temporal point process under a given alternative intensity function. Simulation experiments using synthetic and real epidemiological data show that the counterfactual realizations provided by our algorithm may give valuable insights to enhance targeted interventions.

Counterfactual Temporal Point Processes
Kimia Noorbakhsh Sharif University of Technology kimiaanoorbakhshh@@mall. com
Manuel Gomez Rodriguez
Max Planck Institute for Software Systems manuel wwrellrrrmpppppe
Abstract
Machine learning models based on temporal point processes are the state of the wide variety of applications involving discrete events continuous time. However, these models lack the ability to answer counterfactual questions, which are increasingly relevant as these models are being used to inform targeted interven- tions. In this work, our goal is to fill this gap. To this end, we first develop causal model of thinning for temporal point processes that builds upon the Gumbel-Man structural causal model. This model satisfies a desirable counterfactual monotonic- ity condition, which sufficient to identify counterfactual dynamics the process of thinning. Then, given an observed realization of temporal point process with a given intensity functiinn we develop a sampling algorithm that uses the above causal model of thinning and the superposition theorem to simulate counterfactual realizations of the temporal point process under a given alternative intensity func- tion. Simulation experiments using synthetic and real epidemiological data show that the counterfactual realizations provided by our algorithm may give valuable insights to enhance targeted interventions.
Introduction
In recent years, machine learning models based on temporal point processes have become increasingly popular for modeling discrete event data in continuous time [12]. This type of data is ubiquitous in wide range of application domains, from social and information networks to finance epidemiology. For example, in social and information networks, events may represent users' posts, clicks or likes 3 4 in finance, they may represent buying and selling orders (5): in epidemiology, they may represent when an individual gets infected or recovers [6]. In many of these domains, these models have become state of the art predicting future events given sequence of past events (7].
Building upon the above models, a recent line of work SIII has developed machine learning methods to automate online, adaptive targeted interventions using reinforcement learning and stochastic optimal control. While this line of work has shown early promise, particularly in personalized teaching and viral marketing, there are many high-stakes application in which targeted interventions are unlikely to be automated. For example, in epidemiology, fine-graine mee-grained interventions that are targeted particular sites individuals (e.g., hygienic measures at work places, closures of schools, or contact tracing) are likely to be decided by govermments, policy makers and health authorities, at least in the foreseeable future. In this work, goal is develop machine learning methods that, given the outcome of an intervention implemented in the past, are able to assist decision makers at implementing better interventions in these high-stakes applications.
More specifically, we focus on facilitating counterfactual thinking, a type of thinking that has been argued to help humans correct and improve behavior that has been unsuccessful in the past [2] [3] In counterfactual thinking. given a history of past events that have already occurred, one asks what past events would have instead occurred if certain intervention had been in place. For example, in epidemiology, assume that, during a pandemic, a government decides to implement business
This work was done during Kimia Noorbakhsh's internship MPI-SWS.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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restrictions every time the weekly incidence the (relative) number of new cases is larger than certain threshold but unfortunately the incidence nevertheless spirals out of control. In this case, counterfactual thinking would help the government understand retrospectively to what extent the incidence would have grown had a lower threshold been implemented
Our contributions. Our starting point is Lewis' thinning algorithm (20), one of the most popular techniques for sampling in temporal point processes. Lewis' thinning algorithm first samples sequence of potential events from a temporal point process with a constant intensity that upper bounds the intensity of the temporal point process of interest. Then, it accepts each of these events with pro- bability proportional to the ratio between the intensity of the temporal point process of interest at the time of the sampled event and the constant intensity. In our work, the key idea is to augment the above thinning process using particular class of structural equation models (SCMs) the numbel-Max SCM (21]. This causal model satisfies desirable monotonicity condition and, given sequence of accepted and rejected events, it allows us to reliably estimate what events would have been accepted and rejected under an alternative intensity. Put differently, can be used to answer counterfactual questions about a set of previously accepted and rejected events by the thinning algorithm.
Unfortunately, the above causal model on its own is not sufficient answer counterfactual questions about observed sequence of real events. This is because, in general, real event data is not generated by a thinning process and, as a consequence, it does not include rejected events. However, we are able to overcome this limitation by using the superposition theorem [22] to sample plausible sequences of events that the thinning process would have rejected if had accepted the observed sequence of real events. Then, these generated sequences of rejected events, together with the observed events, can be fed into the above causal model of thinning to sample counterfactual events given an alternative intensity. Importantly, while our causal model of thinning only allows for inhomogeneous Poisson processes (i.e., it requires the intensities of interest to be deterministic), we can still use it to sample counterfactual events from linear Hawkes processes [23], a popular type of temporal point processes with stochastic self-exciting intensities, by exploiting their branching process interpretation [24]. Finally, we evaluate our sampling algorithm using both synthetic and real epidemiological data and show that the counterfactual events provided by our algorithm can give valuable insights to enhance targeted interventions f
Further related work. The literature on temporal point processes related to causal inference has mostly focused on measuring causal influence by means of, e.g., Granger causality (25 28 integrated cumulants 22 30 or Wold processes [1 32], and on predicting quantities related an interventional distribution of interest (33(35) (e.g., conditional average treatment effect (CATE)). However, there exist a few notable exceptions, which have focused on counterfactual reasoning (3) The work by Schulam and Saria (36) focuses on reasoning about the counterfactual distributions of the marks marked temporal point process, rather than reasoning about the counterfactual intensities of the events we do. The works by Ryalen et al. (37] and Roysland [38] focus survival analysis, i.e., temporal point processes that terminate after one event, and are very different to ours at a technical level. In contrast, we focus temporal point processes with multiple events"
More broadly, the literature on causal inference has a long and rich history 40). However, most of this literature has used counterfactual reasoning to predict quantities related to the interventional distribution of interest such as, e.g., the conditional average treatment effect (CATE). A few recent notable exceptions are by Oberst and Sontag (21 and Tsirtsis et al. [41], which have used the Gumbel- Max SCM to reason about counterfactual distributions in Markov decision processes (MDPs), and by Buesing et al. [2 and Pitis et al. [3]. which have used counterfactuals to improve the training of reinforeement learning agents. However, to the best of our knowledge, the Gumbel-Max SCM has not been used previously to reason about counterfactual events in temporal point processes.
2 Preliminaries
In this section, we first briefly revisit the frameworks of temporal point processes and structural causal models 45
Note that existing epidemiological models 1
including those developed in the context of COVID 19 15 are unable to answer such under certain interventions given past.
counterfactual questions -theyyhhhhhyhh can only predict what (avemge) future would
"To facilitate research in this area, we release an open-source implementation of our algorithms and data at htttss///tttuub... http://gguuuuuooo///. con/ Networkss
Leaming/cunnteraatuuuin
"The work by Zhang el al.
39 is contemporary to ours and presents a methods to estimate individual ireatment effects (ITE) on the intensity
temporal point process.

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Temporal point processes. A temporal point process is a stochastic process whose realization consists of sequence of discrete events localized in continuous time. 7 ft, R = E N .t. < 1. A temporal point process can be equivalently represented as a counting process, [(),, which records the number of events before time t. Moreover, in an infinitesimally small time window di around time t, it is assumed that only one event can happen, Le., dt( E 10.
temporal point process is typically characterized via its intensity function A(t) 2 0, which determines the probability of observing an event in [t, t dt),
X(I)d ((dN(() 1} = E(dNN()))
(I)
In general, the intensity A(t) may depend on the history H((t) = {t, E H. < t up to time and its functional form is often chosen to capture the phenomena of interest. Throughout the paper, will consider inhomogeneous Poisson processes, ie.., A(t) = g(t), where g(t) 2 0 is time-varying function, and linear Hawkes processes, i.e.,
X(1) = μ +
Σ 4EH(()
g(l - 4),
where μ 2 0 and the second term, with o 2 0. g()) influence of previous events on the current intensity
2 0 and g(() = 0 for all / < 0, denotes the
Structural causal models. Given a set of random variables X = [X ). a structural causal
model (SCM) C defines complete data- data-geneen generating process via a collection of assignments
X; == (.).,
where PA, S XXX, are the direct causes of X,, U (UI....,U.. Un} are noise variables, and P(U) denotes the (prior) distribution of the noise variables. Here, note that, given an observational distribution P(X P ....... xx), there always exists distribution for the noise variables and functions Si so that P P pč, where pC is the distribution entailed by C. In general, the noise variables U may not be jointly independent if there unmeasured confounder However, in our work, will assume there are no unmeasured confounders and U are jointly independent.
Given SCM C. we can express (atomic) interventions I using the do-operator, e.g., I do(X; x) corresponds to replacing the causal mechanism (PA, Ui) with x. The intervened SCM is typically denoted C² and the interventional distribution entailed by the intervened SCM pº: Moreover, given a SCM C and an observed realization of assignments X = Σ, we can define a counterfactual SCM Cx = where the noise U variables are distributed according to the posterior distribution P(U X = x) and not necessarily jointly independent anymore. Counterfactual statements can now be seen as interventions in a counterfactual SCM Cx =z and, given intervention I, denote the interventional counterfactual distribution entailed by C+ - pe| X==;I :I, However, the posterior distribution of the noise variables may be non-identifiable without further assumptions. This is because there may be several noise distributions and functions 9: consistent with the observational distribution but result in different counterfactual distributions.
In the context of binary random variables, monotonicity is an assumption that avoids the above mentioned non-identifabiiityy issues-it restricts the class of possible SCMs to those which all yield equivalent counterfactual distributions over a binary variable of interest [46] 21]. More specifically, a SCM of binary variable Y is monotonic with respect to binary variable T if and only if the condition
po: (Y = y) Z po:do(T=t') = =y)
implies that pel Y=y, () = v/) O, where yf 4 y.
3 A Causal Model of Thinning
In this section, we first revisit Lewis' thinning algorithm 20 one of the most popular techniques to simulate event data from inhomogeneous Poisson processes. Then, augment this classical algorithm using a particular class of SCMs satisfying the monotonicity assumption, the Gumbel-Mar SCMs (2.) Finally, demonstrate that the resulting algorithm can be used to answer counterfactual questions about a set of previously simulated events.
"The function gt) is often called triggering kemel.
An unmeasured confounder unobserved variable that direct cause (observed) variables X; and X3-
3

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Algorithm It samples a counterfactual sequence of accepted events given a sequence of accepted and rejected events provided by Lewis' thinning algorithm
1: Input: Am (t), Am/ (t), Hm. Hmaxs Amax.
2: Initialize: Hm 0.
3: function ACC(Am (t), Am (f), Hm, Hmaxx Amax)
4: Hmmm +- 1
for Hmmax
6: 2i - 1[t: E Mmm]
x4 p ( (X)
7:
if af
then
9:
Ho.! + H.s U {ti}
10:
end if
11:
end for
12:
return H.m
13:
end function
Let M be a set inhomogeneous Poisson processes of interest and, for each m E M, assume its corresponding intensity function (t) S Amax for all i E g+ To sample a sequence of events Hm from any process m E M, Lewis' thinning algorithm first samples a sequence of potential events Hmax from homogenous Poisson process with intensity Amax. Then, for each event 4, Hmmax it additionally samples a Bernoulli random variable X, with parameter p = p( = Am (m(t))/2mar Amax- Finally, it accepts all the events f; such that X, 1. i.e., Hm {t, E H.max X, 1}. Here, note that the specific choice of Amax does not affect the distribution of accepted events as long as Amax Am (t) for all E M and E R+ Algorithm 4 in Appendix B.1 summarizes the overall procedure, where the parameter is often called the thinning probability.
Given process of interest with intensity (t), Lewis" thinning algorithm is helpful to make predictions about future events. For example, it can be used to compute Monte Carlo estimates of the average number of events ([) at a time l in the future. However, it is not sufficient to make counterfactual predictions, e. e.g., given the sequences of events Hmax and Hm, we cannot know what would have happened if, at time t,, the intensity had been (ti), with * m', instead of Am (t.). To overcome this limitation, we will now augment the above thinning algorithm using a Gumbel- -Mx SCM. More specifically, let C be a SCM defined by the assignments:
X; argmax g(æ, A;, U.) and A; = =>(4),, 2ε(0,1)
where
1o) Ar) + U2
with p(X; - A.) (1))))))))1)))) P(A.) = Am/x Us ~ Gumbel(O, 1), and
t; ~ Amaz. Then, the thinning probabilities in Lewis" thinning algorithm are given by the following interventional distributions over (+
A(4.)
pe p(:0oA.-MM)) p({(t))
(5)
Amax
Under this view, given a sequence of accepted events H.m and rejected events Hmax (mm under an intensity Am (t), as determined by the binary samples {a²}, }. we can estimate the posterior distribution pe| X, K, A; -A..(ta) =do(A,=A,,) (t.)) (U,z)) of each Gumbel noise variable Uz using an efficient procedure 488 (refer Appendix [.3)) Then, we can use these noise posteriors compute an
unbiased finite sample Monte-Cate Carlo estimate of the counterfactual thinning probability, i.e.,
pC|xx 21, (tc) do (..AAAA A.. (t)) (X: x) Bu, (t.) [1r argmax g(ø', An (14), U.),, r'E(0.1)
where we drop do(-) because U, and Am (t)) are independent in the counterfactual SCM. Importantly, the above counterfactual thinning probability allows us to make counterfactual predictions, e.g., given
In this vomtext, (time- varying) intensity X (() 2 (t) (t) for all i M and t E RR, as shown in Theorem in Lewis and Shedler 20
using homogeneous Poisson process intensity Amax. une could any process
This equality follows immediately from Gumbel-Max trick

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Algorithm 2 It samples counterfactual sequence of events given a sequence of observed events from an inhomogeneous Poisson process.
1: Input: Am (t) Xm/ (t), Hm. Amax, T.
2: Initialize: Hm 1 0.
3: function CF(Am (t), Am (f), Hm, Amax, T)
4: Hmaas_ LEWIS (Amax Am Amaxi Amax, I')
"mmax f- Hmax Hm
6: Hmm + AAC((m Amd Hm, Hmnsx, Amax)
7: return Hmm
8: end function
a sequence of accepted events Hm and rejected events H Hm under intensity Am (t), we can use the counterfactual thinning probability to predict which events, among those in H. would have been accepted if the intensity had been Am instead of Am (t). Algorithm summarizes the algorithm. Here, it is important to note that the specific choice of Amax does not affect the distribution of counterfactual events as long as Amax 2 Am (1) for all m E M and / R+, similarly as standard thinning (refer to Appendix A.I for proof).
Finally, is important note that, by using Gumbel-Max SCMs, our causal model of thinning satisfies the monotonicity assumption [46] 21]. discussed in Section and thus the counterfactual thinning probability does not suffer from non-identifiability issues. More formally, we have the following proposition (proven in Appendix A.2
and C be the corresponding causal model of thinning. Then, if (ta) 2 Amm (t,). it holds that Pe| (==0,A==Am(tt))) (do: (ta)) (X, 1) 0. Con- (X, = =)
Proposition 1 Let M = ()m(t),
versely, if (m (t.) S (t,), it holds that pC| x,=1,,,= (.., (t)::
The above result directly implies that, if potential event t E Himax was rejected under Am (t)...e. & Hm. then in counterfactual scenario, if Xm (t) S (t), it is impossible that the event is accepted under (t). i.e., t4 E Hm. Conversely, Am (l), i.e., I; E Hm. then in a counterfactual scenario, if Am (L) 2 Am (1), it is impossible that the event is rejected under Am (1), i.e., 4 & Hm..
potential event f. Hmmax was accepted under
Sampling Counterfactual Events
In this section, we develop a sampling algorithm that, given an observed realization from temporal point process with a given intensity function, it uses Algorithm and the superposition theorem [22] to generate counterfactual realizations of the temporal point process under a given alternative intensity function. To ease the exposition, we first focus on inhomogeneous Poisson processes and then generalize our algorithm to linear Hawkes processes.
Inhomogenous Poisson processes. Assume we have observed a sequence of real events Hm and this sequence can be accurately characterized, observationally, using an inhomogeneous Poisson process with intensity (t) = g(t), where g(t) 2 O is a time-varying function. In reality, this sequence of events has not been generated by Lewis' thinning algorithm but by natural phenomena of interest. As result, we cannot directly apply Algorithm since requires both a sequence of accepted and rejected events. However, would have rejected if it had accepted the observed sequence of events.
[ find plausible sequences of events that Lewis' thinning algorithm
By construction, the intensity of accepted and rejected events Aaccepted (4) and Arejectea(() provided by Lewis' thinning algorithm should satisfy that Amax = Ascoopeddt) + Arejected Then, if Asccopted (t) = (t), Amax 2 max; Am (t) and Haccopeed Hm. by the superposition theorem, plausible sequences of events that Lewis' algorithm would have rejected just by sampling from the intensity (tejected together with the sequence of observed events, can be fed into Algorithm to sample sequences of counterfactual events given an alternative intensity Xm (t). Algorithm 2 summarizes the algorithm, where LEWIS(-) samples a sequence of events using Algorithm in Appendix B. and Acc(-) samples counterfactual sequence of accepted events using Algorithm [
can find
Xmax (t). Then, these generated sequences of rejected events,
Linear Hawkes processes. Assume we have observed sequence of real events Hm and it can be accurately characterized, observationally, using a linear Hawkes process with intensity Xm (t)
5

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Algorithm 3 It samples counterfactual sequence of events given a sequence of observed events from a Hawkes process.
1: Input: Pmm am.. 9m (t), Mmns Om's Im (t)(t), Hom Amax, T.
2: Initialize: Hm = 0.
3: [Hm.s} &- ASSIGN(Hmi (Hm, (m)
4: Hm'oo + CF(Mmo Tin (t), Mm Amax: Hmm' Hmm' U Hm
6: for 1 E Hn do
7:
if E 'Hmm' then
8: Hm'n + CF(mma(t), 7mm's Hmms Amax. 2)
9: Hm! f- Hmm U Hols
10: end if
11:
end for
12: H 4- Hmm Hm
13: while [7d] > 0 do
14: + mingEN
15: Hm... LEWIS(7Tm' (1), Ammee T)
U {(x) 17: Hm' + Hmm U Hma
16: t H
18: end while
19: return Hmm!
parameterized by Pm. Om and (.),)), as defined in Eq. 2 If our goal is to sample sequences of counterfactual events Hmm' given an alternative Hawkes intensity (t)(t) parameterized by pm', O'm' and 9 (-), we cannot proceed similarly as in the case of inhomogeneous Poisson processes. This is because, to sample plausible rejected events within Algorithm 2 we need to pick value of Amax that upper bounds both (!) and (!), otherwise, Algorithm might break because the thinning probabilities p used by the causal model of thinning could be greater than Unfortunately, since Am (t) depends on the counterfactual history Hmm' we aim to sample, we cannot know its maximum value at the time we sample the rejected events. However, we can overcome this challenge by resorting to the branching process interpretation of Hawkes processes (24).
More specifically, we can view any linear Hawkes process as a superposition of several temporal point processes, i.e., process with constant intensity (t))) = and, for each event f; E H. process with intensity (t) = ag t.). Under this view, we can naturally derive the following thinning algorithm to sample from linear Hawkes processes 49 First, we sample a sequence of
events 4 from the process with intensity (4). Then, for each sampled event 4, we create process with intensity (t))) and sample a sequence of events ij independently from each of them using Algorithm 4 in Appendix B.T. Further, for each of these sampled events (j. we again create another set of processes with intensity "r (t) and sample from them independently, and continue recursively, By the superposition theorem, it readily follows that the overall sequence of sampled events is a valid realization of the original Hawkes process, Algorithm 5 in Appendix B.2 summarizes the algorithm.
Importantly, all the intensities 7/ (t) we sample from are bounded by max max, ag(t)). As result, given two intensities of interest (t) and Am (t), we can sample sequences of counterfactual events by running Algorithm independently for each of the above processes with
Amax Z max(Ums Mm', max am.mm()) max aem' 9m (f)}.
However, to do so, we also need to assign each observed event 4, E Hmm to one of the above processes with probability (L,) Eke; (m,) which is the probability that the process has caused the event [50]. Algorithm summarizes the resulting algorithm, where ASSIGN(-) returns the observed events Hm..j assigned to each process j. CF(-) samples a counterfactual sequence of events using Algorithm 2 and LEWIS(-) samples a sequence of events using Algorithm 4 in Appendix B. Within the algorithm, it is also worth noting that lines 6- I need to go through the observed events 1 in chronological order and, for each L,, the algorithm only accepts counterfactual events from the corresponding process with intensity 7m if the event lj has been previously accepted in the counterfactual realization. Moreover, lines 12-18 18 sample a sequence of events for each of the processes triggered by the counterfactual events that did not exist in the observed sequence of events.

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199
4.2
Intensities
(b) Low # of events
(c) Medium # of events
(d) High # of events
Effect of interventions in two inhomogeneous Poisson processes. Panel (a) shows the intensities of the original and the intervened processes (in blue and green, respectively) and a window of interest (dashed vertical lines). Panels (b-d) show the difference in average number of events over time between the counterfactual and the original realizations, where each row corresponds one process and we group the original realizations in three quantiles according to their overall number of events the time window of interest. Positive (negative) differences are shown in green (red). In each experiment, we sample 1, 1,000 realizations from the original process and, for each of these realizations, we sample 100 counterfactual realizations from the intervened process.
Figure 1:
5 Experiments on Synthetic Data
In this section, feed Algorithms and with realizations of synthetic inhomogeneous Poisson processes and linear Hawkes processes and investigate to what extent the counterfactual realizations under alternative intensity functions differ from the original realizations fed to them g
Experimental setup. consider the family of inhomogeneous Poisson processes M(Φ, a; T) para- meterized by weighted combinations of RBF kernels, i.e., X(t) E, by exp (=ay r,)) 2 where Φj. aj. Ty 2 0, and the family of linear Hawkes processes M o; w) defined in Eq. with exponential triggering kernels gt) exp(-wt). Moreover, we experiment with simple interventions under which, for inhomogeneous Poisson processes, one of the RBF kernels, picked random, change its amplitude, i.e., im = max(ám. + 0), and, for Hawkes processes, the parameter a change its value, i.e., amm max + c, 0), where e (),
In each experiment, we first sample 1,000 realizations from a process with one set of parameters using Algorithm 4 (or Algorithm (5) Then, carry out the above mentioned intervention and, for each the sampled realizations, we use Algorithm 2 (or Algorithm 3 to sample 100 counterfactual realizations under the resulting alternative set of parameters. Finally, realizations in three quantiles according to their overall number of events in a time window of interest and, for each quantile, we look the number of events in the corresponding counterfactual realizations in the same window of interest. In all experiments, Algorithms [ use 100 samples ==(t)) from the posterior distribution pc| X; A,= -areAA;A((ttt do(A, A.d (f+)) (U,) of each Gumbel noise variable
partition the original
UZ. to estimate the counterfactual thinning probabilitics pe X,=x,A ; do(AA;=,, (t))) (X,).
Results. Figure I summarizes the results for two specific inhomogeneous Poisson processes under- going one of the above mentioned interventions- we found qualitatively similar results for other inhomogeneous Poisson processes. In the top row, the results show that, under our model, inter- vention that increases the original intensity of the process, by increasing the amplitude one the RFB kernels by approximately additional half does not have the same effect across realizations. In realizations with low (high) number of events, the average number of counterfactual events in the window of interest increases up ~60% (~40%) over time with respect to the average number of events the original realizations. In the bottom row, we find that this is also true for an intervention that decreases the original intensity of the process, by approximately halving the amplitude of another of the RBF kernels. However, the difference is smaller in relative terms.
Figures 2 summarizes the results for specific Hawkes process undergoing two of the above men- tioned interventions-wer found qualitative similar results for other Hawkes processes. We find that,
All experiments performed machine with 48 (ntel(R) Xeon(R) 3.00GHz CPU and L.TTB.

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Xonter ofw.ssofrttttttttre
Midius
Low
Medlich
Quartta
(a) Self-excitation increases
(b) Self-excitation decreases
Figure 2: Effect of interventions in Hawkes processes. Panels summarize the distribution of the number of events per counterfactual realization corresponding to original realizations with a low (red; |Hmm| E 00, 9 E Hm 5.04), medium (blue; [mmll E [10, 22). E[[ 17.17) and high (green; [mm| E [25, 59). E[jHm]] 36.17) number of events. The horizontal lines within the boxes indicate average value, the boxes indicate 25% % and 75% quantiles, the whiskers indicate 5% and 95% quantiles and the points are outliers. Here, we sample 1,000 realizations from the original process, sample
with parameters Pmm 1, am 1, and Wm 1, and, for each of these realizations, 100 counterfactual realizations from each of the intervened process. In panel (a), the intervened process has parameter 0'm" 1.44, in panel (b), it has parameter O'm'' 0.75 and, in both panels, the remaining parameters Hmm = fmm and amm = amm- We set the time horizon T
similarly as in inhomogeneous Poisson processes, the interventions do not have the same effect across realizations. However, in this case, the difference among them is more stark while the average number of counterfactual events (a) increases by 115% and (b) decreases by 11% for realizations with low number of events, it (a) increases by 216 and (b) decreases by 21% for realizations with high number of events. Moreover, realizations, especially when O'n' > Om- For example, while the original realizations with a low number of events never contained more than 9 events, there exists counterfactual realizations with more than 200 events. This is due to the self-exciting property of Hawkes processes by which counterfactual events may trigger the emergence of additional counterfactual events, shown as yellow dots in Figure in Appendix C
also find that, there is high variability across counterfactual
Experiments on Epidemiological Data
In this section, we run a (simple) variation of Algorithm 3 (refer to Appendix B.4 to quantify the effect of interventions on a networked Susceptible Infectious-Recover model [5I) fitted using real event data from an Ebola outbreak in West Africa in 2013-2016 [52
(SIR) epidemiological
Experimental setup. We build upon the networked Susceptible- Susceptible-ifeectious Infectiouu--Recoveree (SIR)
epidemiological model introduced by Lorch et al. [51], which is based on temporal point pro- cesses. Given contact network E), represent the times when each node gets infected and recovered using a collection of binary counting processes (() and W (4) and we track the current state of each node using collection of state variables X(t) Y Yt)
(t), where X,()) 1 indicates node E V is infected at time and Xx(t) = O indicates it is susceptible or recovered. For each node i V, we characterize the above counting processes using the following intensities
(0 | H(O) = (1 - X((I)
(X)(d)dt and EldW:(!) H(O) 5 (;)dddt, (6) (E.))
Σ
where note that the counting process Y (t) can be viewed as (networked) multidimensional Hawkes process with stochastic triggering kernels defined by step functions. Refer to Appendix D for more details on how we set the parameters ,3 and ó and how we generate the contact network g.
simulate realistic outbreak by sampling realization from the above fitted model. To this end, for each real recorded case up to January 1, 2014 (52), we sample a seed node at random from the same district as the observed case Figure g Appendix D shows the geographical distribution of seed infections and overall cumulative number of infections in the outbreak. Then, given this sampled realization, we quantify the effect of two types of interventions by sampling counterfactual realizations using a variation of Algorithm (refer to Appendix B.4):
In experiments,
The was recorded on 26, By Jamuary 2014, recorded different districts.

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Numbor activw caurse
1200
900
Thrssted -07
Thraazz +22
300
100
[Mayy]
(a) Contacts reduction highest incidence district
1200
1.000
Ohamates Decrdaetiv 10% duttee re ductii
Olscuates reduttto reductio 20%
Nomber ofantrr
reductoo
200
092
reduction
redoctio
100
200
203
100
160
<10yyl
Threshold 300
Threshold 600
Threshold 900
(b) Contacts reduction in all districts
Figure 3: Effect of interventions where individuals reduce their contacts after the overall number of active infections reaches varying threshold. In all figures, the black line ("Observation") corresponds outbreak sampled from the fitted SIR model, the remaining lines correspond to counterfactual realizations for this outbreak under different interventions, and the shaded regions correspond to 95% confidence intervals. For each threshold and reduction level, we repeat the experiment five times.
2x 4.00
300 350
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Threshold = 300
Threshold = 600
Tireshold = 900
Figure 4: Geographical distribution of the overall cumulative number of infections per district in counterfactual outbreaks where all individuals reduce their individual contacts by 25% after the overall number of active infections reaches certain threshold.
Reduction of number of contacts: individuals reduce their individual contacts after the overall number of active infections reaches certain threshold. In one scenario, only individuals from the district with the highest incidence reduce their contacts within the district and get isolated from all other districts and, in another scenario, everyone reduces their individual contacts. Vaccination: a percentage of the overall population receives a vaccine with a certain level of efficacy. Within model, we measure vaccine efficacy in terms of reduction of the value of the parameter B, which controls the infection rate between individuals.
In each experiment, to estimate the average and confidence intervals of the outcome of interest (e.g., number cases), we sample 20 counterfactual realizations.
Results. Figures 3 summarize the results for the interventions where individuals reduce their individual contacts alter the overall number of active infections reaches certain threshold. Our results suggest that, at all threshold levels, reducing the individual contacts across all districts, even by just 5%, would have been more effective than isolating and reducing the contacts by 50% in the district with higher incidence. Moreover, we also find that, for lower threshold values, the counterfactual outbreaks would have spread to fewer districts and overall number of infections would have been significantly lower. Figures 5 summarize the results for the interventions where a percentage of the overall population receives a vaccine with certain level of efficacy. Our results suggest that high level of vaccine effectivity would have not been sufficient to reduce the number of infections if

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110
Figure 5: Effect of interventions where a percentage of the overall population receives vaccine with a certain level of efficacy. The figure shows the average reduction in the cumulative number of infections under each intervention with respect to outbreak sampled from the fitted SIR model.
200 350
200 -20
150
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250 300
150 2000
50 100
10% vancinated 50% reduction in /
20% vaccinated
80% vaccinated 60% reduction in
90% reduction in /
Figure 6: Geographical distribution of the overall cumulative number of infections per district tin counterfactual outbreaks where a percentage of the population receives a vaccine with a certain level of efficacy.
the percentage of the population who had received the vaccine was low. For example, if less than 20% of the population had received the vaccine, even a vaccine with a 90% effectivity would have been unable to reduce the infections by >70%. If >80% of the population had received the vaccine, a vaccine with just a 60% effectivity would have reduced the infections by >95%. Finally, we find that, similarly as in the scenario where individuals reduce their individual contacts. the reductions in the overall number of infections would have also led to lower geographical dispersion.
7 Conclusions, Limitations and Future Work
Since counterfactual reasoning lies within level three in the "ladder of causation" 45 unable to validate our counterfactual predictions using observational nor interventional experiments. That being said, we have made an intuitive assumpiion-moootoniciio of the world-ood causal model of thinning- haining--wich which specifies how changes on the intensity function of temporal point process may have lead particular outcomes while holding "eveeyythinn else" fixed, similarly as previous work [1 41 In this context, it would be worth to understand the sensitivity of counterfactual realizations to the specific choice of SCM. In our work, we have focused on temporal point processes, however, some of our ideas can be readily extended to spatial point processes. Moreover, it would be interesting to extend our methodology to support nonlinear Hawkes processes and neural Hawkes processes [53] (4).. To this end, the main challenge is that there is no branching process interpretation for nonlinear or neural Hawkes and thus we would have to find alternative ways to bound the value of the relevant intensity functions. Finally, it would be important to carry out a user study in which the counterfactual realizations provided by our algorithm are shared with domain experts (e.g.. epidemiologists) and evaluate our sampling algorithm using other real datasets from other applications such as climate change, recommendation systems and voting data.
we have been
the causal mechanism
Acknowledgments and Disclosure of Funding
We would like to thank William Trouleau for sharing with us pre- -ropeceeseed version of the Ebola dataset as well as the fitted stochastic block model we used in our simulation experiments and thank Stratis Tsitssss and Abir De for fruitful discussions. the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 945719).
Rodezz-odrriguz acknowledges support from
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Checklist
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(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] Refer to "Remarks" in Section and Section 7
(c) Did you discuss any potential negative societal impacts of your work? [Yes] We cannot think of any potential negative societal impact for our work.
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(a) Did you state the full of assumptions of all theoretical results? [Yes]
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14


Reviews:
[{"review_id": "euZCsIgReT", "reviewer": "Reviewer_nA6T", "summary": "In this paper, the authors proposed a simple but solid method to answer counterfactual questions for temporal point processes.\nIn particular, the authors focused on the counterfactual question that whether a historical event will happen if the corresponding intensity is changed. \nFor the inhomogeneous Poisson process, the authors demonstrated that via sampling events with a rejection-related intensity (i.e., lambda_max - lambda_accept) and superposing the events to observed ones, the counterfactual question can be answered simply based on a thinning process.\nMoreover, leveraging the fact that Hawkes process can be implemented as the superposition of various inhomogeneous Poisson process, the proposed thinning process can be extended to answer the counterfactual question of linear Hawkes process — just applying the method to each branch of the Hawkes process.\nA simulation experiment on epidemiological data demonstrates the rationality of the proposed method.", "questions": "No more questions.", "limitations": "(1) The method is focusing on inhomogeneous Poisson process and linear Hawks process. Whether it can be applied to other TPPs is unknown and needs to be discussed in depth.\n\n(2) Besides epidemiological data, it would be nice if the authors can discuss more potential applications, e.g., recommendation systems, and voting data.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n(1) The paper is well-written and the idea is easy to follow.\n(2) The idea is simple and theoretically supported, and the method is easy to implement. Additionally, Proposition 1 fits our intuition. \n(3) The application of epidemiological data is important and interesting.\n\nWeaknesses:\n(1) If my understanding is correct, the current application scenarios are limited to inhomogenous Poisson process and linear Hawkes process. It would be nice if the authors can discuss how to extend the proposed method to more complicated TPPs, e.g., neural Hawkes, or nonlinear Hawkes.", "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "OYwAAPNKTZ", "reviewer": "Reviewer_7bXC", "summary": "The authors propose an approach for estimating counterfactuals in temporal point processes.\n\nIn the proposed approach, a temporal point process is modeled using the Gumbel-Max structural causal model under the assumptions of monotonicity and binary random variables. The error terms in this model are all independent of each other, which implies Markovianity, i.e., no unmeasured confounders). Under those assumption, counterfactuals are always identifiable. \n\nThe authors' contribution is a sampling algorithm based on the Lewis' thinning algorithm and the  superposition theorem. Lewis’ thinning algorithm is a popular method to sample from the a temporal point process. They authors use the superposition theorem to sample plausible sequences of events that the thinning process would have rejected if it had accepted the observed sequence of real events. The proposed sampling algorithm was evaluated using both synthetic and real epidemiological data.\n", "questions": "No questions.", "limitations": "In line 103, SCMs are defined with $U_i$ as jointly independent noises, which implies a very particular case of SCMs where there is no unmeasured confounders (i.e., a Markovian system). This is not the general definition of an SCM (as in Pearl, 2009) and should be clarified as an additional (and strong) assumption of the work.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper addresses the important and relevant problem of identifying and estimating counterfactuals in temporal point processes. \n\nHowever some assumptions are not clearly stated in the paper. The assumption that the $U_i$ variables are jointly independent noise is equivalent to the assumption that the system is Markovian (i.e., absence of unmeasured confounders between all pairs of variables). Moreover, the proposed model makes the assumption that $U_i$ is an additive noise. \n\nThose assumptions are not part of the general definition of an SCM (as in Pearl, 2009), but are being used in the proposed sampling algorithm. They must be clearly stated as additional assumptions of the method and a their plausibility should be discussed.  Those are strong assumptions, most likely violated in many practical setting, so it is a critical limitation of the method. Further, the empirical study should address the robustness of the method under violation of those assumptions. \n\nWithout a clear statement of the underlying assumptions and a more detailed empirical study assessing robustness against violations of the assumptions, it is hard to judge the soundness and relevance of the proposed method.", "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "DlIsCKjFEO0", "reviewer": "Reviewer_E24B", "summary": "This paper proposes an approach to address counter-factual queries in certain types of temporal point processes, such as what would have happened if the conditional intensity was different in the past. This is a query that has not been well studied in TPPs, due to several technically challenging issues. The proposed work essentially relies on leveraging the thinning algorithm together with a Gumbel-Max structural causal model. There is also a sampling algorithm that uses this causal model of thinning together with the superposition theorem to simulate counterfactual realizations of TPPs. A limited application is studied in the main text for experiments.", "questions": "I have posed some questions previously but here are a few more questions and comments:\n\nCould the authors please clarify that the proposed ideas work for univariate TPPs? What classes of TPPs are covered by the approach?\n\nWhy is [4] cited on line 21?\n\nThe authors make an incorrect distinction in the second paragraph of Section 1. How does it matter whether a machine makes a decision or a human? And how does it matter if the stakes are “high” or “low”? A decision is a decision, and I do not understand how counterfactuals are needed for “high-stake” decisions but not “low-stake” decisions. This entire paragraph needs to be rewritten, in my view. And the authors need to clearly explain the value of having a model that enables counterfactual queries.\n\nThere are several concepts introduced on page 2 that would not be easily understood by reader not familiar with TPPs.\n\nI’m surprised that Pearl’s work and related work is not cited at all in Section 1. It seems to be left for much later.\n\ng_i should probably be f_i on line 118.\n\nMonotonicity is not clearly explained on page 3.\n\nThe Gumbel-Max trick mentioned in footnote 4 needs an explanation.\n\nIn lines 173-177, should “unlikely” be “impossible”, based on the probabilities in Proposition 1?\n\nIn my opinion, the application studied is unnecessarily involved, for what it ultimately illustrates. It is far from a real application and more like a case study that is mildly real-inspired. I understand of course that there is a general challenge around demonstrating counterfactual analysis, but I wonder if a variety of purely synthetic data experiments in the main text would have been more suitable. ", "limitations": "The authors need to write a lot more clearly about the many limitations of the work.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\nI think this paper has several new ideas for how to handle counterfactual queries in TPPs. It brings together previous ideas related to TPPs (such as thinning and superposition) and structural causal models (such as a recent Gumbel-Max version) but I think it does so in an innovative way. There are many non-trivial aspects in my view, such as how to handle the fact that “rejected” events from the thinning algorithm are not seen in the observational data. Thus I believe the work is sufficiently novel and significant.\n\nAlthough there are a few details in some sections which I could not completely follow, I’m generally favorable towards the paper. I also however have some concerns and many suggestions, which I outline below.\n\nWeaknesses (with questions and suggestions):\n\nFirstly, the title of the paper is over-encompassing. There are several assumptions made, and as far as I understand, the approach does not necessarily work for all TPPs. A specific title that is more descriptive would be appropriate. I highly encourage the authors to propose and use a different title. For me, one of the major weaknesses of the work is the lack of detail around the limitations and assumptions (even beyond the title). The authors need to be clearer about the scope of the work and the implications of the assumptions.\n\nA fundamental question for the authors: why do counterfactuals enable for better targeted interventions? Shouldn’t it be sufficient to have an interventional model if one is only interested in interventions? I found some of the motivation for the work on counterfactuals to be unfounded. Counterfactuals are clearly important for some sort of analysis, but I would like to better understand from the authors how it helps them in their problem setting.\n\nI find some of the positioning of the related work to be unclear and even incorrect. For instance, Didelez (2008) is clearly related to Granger causality for multivariate TPPs, yet it is categorized in a separate bucket – I do not understand what “predicting quantities related to an interventional distribution” means. Is this a reference to average treatment effects? Could the authors please clarify? I recommend the authors refer to other work on Granger causal models as there is a lot of literature on graphical representations of TPPs. The ones that are cited are more neural network inspired than founded on graphical model foundations. Also, the authors should do a better job of mentioning why some of the prior work (such as by Roysland) is not directly relevant.\n\nI found the experimentation section to be on the weaker side. I am surprised that synthetic experiments were placed in the Appendix, given that the paper is about counterfactual analysis. To me this is a sign that the scope of the work is quite narrow. This is not a major problem for me, but I think this should be made more explicit.", "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "2Ze42mD4Z5x", "reviewer": "Reviewer_Ddj9", "summary": "This paper proposes a causal model of thinning for temporal point processes. Given a set of observed events for a point process one can use the superposition theorem to sample plausible sequences of rejected events. Given the observed events and the rejected events, a SCM can be used to sample counterfactual events. These are realisations of the point process we would observe under alternative intensity functions. A Gumbel-Max SCM is used to ensure identifiability of the counterfactual distributions. ", "questions": "See section above.", "limitations": "Yes, I cannot identify potential negative societal impact for this work.\n", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is well written and easy to follow. This is one of the few papers I have read combining point processes with causality which makes it novel and interesting despite being quite simple. The paper would benefit from clarifying the points given below: \n\t\n- Section 3. I found the last part of this section (from line 153), which I believe to be the most important from a causality point of view, quite confusing. Given that the authors use the language of structural causal models to frame the problem, it would be useful to add a causal graph representing the data generating mechanism. This is important as it determines how the equation under line 157 is computed and what type of adjustment the do-calculus requires. For instance do(.) is dropped ‘because U and $\\lambda$ are independent in the counterfactual SCM'. This is not evident from your discussion and it could be clarified using causal graphs. In addition, a short sudo algorithm describing how to sample from the distribution after line 157 would clarify the paper.\n- Line 155. How you estimate p(U) is very important in this paper as it also justifies the choice of the Gumble max SCM. This should be explained in the preliminaries or in the appendix.\n- Section 4. Depending on the value of $\\lambda_{max}$ the number of rejected events consistent with $\\lambda_m$ changes. How would that change the counterfactual realization that you would generate from Alg 1? Would that change the distribution of $U_i$ and therefore the distribution at line 7 of the algorithm? \n- In Alg 1 the function ACC(.) takes $\\lambda_{max}$ as input, where is that used? I don't think you need it once you have $H_{max}$\n- Could you clarify why we need to restrict the SCM to be the Gumbel max. Is this to ensure that given the data from observational distribution we are sure that we can identify the counterfactual distribution correctly?\n- What is preventing you from working with points in a higher dimensional space? What about spatial point processes for example?\n\n*Experimental section:*\n- I liked the application the authors look at in this paper as it highlights the importance of the problem addressed. However, assessing the behavior of the proposed methodology with real-world data is difficult. I would prefer to see at least one experiment for the synthetic settings in the main paper. \n- Given that you assume the intensity to be known, the only approximations you are introducing are the estimation of the distribution of the noise variables and the Monte-Carlo approximation of the counterfactual thinning probability. Given these approximations, I wonder if one could assess how close the generated counterfactual realizations are to the ground truth. In a synthetic setting you can obtain counterfactual realizations from a known SCM and compare them with those you generate.\n\n*Minor:*\nLine 150 what do you mean by $t_i \\sim \\lambda_{max}$? I think in thinning the total number of points are sampled from a Poisson and then the points are uniformly distributed?\n", "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept