Causal Structure Learning in Hawkes Processes with Complex Latent Confounder Networks
Authors: Songyao Jin
Biwei Huang
Venue: ICLR 2026 Oral (2026)
Keywords: Hawkes processes
causal discovery
latent subprocess model
structure learning
time series
Primary area: causal reasoning
TL;DR: We propose a method to uncover causal relationships in partially observed multivariate Hawkes processes, despite the presence of latent subprocesses, using a discrete-time representation and a two-phase iterative algorithm.

Abstract: Multivariate Hawkes process provides a powerful framework for modeling temporal dependencies and event-driven interactions in complex systems. While existing methods primarily focus on uncovering causal structures among observed subprocesses, real-world systems are often only partially observed, with latent subprocesses posing significant challenges. In this paper, we show that continuous-time event sequences can be represented by a discrete-time causal model as the time interval shrinks, and we leverage this insight to establish necessary and sufficient conditions for identifying latent subprocesses and the causal influences. Accordingly, we propose a two-phase iterative algorithm that alternates between inferring causal relationships among discovered subprocesses and uncovering new latent subprocesses, guided by path-based conditions that guarantee identifiability. Experiments on both synthetic and real-world datasets show that our method effectively recovers causal structures despite the presence of latent subprocesses.

Published as a conference paper at ICLR 2026
CAUSAL STRUCTURE LEARNING IN HAWKES PRO- CESSES WITH COMPLEX LATENT CONFOUNDER NET- WORKS
Songyao Jin University of California San Diego soj007@ucsdd.euu
Biwei Huang
University of California San Diego
bih007@ucado
ABSTRACT
Multivariate Hawkes process provides a powerful framework for modeling tempo- ral dependencies and event-driven interactions complex systems. While exist- ing methods primarily focus uncovering causal structures among observed sub- processes, real-world systems are often only partially observed, with latent sub- processes posing significant challenges. In this paper, show that continuous- time event sequences can be represented by a discrete-time causal model as the time interval shrinks, and we leverage this insight establish necessary and suf- ficient conditions for identifying latent subprocesses and the causal influences. Accordingly, we propose two-phase iterative algorithm that alternates between inferring causal relationships among discovered subprocesses and uncovering new latent subprocesses. guided by path- conditions that guarantee identifiabil- ity. Experiments on both synthetic and real-world datasets show that our method effectively recovers causal structures despite the presence of latent subprocesses.
INTRODUCTION
Understanding causality in complex systems is essential across diverse scientific and practical do- mains, including social networks (Zhou et al., 2013), neuroscience (Bonnet [2022]. and finance (Hawkes 2018). Multivariate Hawkes processes (Hawkes) 1971), with their ability to model tempo- ral dependencies and event-driven interactions, have emerged as powerful tools for capturing these
dynamics. A majority of existing approaches (Xu et al.] 2016) Salehi et al. 2019 [dé et al.) 202T) learns Hawkes processes by using Granger causality (Kim et al 2011)) and fitting continuous-time event sequences via maximum likelihood (Veen & Schoenberg 2008). Another line of work reduces reliance on high-resolution event timestamps by performing likelihhod-barr based estimation directly on pre-binned counts (Shlomovich et al.; 2022 Cai et al. 2022 Qiao et al. 2023).
Almost all existing methods, including the two lines above, implicitly assume causal sufficiency: all task-relevant subprocesses (i.e., event sequences) fully observed, and the goal is to uncover causal structure only among those observed subprocesses. In practice this assumption is often vi- olated: many real-world systems are only partially observable, with some event sequences entirely unmeasured. For example, in neuroscience, limitations of neural recording leave many neurons unobserved even though they influence recorded neurons (Huang) 2015), obscuring the true causal structure. Identifying such latent subprocesses is crucial for reliable causal discovery, particularly when they act as confounders. Even though latent subprocesses cannot be directly intervened upon, failing to account for them can create spurious causal edges among observed subprocesses and lead to incorrect conclusions about the system's 's dynamics. (Shelton al. (2018) proposes method to impute missing event times within an observed subprocess via posterior sampling, but this does not handle entirely unobserved subprocesses unless one specifies their existence and number in advance, an impractical requirement.
In this paper, we target the general scenario partial observability: we seek to recover the causal structure among both observed and latent subprocesses without prior knowledge about whether la- tent subprocesses exist, how many there are, or where they connect. We first show that, as the time interval shrinks, the multivariate Hawkes process admits discrete-time linear causal representa-

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tion. Leveraging second-order (cross-covariance) statistics of this representation, show that the causal graph is virtually identifiable, without prior knowledge of latent subprocesses, when each latent subprocess has suitable observed surrogates. Our main contributions are:
To the best of our knowledge, we provide the first principled framework that identifies latent sub- processes and recovers causal structure in continuous-time event sequences without prior knowl- edge of the existence number of latent subprocesses.
By showing that multivariate Hawkes processes can be represented by a linear causal model over discretized variables, we derive necessary and sufficient conditions for identifying latent subpro- cesses and inferring causal influences.
We develop two-phase iterative algorithm that alternates between causal-structure recovery and latent-subprocess discovery, using rank tests on cross-covariance matrices of observed discretized variables; requires no prior knowledge of latent components.
2 RELATED WORK
Related work relevant to this paper are two areas: Hawkes processes and causal discovery. We briefly review the most relevant works here; more detailed discussion is deferred to Appendix
Hawkes Processes. Hawkes processes provide a flexible framework for modeling temporal depen- dencies among events (Hawkes) 1971 Laub et al. 2015). Existing methods for learning Hawkes structures from continuous-tim data predominantly rely on likelihood based estimation, using ei- ther parametric kernels (e.g., exponential, power-law) (Xu et al. 2016) or nonparametric procedures (Lewis & Mohler 201T). often combined with sparsity regularization (Zhou et al. 2013 Ide et al.] 2021). The NPHC method (Achab al., 2018) estimates integrated kernels via mooheng methods formulate causal dis- covery as selecting the Hawkes network that minimizes an minimum description length (MDL) or minimum message-length (MML) criterion (Jalaldoust et al. 2022 FllavčckováSShhnndle et al.) 2024]. These approaches are effective when all subprocesses are observed, but they do not ac- count for latent components. When only binned event counts are available, another line of work fits Hawkes models by maximizing likelihood over bin counts (Shlomovich et al., 2022, Cai et al.) 2022 Qiao et al. 2023), but such methods again presuppose full observability. In contrast, al- though our framework also operates on discretized data, departs from likelihood- based fitting: by leveraging the autoregressive representation of Hawkes processes, exploit low-rank patterns in cross-covariances, thus enabling the identificatinn of latent subprocesses. Unlike bin-count likeli- hood methods such (Shlomovich al. (2022), which reconstruct the likelihood of binned counts without modeling structural relations among bins, our approach establishes a linear causal represen- tation of the discretized Hawkes process. This structural correspondence is fundamentally different and is elaborated in Appendix A.I
providing a nonparametric alternative. Recent compression-
Causal Discovery. Existing causal discovery methods are primarily designed for settings where variables follow deterministic relations, rather than stochastic event-driven dynamics such as Hawkes processes. Classical approaches for i.i.d. data include constraint-based (Spirtes al. al.] 2001). score-based (Chickering, 2002). and functional approaches (Shimizu et al.] 2006). Extensions to handle latent variables include based methods that identify hidden structures in linear models (Huang et al.] [2022] Dong et al.) (023). While these allow latent variables, their guarantees typically hold only up to equivalence classes and rely on restrictive structural and cardinality assumptions that are incompatible with Hawkes dynamics. For time-series data, LPCMCI (Gerhardus & Runge) 2020) adapts autocorrelation and exogenous latent variables, assumptions that are violated in Hawkes processes where dense cross-lag dependencies and endogenous latent subprocesses naturally arise. Although method also leverages second-order (rank) statistics, it differs in two key respects. First, it tar- gets subprocesses in multivariate Hawkes systems rather than static variables and accommodates both endogenous and exogenous latent subprocesses. Second, it establishes identifiability through time-aware rank constraints specifically tailored to Hawkes dynamics, avoiding the infeasible as- sumptions underlying existing rank- or independence-tess based approaches.
iatiiiional-ndeeeedeeeee
testing to temporal domains. However, assumes weak
PARTIALLY OBSERVED MULTIVARIATE HAWKES PROCESS- CROCESS-BASED CAUSAL MODEL
This section introduces the causal modeling framework for partially observed Hawkes processes and establishes the key definitions that support the subsequent results structure discovery.
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3.1 MULTIVARIATE HAWKES PROCESS
A multivariate Hawkes process is a self-exciting point process modeling temporal dependencies among events via set of counting subprocesses Ng [N where N:(1) records the number of type-i events up to time / (Hawkes) 1971 Laub et al. 2015). For each i E ......,], the intensity
of subprocess N, that governs the event- centerriggering behavior
j=1 EL
(.(t) = ui +
0is (
5) (s),
(I)
where μ4 is the background intensity, and (15(s) (8) 2 0, (0, oo) is excitation function, which measures the decaying influence of historical type-j events on the subsequent type-i events and piecewise continuous. (Strictly) stationarity requires the spectral radius of the influence matrix E Rixt with entries Pij Jo (s)ds to be less than one (Bacry & Muzy 2016). A detailed exposition of Hawkes process is deferred Appendix B
See also Fig. Ta for illustration.
We are interested in identifying, for each subprocess N,, the minimal set of subprocesses Pg S Ng such that (t) depends only the historical events of the subprocesses in Pg and not on others. Formally, this corresponds to f Φty(t 8) (s) > 0 for each N, E Pg. and zero otherwise. In this case, N, said to be locally independent (Dieelez, 2008 of Ng (Po given Pg.
3.2 MODEL DEFINITION
To formalize our framework, we define graphical causal model for multivariate Hawkes processes, where nodes represent subprocesses and directed edges correspond to nonzero excitation functions. The goal is to recover the causal structure among both observed and latent subprocesses.
Definition 3.1 (Partially Observed Multivariate Hawkes Process-basse Causal Model (PO-MHP)). Let G = (Ng,&g) be a directed graph, where each node N₂ E Ng represents a subprocess in a multivariate Hawkes process. A directed edge Esj E Eg exists iff Jo Φi - s) d > 0. The node set Ng consists of p observed nodes Og = {O:)e= and latent nodes Lg = {L,) which correspond to the observed and latent subprocesses, respectively.
The PO-MHP model naturally allows cycles and self-loops that typically challenging analyze Claassen & Mooij 2023). Two subprocesses N; and N₂ form a directed cycle there exist directed paths from N; to Nj and from N₂ back to N₂- Any subprocess N; has self loop if f da(l s) (s) O. Furthermore, it allows for the presence of latent subprocesses, and directed edges may exist between any pair of subprocesses, whether observed or latent; both N, and N, may be either observed latent. To the best of our knowledge, it is the first principled work investigate such a general structure in Hawkes processes.
Definition 3.2 (Cause and Effect). For any N,, NN E Ngg, if there exists a directed path from N; to Nj. then Nj is said to be a effect of N;. and N; is said to be cause of Nj.
Definition 3.3 (Parent-Cause Set). For N; NNg, the minimal set Pg S Ng \ {N,} is called its parent-causs set every directed path from nodes in Ng' ([V[ N; passes through some node in Pg. In the special case where N; has self-loop, it is also included in Pg.
Proposition 3.4 (Parent-Cause Set and Local Independence). Subprocess N; is locally independent (defined in Section 3.1 of Ng Pg given Pg if and only if Pg is the parent-cause set of N in Ng..
Proposition 3.4 is the PO-MHP analogue of the local Markov property: loops, conditioning on the parent-cause set Pg screens off N; from all other subprocesses.
with cycles and self-
4 STRUCTURE IDENTIFICATION IN PARTIALLY OBSERVED HAWKES PROCESSES
In this section, we formalize partially observed Hawkes processes in discrete- ciscrete-time framework and develop rank-based tests on observed counts that enable identification of the summary causal graph, including relationships mediated by latent subprocesses.
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N.
(a)
(b)
(c)
(d)
Figure 1: Figure I: Illustration of multivariate Hawkes processes. (a) Point process representation
with three subprocesses N1, N2, N3. where the continuous timeline is partitioned into intervals of length A. (b) The corresponding summary causal graph, the central object of this paper, with causal relations N1 + N2 Ns and self-loops all nodes. (c) The window causal graph, showing the underlying time-lagged causal mechanism: each node denotes the count in one interval of length A, modeled weighted sum of lagged parent nodes plus noise (Eq. I (d) A minimal example with a latent subprocess L1 confounding O₃ and O2, highlighting the primary focus of this paper.
4.1 FROM CONTINUOUS-TIME TO DISCRETE-TIME REPRESENTATION
Directly inferring the causal structure among subprocesses in multivariate Hawkes process is chal- lenging, particularly with latent subprocesses, Eq. [ defines a continuous-time stochastic process where (t) is the expected history-dependent instantaneous rate. Instead of relying on conventional moximum-likelihooe fitting, present an explicit correspondence between Hawkes dynamics and a specified discrete-time linear autoregressive causal model, which enable identification of latent subprocesses and the causal structure by applying statistical tests only observed count data.
Theorem 4.1 (Hawkes Process Linear Autoregressive Model), Let Ng {NoY-= be sta- tionary multivariate Hawkes process with background intensities {1a)== and excitation functions {din (o))())) Define the discretized event count in the n-th time window of size A E (0, (0
N(") N₁ (nA) N; - (n)1)))) with N. 0,
where 8 depends on the moment structure of the process. Then, as - 0, the Hawkes process admits the linear autoregressive representation
n
N((1)
(()) Ej
N(n-k) + (()) + 0000 Zt
j=1k=1
where 000) = - μa is the background parameter, 0 cient, and e( denotes the n-th realization of serially uncorrelated white noise sequence.
kA ((= 1)a (s)(s)ds is the excitation coeffi-
Theorem shows that each current-bin count N((") (referred to as variable hereafter) linear combination of lagged counts (N/m=>))kE(L (E
(E.......)) plus noise. The discretized variables therefore encode the causal structure of the underlying continuous time subprocesses. As illustrated in Fig. directed edge N₂ N, in the summary graph corresponds to edges from all lagged variables of
N2 into N(") in the window graph. This summary-window correspondence allows us to infer the causal structure among subprocesses by testing structural relations among the discretized variables. The proof Theorem 4.1 appears in Appendix G
In practice, is unnecessary include all lags kt 1,...,.7 Because the excitation function (s)(s), which serves decay kernel, typically has finite support, the coefficients e(t)) _ diy ds
vanish for large distant lags (((=k)) therefore have little influence on !²("), We define the small- est cutoff K at which this occurs as the number of effective lags. Accordingly, we truncate to at most m lags with m > K, a standard finite-windww practice in time-series estimation (Hyvärinen et al. 2010 Peters et al. 2013), though they assume acyclic summary graphs and understate latent com- ponents. A practical way to estimate is to retain lagged variables whose correlations with currett variables remain statistically significant. Moreover, bin width A small relative to the support of
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(b)
(c)
(d)
Figure 2: Examples of causal graphs with latent confounder subprocesses. (a) Summary graph where O1, O2. O3. O4 are observed and L1 is latent. (Unlike Figure Id O1, O2 are shown without self-loops simplify the derivation.) (b) Corresponding window causal graph among O1. O2. and L1 with two effective lags. (c) More complex case where L1 connects O₁, O2 via intermediate latent subprocesses T2, L3. All subprocesses have self-loops except La and Lg. (d) An even more intricate case, extending (c) with more complex intermediate latent subprocess paths and an additional edge O2 + L1. All subprocesses except the intermediate latent ones have self-loops.
the excitation function is sufficient; Appendix Q.3 for empirical illustration. Data sparsity of- ten offers intuitions: sparse event sequences correspond to excitation functions with larger effective support. In practice, we recommend selecting A via simple grid search and choosing a value from the stable range, where the recovered structures remain robust to small perturbations in A.
4.2 STRUCTURE DISCOVERY THROUGH RANK CONSTRAINTS
In this section, we link second-order statistics of Hawkes data to variables in the window causal graph, which in turn enables identification of the summary causal graph- even with latent sub- processes. Under the linear representation in Eq. 2 with white noise, the causal structure induces characteristic low-rank patterns in cross-covariance matrices observed discretized variables.
4.2.1 FROM OBSERVED PARENTS TO LATENT CONFOUNDER SUBPROCESSES
To illustrate, consider Fig. Although the summary causal graph may contain directed cycles and self-loops, the associated window causal graph a directed acyclic graph (DAG): by construction, future events cannot causally influence the past, reflecting the intrinsic temporal (Granger-causal) directionality (Shojaic Fox 2022). In this example, all three subprocesses are observed. In the summary graph, the parent-cause set of N1 is Pg (N, (e.2)- In the window graph, consider the observed variable set N JE (iE(1.23)
where is chosen to be least the number of effective lags. Conditioning the lagged variable set P. (N (e-mm.... renders zy()) (iE(1.2) d-separated from the remaining variables R. = N. (Pa U Ar(")). Consequently, the rank of the cross-covariance between N(()) U.P. and R. UP. equals [Po]. The theorems below formalize this connection between rank constraints and the window -gwaphrrph structure, which then leverage to identify the summary causal graph, including cases with latent subprocesses.
Lemma 4.2 (D-separation and Rank Constraints in the Window Graph), Consider the window causal graph of a PO-MHP. For any disjoint variable sets A,, B, and C,, C. d-separates A. and B., and only if rank(EA,,UC., B,UC,) |Col. where ΣA,UC., B.UC, denotes the cross- covariance matrix between A, U C, and B U C., and |C,l is the cardinality of C..
Proposition 4.3 (Identifying Observed Parent-Cause Set). Consider PO-MI1P with observed sub- processes OG {O.) The followings are equivalent:
In the summary graph, the set Pg Og is the parent-cause set of the subprocess O1- In the window graph, with the observed variable set O = jn/n-m.....
(1e(1.2....)) Pg is the minimal
set such that lagged variable set P. = (0(5) je(-)).ee_m..._... 1) contains all parent variables of the Jo.EPs
current variable )
Pg is the minimal set such that variable set P,, d-separates O (m) from the rest O, O, (*)
Pg is the minimal set such that tank(Eog)) uP.. o, or(o(()

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The criterion in Proposition 4.3 involves only observed variables. Whenever an observed subprocess O1 satisfies this criterion, its parent-cause set Pg is uniquely determined (by the minimality condi- tions in items 2-4), and O₁ is locally independent of all other observed subprocesses given Pg. As we show next, these four equivalent statements remain valid even with latent subproessses.
One type of latent subprocess is the intermediate latent subprocess, which, if it exists, may lie on directed paths between an observed subprocess and each of its identified parent-cause set. In general, such intermediates are unidentifiable and typically omitted, since their effects can be ab- sorbed by the identified parent-cause set. Nevertheless, owing to the specific structure of discretized Hawkes processes, once the observed parent causes of an observed subprocess have been identified via Proposition 4.3] it is possible to quantify the number of intermediate latent subprocesses along these paths. Detailed statements and proof' are deferred to Appendix C Unless otherwise stated, for notational simplicity we do not consider intermediate latent subprocesses into parent-cause set.
Another type of latent subprocess is the latent confounder subprocess, key focus of this paper. It is latent subprocess that must be included in the parent-cause set to render its observed effect subprocess locally independent of others. For example, in Fig. [2 the structure O + 11 + O₂ makes L a latent confounder of O₃ and O2; in this case, O₁ and O2 are locally independent only when conditioning on L1_ Proposition 4.3 will not identify the parent-cause set of either O2 or O2, since such confounders are unobserved. Furthermore, note that Eq. does not specify the excitation function. We impose the following mild constraint on it to facilitate identification.
Assumption (Excitation Function). We consider that the excitation function takes the form (1)(8) = dzyte' (s),, Vi,j E (I..... .I), where di is constant capturing the peer influence between event types and j. and w(ss) is a common decay function depending only on the time lag s.
This assumption is quite general. For instance, the widely used exponential decay function ane Ss (Zoou et af. 2013) falls into this class. Other examples include normalized linear decay, nor- malized logistic decay, and related forms (Burt) 2000)) Moreover, following standard practice in causal discovery (Spirtes) [013 Huang et al.) 2022), we also adopt the rank-faithfulness assumption for Hawkes processes. Intuitively, this assumption rules out pathological parameterizaaions where causal relationships cannot be identified. It holds generically with infinite data, since the degener- ate cases where fanh-faitsfulness fails constitute a set of Lebesgue measure zero (Spirtes) 2013).
Further details are deferred to Appendix D
Intuitively, latent confounder leaves a characteristic footprint in the second-order statistics of the observed subprocesses. When the same latent subprocess drives multiple observed subpro- cesses through the Hawkes excitation kernel, its contributions to their current values become aligned across time lags, that the corresponding rows in suitable cross-covariance matrices lie in a low- dimensional subspace. This alignment manifests itself as a rank deficiency that cannot be explained by any purely observed parent-cause set. In the following, we make this intuition concrete by ana- lyzing simple examples and then distilling the structural patterns into general rank-based conditions for identifying latent confounders.
4.2.2 CHARACTERIZING LATENT CONFOUNDERS VIA RANK CONSTRAINTS
Building on the above intuition, we now characterize when latent confounder subprocesses leave identifiable low-rank signatures in cross-covariance matrices, moving from illustrative examples to general conditions. Given the excitation function (1y(s) dsyto(s), the excitation coefficients in
Eq. are g5h) { (s) ds aez f(K-1)A w(s) ds, where the integral term f+a ((-11) w(s) ds depends only on the time lag kt Consider the summary graph and its corresponding window graph
with m = 2 laggged variables, as illustrated in Figs. Figs.2alann [2j 2b According to the linear causal model
in Eq. 2 the structural equations for the current variables O and
Ol²)) are
L
(x)
000)
doth Jo w(s) da
Go14 1 20 (xs)
E
+
E
(3)
Co2
dozhi Jo w(s) ds
Gogls Ja w(s)
It is evident that the coefficient matrix E has rank 1. Consequently, the cross-covariance matrix satisfies
rank E, " ,of"), h 1o(9) pe(npp.
1. For details, see Proposition 4.5 and its proof in Appendix K
he(3.)
This indicates single latent confounder subprocess T.1 that serves as the parent cause of both O₂ and O2, such that, conditional L1, {O1, O2} locally independent of (Os, Oa).

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However. if O2 and O2 in Figure 2a also have self-loops (as illustrated in Figure Td the rank of the coefficient matrix is no longer The self-loops generate additional indirect causal cf- fects propagated from the lagged latent variables (T.² be he(n-m....n-I) through the lagged observed variables "o to the current variables Ol") and O Since these lagged ob- (e(222) served variables are available, we include them in the cross-covariance matrix, which yields rank Me
2m + 1, where 2m corresponds the lagged variables of the two observed subprocesses O₂ and O2, and corresponds to the latent confounder subprocess. See Appendix E for further details.
(oll (ee")ettt je " "u yefn-m...n-1))
(3.4).)
(1.(1.3)
Furthermore, in more complex scenarios, as illustrated in Figs, 2c and 2d the causal pathways from the latent confounder L1 to the observed subprocesses On and O₂ become increasingly intricate. To address such cases, we formalize the symmeiric path situation, which precisely characterizes those graphical configurations that induce rank deficiency in certain sub-covariance matrices of the observed subprocesses. It underpins the sub- sequent theorems by establishing a one-to-one correspondence between the underlying graph structure and its observable statistical properties.
Definition 4.4 (Symmetric Acyelic Path Situation). Consider latent confounder L1 and observed effect subprocess Oş1- The following conditions define the symmetric path situation:
There exist directed paths from La to each subprocess in Ogi such that each path consists exclusively of intermediate latent subprocesses (i.e., no observed subprocesses appear along these paths). and neither L nor any subprocess in Oşi appears as non-end point along these paths.
All such directed paths have the same length, meaning they contain the same number of intermediate latent subprocesses.
All such directed paths are acyelic. Naturally, none of the intermediate latent subprocesses involved have self-loops.
The structure in Fig. 2c satisfies Definition 4.4 where the latent confounder Ln connects On and O2 through the intermediate latent subprocesses L2 and T.3, respectively. both without self-loops. If one intermediate subprocess is removed (e.g.. Ia), the condition in Definition 4.4 is violated, since the path from La to O would then include one intermediate latent subprocess L2. w whereas the path from L1 to O2 would include none. Similarly, in the more complex structure shown in Fig. 2d the core structure formed by the blue causal edges satisfies Definition 444 and the addition of the green edges still preserves this property. However, adding an extra edge, for instanee, from Ls to La, would break the condition, as it would introduce both asymmetry and cycles into the paths. The subsequent thcorems will leverage this path condition to formally characterize graph structures involving the identification of latent confounder subprocesses.
Proposition 4.5 (Identifying Latent Confounder from Observed Effects). Consider a PO-MHP with citation function thi (8) asyu((s) under rank faithfulness. The system consists of observed sub- processes Og denote the set of corresponding observed variables. For any observed subprocesses {O,,
{0:)=-1 and potemtially latent subprocesses. Let O, {0 (n)))se(--me (n-mm......))) (iE(L.....))
2m + L. if and only if there exists a latent confounder subprocess L1 in the parent-cause set of {O1, 02) such that conditioning on P = In (Oi)ect)))))))) renders 02) locally independent of nonempty set Og (P'. and L1 with {O1, O2) satisfy the Definition 4.4
rank [ (oo())ye(r.. m....,,, ouitof"" (((1))))))m))))))) (icl1.2)
Proposition allows us to infer the existence of a latent confounder from its observed effects. This naturally raises the question: How can we systematically infer the remaining causal relations involving these inferred latent subprocesses? This challenge is illustrated in the four summary graphs of Fig. 3 In the following, we show how the observed effects can serve as surrogates for their latent confounders, thereby enabling recovery of the remaining causal structure.
Definition 4.6 (Ohserved Effects as Surrogates). For each latent subprocess L, inferred from its observed effects [O1, O2). we designate one of its observed effects, denoted De(L,) = O1, an observed surrogate of L1. The surrogate chosen such that there exists directed path from Lı De(La) that does not traverse any other observed subprocess. We further define Sib(De(L1) as the set of observed siblings of De(L1), L1). consisting of all known other observed subprocesses influenced by Li through paths that likewise do not pass through any other observed subprocess.
For any observed subprocess Oz, we adopt the unified notation De(O,) O₂ and, correspondingly, Sib(De(OO)) 0. Moreover, Sib(De(Lz)) captures the minimal set of observed subprocesses required to isolate the local inffuence of Li on the rest of the system, exeept through De(Li).
Theorem 4.7 (Identifying Parent- Parent-Cause
Cause Set with Latent Confounder Involved), Consider PH-MHP with excitation function (s) = as))(() under rank faithfulness. The system Ng == Og U Lg consists of observed subprocesses Og = {0.) and inferved latent confounder processes Lg whose parent-cause
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(a)
(b)
(d)
Figure 3: Illustrative examples of interactions among inferred latent confounder and the remaining observed subprocesses. In (a)-(c) assume I1 has been inferred from its observed effects (O1, O2)- (a) O3 causes L1- (b) Both L1 and Og cause O4- (c) L1 causes La, where L4 can be inferred from {Os, O4}. (d) L1 serves as the latent confounder of both inferred latent confounder L2 and L3.
sets remain to be identified. Let O (00t0)
(iEnmmiiim...... denote the corresponding observed variable TiE (1....)))) (i
set. For a subprocess N₁ E Ng and a candidate parent-cause set P''s S Ng.. when either N₁ is latent, or P'g contains latent subprocesses, or both, the following condition holds: Pgg is the minimal set such that
rank -...l where A = (De(NN)
m.....) U4O(()
() (30))))11 (e-j(emm....... (n-m......
L.EE's and B = O. (De(N))) U (De(L.)))))L (2.EE)
10,EP%
and only i/ is subset of the true parent-cane cause set of N₂ such that: (i) conditioning Sg == PP U (e()) U
(o."))eetn m.....)....
(Sib(De(L.)) Li)) ( renders N, locally independent of nonempty set (0,E8.(DD(MM)) (Sa(De(L.)
) U Sib((De(NN)
Soa (ii) for each L, E P/, the latent confounder Li with observed effects (De(Ni), De(L.)) satisfies
Definition 4.4 and (ii) all possible observed surrogates of N; Og have been identified and added into the observed sibling set.
we identify arbitrary causal relations among both observed and inferred latent subprocesses. This naturally raises a final question: How can we further discover new latent subprocesses that are causally related to inferred latent subprocesses, as in Fig. 3d As shown below, the observed surrogate can still be leveraged for this purpose.
With Theorem 4.7 (and Proposition 4.3
Theorem 4.8 (Identifying Latent Confounder from Latent Confounder) ). Consider a PO-MHP with excitation function Φ1y (s) = GG (s) under rank faithfulness. The system Ngg == Og U Lg consists of observed sub- processes Og = 1O and inferred latent confounder subprocesses L& whose parent-cause sets remain jein-m.....n) denote the corresponding observed variable set. unidentified by Theorem 4.7 Let O == {O(00) (2.(......))
For any two subprocesses NI. N2 C Ng (either observed or latent), rank (B.), = |Aul- 1, where A == (De(N.) ( ( (e(mmm.....) u ( (n (n-m.......)
Jo (O.ESS)(Pe(N ( (JsSs)())(NN))) and B, == O, B>\(De(Ni)) )(m), (e(Na)(m) ifand only if there exists a latent confounder subprocess L1 in the carenttccuse set of N22 such that: (i) conditioning on Pg- = -T1 U (Na)eet U (SSib(DeeN.)))
(iE(1.2)
renders { (N1, N2) locally independent of nonempty set Ng NPg: (ii) Lı with (Pe(N:), De()) satisfies Definition 4.4 and (fii) all possible observed surrogates of {N1, No in Og have been identified and added into the observed sibling set.
Theorem 4.7 and Theorem 4.8 are extensions of Proposition 4.3 and Proposition 4.5
respectively. These extend the framework by replacing latent subprocesses with their observed surrogates when evaluating the rank of the relevant cross-covariance matrices. Equipped with these four key theorems, we are now ready to present the discovery algorithm in the next section.
RANK-BASED DISCOVERY ALGORITHM
In this section, we present two-phase iterative algorithm that leverages the identification theorems to progres- sively recover causal relationships among discovered subprocesses and to uncover new latent subprocesses. Let Ag denote the active process set, consisting of subprocesses whose parent causes remain unidentified. Initially, Ag is set to the observed subprocess set Og and is updated throughout the procedure. Moreover, due to cycles in the summary causal graph, observed subprocesses previously identified as effects may still act as causes for other subprocesses in Ag. and thus remain under investigation. The overall procedure is in Algorithm] T
Phase I: Identifying Causal Relations Each iteration begins with Phase which identifies the causal structure under- investigined subprocesses (both observed and latent) in Ag. In this phase, we systematically iterate over each subprocess in Ag and test its parent causes using the current set Ag U Og. If subprocess's 's carent-cause set is fully contained by this set, it can be identified using Proposition 4.33ioo Theorem 4.7 Once identified, the subprocess is removed from Ag. The phase continues until no further updates occur. Detailed steps are given in Algorithm 2 in Appendix 0.1
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Algorithm 1 Two-Phase Iterative Discovery Algorithm
Input: Observed subprocess set Og
Output: Causal graph G
1: Initialize partial causal graph G == 0 and active process set Ag == Og.
2: repeat
3: (G.Ag) +- Identifying Causal Relations AG, Og).
// Phase I // Phase II
4: (9, < Discovering New Latent Subprocesses Ag, Oz).
5: until Ag empty updates occur.
6: return g
Phase II: Discovering New Latent Subprocesses When additional subprocesses in Ag be resolved in Phase I, the algorithm proceeds Phase II. Here, we search for new latent confounder subprocesses by exhaustively checking all pairs in Ag using Proposition 4.5 and Theorem 4.8 Identified latent confounders are merged if pairs overlap in subprocesses, implying shared latent parent cause. Ag is then updated by adding the latent subprocesses and removing their effects. After that, the algorithm returns Phase I. This procedure continues until Ag empty or unchanged. Details are provided in Algorithm Appendix O.2
Theorem 5.1 (Identifiability of the Causal Graph). Consider PO-MHI with excitation function (s)(s)
auy (s) under rank faithfulness. every latent confounder subprocess, along with all its observed surrogates (2 2), satisfies Definition 4.4 then causal graph over observed and latent confounder subprocesses can be identified. When no latent subprocesses exist, the causal graph is fully identifiable using only Phase
Morcover, the computational complexity depends the number of subprocesses (including latent confounders) and the density of the underlying causal graph, which together determine the number of iterations required for complete graph discovery. A detailed complexity analysis is in Appendix P
EXPERIMENTS
Synthetic Data We compare our method against six strong baselines. SHP Qiao et al. 2023 and THP Cai
et al. 2022) are likelihood-basee based approaches for discretized (binned) Hawkes data, while NPHC (Achab et al.
2018 cumulant-based method for original Hawkes data. Since existing Hawkes-based methods cannot identify latent subprocesses without prior knowledge, we additionally include two rank-based approaches de- veloped for Li.d. linear latent models, Hier. Rank (Huang et al.) (2022) and RLCD (Dong et al.] [2023). as well as LPCMCI (Gerhardus & Runge 2020). time-series baseline that accounts for exogenous latent confounder variables, though all three rely strong assumptions
satisfied in our setting (See Appendix A For these methods, we apply them to discretized Hawkes data. our method, we evaluate both event sequences gen- erated by the Hawkes process in Eq. []] and data generated directly from the discrete-ting time model in Eq. (2). We test on six synthetic graph families: the fully observed graph in Fig. [bb and five structures with latent sub- processes in Figs. 2a and 3a 3d Results are reported as average Fi-scores over ten runs personal CPU machine. Additional expermeenal details and extended results (larger graphs, sensitivity A, and robustness to fattffiithullness violations) are provided Appendix Q As shown Fig. 4 our method consistently out- performs all baselines. Notably, latent cases typically require larger sample sizes: because the spectral radius of stationary Hawkes process is < 1, causal influences attenuate along latent paths, making reliable detection more demandig...
Ours (Discrete) Lowel
Outs (Howkes)
SHP
NPHC
THP
Hicr. Rank
RLCD
LPCMCI
1.0
Lave 2
Case 3
1.
Cave 4
2.5
2.5
0.2
2.-
24-
0.4
32-
12
02
30003
50550
0000e
18
30303
50050
0C0C0
30303
50000
00380
30302
50000
6038C
Semples
Figure 4: Fl-score comparisons for first four synthetic causal graphs (Cases 1-4), corresponding to the structures in Figs. lb
2a
3a
and 3b
See Appendix Q.3 for additional cases.
Real-world Data We evaluate our method on a public cellular network dataset Qiao et al. 2023 with expert-validated ground truth. The original dataset contains 18 alarm types from 55 devices ( 35k events over eight months). though not every device exhibits all alarms. We focus on device_id=6, which in- cludes the alarms relevant to the subgraph of interest. For evaluation, consider a five-alarm subgraph (AT amm i dds=0=3 and 7) and treat A arm d as latent by manual exclusion. Notably, Alarm i and

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Figure 5: Inferred causal subgraph from the cellular network dataset, where Alarm id=7 is suc- cessfully identified as latent subprocess.
Al rm-id=3 are observed effects of this latent subpracess, enabling its recovery from observed data. Our inferred graph (Fig. [) successfully recovers the latent subprocess and its primary influences, Furthermore, on this sub dab-dataset our method quantitatively outperforms representative baselines; Appendix Q.4 for details.
7 CONCLUSION AND FUTURE WORK
We presented principled framework for structure learning in partially observed multivariate Hawkes pro- cesses (PO-MHIP) capable uncovering both causal relationships among observed subprocesses and latent confounder subprocesses influencing them. By leveraging sub-covariance rank conditions and a tailored path constraint, we derived necessary and sufficient conditions for identifiability and designed a two-phase iterative algorithm that reconstructs the full causal graph with guarantees. One future rescarch direction excitation function, for example, by introducing node-spe specifcc decay rates, broaden its applicability. Another direction is to develop discovery algorithms with lower complexity. Furthermore, more diverse real-world datasets can be applied to gain deeper domain insights.
relax the
ACKNOWLEDGMENTS
The authors extend their sincere gratitude to the anonymous reviewers for their insightful feedback and con- structive suggestions. Biwei Huang would like to acknowledge the support of the National Science Foundation (Grant No. DMS-2428058).
ETHICS STATEMENT
This work makes primarily methodological contribution, proposing framework for structure learning in partially observed multivariate Hawkes processes. The study does not involve human subjects, personal or sensitive data, or applications with direct societal or ethical risks. Our experiments are conducted on synthetic data and publicly available dataset with expert-validated ground truth. We therefore believe that the research does not raise ethical concerns beyond standard scientific integrity, and fully consistent with the ICLR Code of Ethics.
REPRODUCIBILITY STATEMENT
We have made significant efforts to ensure the reproducibility of our results. All theoretical results are stated with explicit assumptions, and full proofs are provided in the appendix. Experimental details, including data preprocessing steps, and additional results (e.g., sensitivity to A and robustness analyses), are included in the appendix and supplementary materials. Synthetic data generation procedures fully specified, and the real-world dataset
publicly available.
LLM USAGE STATEMENT
Large Language Models (LLMs) were used only as general- purppse tools aid in polishing the writing and improving clarity presentation. They were not involved research idcation, methodological design, experi- ments, and analysis. No LLM qualifies for authorship.
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Appendix
A
Detailed Related Work
14
Detailed Relation Binned Hawkes Estimation
15
A.2 Detailed Relation Rank-Based Latent Discovery
16
Multivariate Hawkes Process Details
16
Identifying Intermediate Latent Subprocesses
16
Rank Faithfulness for the Hawkes Process
18
Accounting for Self-Looped Observed Subprocesses under Latent Confounder Influence
18
Proof of Proposition 3.4
19
Proof of Theorem 4.1
19
Proof of Lemma
21
Proof of Proposition 4.3
21
J Preliminaries for Proofs of Proposition 4.5 and Theorems 4.7 and 4.8
21
Proof of Proposition 4.5
22
Proof of Theorem 4.7
24
M Proof of Theorem 4.8
24
Proof of Theorem 5.1
25
Details of identification algorithm
26
Phase
26
0.2 Phase
26
P
Computational Complexity of the Algorithm
26
More Details of Experiments
27
Q.1
Synthetic Data Generation and Implementation
27
2 Evaluation Metrics
28
Q.3 Additional Synthetic Experiment Results
28
Q44
Analysis of Real-world Dataset Results
31
A
DETAILED RELATED WORK
We review related work in three areas: point processes, Hawkes processes, and causal discovery.
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Point Processes large body of research studies temporal dependencies in general point processes. Meek (2014) introduced a graphical framework based on 5n ssepaaaaioo and process independence to connect graph- ical representations with statistical properties. GGuaaarddaaa et al. (2011) proposed a one-dimensional point process model with piecewise-constee constant conditional intensity, enabling closed-form Bayesian inference of tem- poral dependencies. [Chwialkowski & Gretton (2014) developed a kerne--aased independence test for general random processes, offering nonparametric perspective.
Other works focus on specific structural settings. [Basu et al. 2015) studied Granger causality for discrete transition processes with grouping structures. Daneshmand et al. (2014) proposed a parametric cascade gener- ative process for continuous-time time diffusion networks. In the setting of marked point processes, (eeeessss,DDdeeez (2008) introduced graphical models capable of capturing local independence across marks, generalizing dependency analysis in complex systems.
Hawkes Processes Hawkes processes (Hawkes 1971 Laub et al. 2015) form a prominent class of self-
exciting point processes that model how past events influence future occurrences. A major rescarch line cs- timates Hawkes structures from continuous-timg event data, typically via likelihood-based methods. These approaches assume parametric excitation kemels such exponential or power-law (Zhou al. 2013 Zhao etal. 2015), or adopt nonparametrie procedures (Lewis & Mohler 201 [ [uo et al. 2015), often combined with sparsity or low-rank regularization (Xu et al. 2016 Idé et al. (021). p. The NPHC method (Achab et al. 2018
provides a nonparametric alternative, estimating integrated kernels through moment- matching. While effective when all subprocesses are observed, these approaches do not address the presence of latent components.
moment-matchingg
When only discretized or binned counts are availahle, another line of work fits Hawkes models directly from bin data. (Shlomovich et al. (2022) proposed an EM algorithm with importance sampling to recover parameters from binned observations without precise timestamps. Qiao et al. (2023) introduced SHP, which learns causal structure from binned event sequences using sparsity-regularin
likelihood, [Cai et (2022) developed THPs, which impose topological constraints to recover causal influences from discrete sequences. These binned- likelihood approaches, however, assume full observability and do not infer the existence or number of latent subprocesses.
Causal Discovery Causal discovery aims to infer causal relations from data, traditionally under i.i.d. as- sumptions with DAG structures (Pearl (2009). Classical approaches include constraint-based methods (e.g.. Spirtes al. (2001)) I. score-based methods (e.g.. GES Chickerin (2002), ) and functional approaches (e.g.. LiNGAM Shimizu et al. (2066). Latent variables pose major challenges. leading to extensions such as FCI and its variants (Spites ct al.] 1995 Colombo et al. 2012 Claassen ct al. (2013), which use conditional inde- pendence to infer partial structures under exogenous latent confounder variables.
More recent advances explore settings with causally related latent confounder variables. [Tuang et al. (022)) and Dong et al. (223) identify equivalence classes linear models using second-order (rank) statisties. However, these methods typically rely on restrictive assumptions-succe as hierarchical latent structures or cardinality constraints -hhtt are incompatible with Hawkes dynamics. In Hawkes- dense across lags, observed surrogates are often fewer than effective latent "parents," and endogenous latent confounder subprocesses arise naturally. Other works, such as Xic al. (2020) 2022] and Jin et al. a (2023), leverage higher-order statisties to improve identifiability, but they still assume i.i.d, data, which may introduce spurious dependencies once temporal dynamics are ignored.
time series, autoregressive
representations
Causal discovery has also been extended to sene-series domains. Approaches such as SVAR- LiNGAM (Hyvärinen al. 2010) and PC-style temporal methods like PCMCI (Runge (220)) and LPCMCI (Gerhardus) & Runge| (2000) rely on conditional independence tests over lagged variables. Yet these methods presuppose weak autocorrelation and exogenous latent variables, assumptions that are violated in Hawkes processes, which exhibit dense cross-lag interactions and endogenous latent subpracesses.
A.1 DETAILED RELATION TO BINNED HAWKES ESTIMATION
Shlomovich et al. 2022 address parameter estimation for binned Hawkes processes via modified EM algo-
rithm when only bin counts N. (N + 1)A)- - (ttA) are observed and exact event times are unavailable. The bin counts are treated as observed data and the unobserved event times T as latent variables (their Eq. 6). Because direct Monte Carlo sampling of 7 is intractable in Hawkes models, they employ importance sampling to simulate within-bin timestamps that match the observed counts, therehy maximizing the (binned) likelihood (see their Sec. 2).
Our goal and methodology differ. Leveraging the link between INAR and linear autoregressive models, The- orem [.I. establishes an explicit linear structural representation for discretized multivariate Hawkes processes. This connection enables causal discovery directly over binned variables including the identification of latent confounder suhproeesses-witt identifiability guarantees (Propositions 4 and 4.5 Theorems 4% and 4.8)
In contrast to likelihood maximization based simulated event times, our framework uses time-aware rank
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constraints cross-covariances to recover causal structure. To the best of our knowledge, prior work has not provided direct, thcoretically grounded reduction from Hawkes processes linear structural models for the purpose of causal discovery.
A.2 DETAILED RELATION TO -AAED LATENT DISCOVERY
Huang et al. 2022 (and related works by Xic et al. 2022 and Dong et al. 2023 study latent structure discovery under L..d. assumptions and continuous variables. Our problem differs substantively: we aim to recover causal structure among observed and latent subprocesses in multivariate Hawkes processes, where each subprocess is a point process and inference is performed on discretized representations.
Different Data Domain and Causal Assumptions Huang et al.
2022 (and Xie et al. (2022) ) assume
a latent hicrarchical structure, specifically: (i) there are no direct causal links among observed variables, and all dependencies among observed variables arise exclusively from their latent confounder variables; and (ii) observed variables cannot cause latent variables, i.e. endogenous latent confounders
ruled (see Eq.
Definition in Huang al. (002), and Eq. 1, 2 and Definition in Xie et al. (2022)). Neither assumption is needed in our framework. We allow both direct observed-to-obsee in our paper) and the existence of endogenous latent confounder subprocesses that can be caused by observed subprocesses (see Theorem 4.7 in our paper).
edges (see Proposition Proposition[4.3
Cardinality Requirements vs. Hawkes Density Huang et al. (2022) Xie 2022), and Dong et al. (2023) rely on a cardinality condition of the form |children| > parents) for certain latent sets (ef. Definition 4 in Huang et al.] (2022). Condition in Xie et al.] (2022). Definition 5 in Dong et al. (2023). ). This is gener- ally incompatible with discretized Hawkes processes, whose autoregressive representation inherently dense (Eq. 2 in our paper): if a latent I,1 causes O2, then each discretized variable O is influenced by many lags of L1 (potentially hundreds thousands in practice), making the required |children| > |parents] condition fail systematically. Our method avoids such cardinality assumptions: leveraging the separable excitation (Assump- tion M. we place lagged observed variables on both sides of carefully chosen cross-covariance blocks so that rank deficiency reliably signals latent confounders (lines 199-216; Proposition 4.5 Theorem 4.8).
Time-Aware vs. Li.d. Causal Discovery The above i.d. methods do not exploit temporal order and, in principle, can test variables time putative parents of variables time 1. Our procedure explicitly time-aware: candidate parents for are restricted to appropriate lags (Propositions 443 and 4.5
Theorems 4.7 and [48). aligning identification with Hawkes dynamics. This distinction mirrors PC (Siites ct al. 2001 (L.id.) vs. PCMCI (Runge) 2020) (time scries).
B MULTIVARIATE HAWKES PROCESS DETAILS
Before introducing multivariate Hawkes process, first describe the temporal point process and counting process briefly. temporal point process is a random process whose realization consists a list discrete events time (Ti, T2....} taking values [0, oo). Another equivalent representation the counting process, N1 = [ E [0, x.)). where N records the number of events before time t and (1 = 0. A multivariate point process with / types of events represented by I counting processes V on probability space F2, N; = (N.(t) E [0, oo)), where (t) is the number of type-i events occurring before time & and N₂ = O. U {1.....+} (sometimes abbreviated as [?]) represents the set of event types. Ω [0, oo) x U is the sample space. F F(t) is filtration, that is, non-descreasing family of r-algebras which for each time point IR, represent the set of event sequences the processes realize before time P is probability measure. Point processes can be characterized by the conditional intensity function, which models patterns of interest, such as self-triggering or self-correcting behaviors (Xu et al. 2015). The conditional intensity function is defined as the expected instantaneous rate of type-i events occurring at time 1, given the event history:
BIN. h) (.())
Ax(t) lim
h
where H(t) ilt <ti E U} collects historical events of all types before time t. The multivariate Hawkes process is class of multivariate point processes characterized by a self-triggering pattern as defined
in Eq. I
C IDENTIFYING INTERMEDIATE LATENT SUBPROCESSES
As shown in the summary causal graph in Fig. 6a| To is intermediate latent subprocess on the directed path fromthe observed subpracess O2 to Og. According to Proposition 4.3 J, is not identifiable and its effect
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of
0
o
0
(a)
(b)
Figure 6: Example of an intermediate latent subprocess on the directed path from O₂ to O (a) The summary causal graph, where L1 is the intermediate latent subprocess. (b) The corresponding window causal graph with two effective lags.
attributed O2, leading the inference that O₂ the parent cause of O3. This because the influence of indistinguishable from that of O₂ and can be effectively merged into Oz.
Consider now the corresponding window causal graph shown in Fig. 6b The observed variable set is given by
Og = {0 jE where m = exceeds the number of effective lags (which is in this example). (1,23,)))
Instead conditioning on three lagged variables (0)--1) 0 of"""") O2, we exclude Ofa-1) and condition only (0,-=2) Ofn-3)). }. this case, O becomes separatedd from the remaining variables O This property arises because, due the presence the intermediate latent subprocess L1, O no longer has direct influence on O!"). The following corollary formalizes general method for identifying the number intermediate latent subprocesses that may exist between an observed subprocess and each of its inferred observed parent causes.
Corollary C.I (Identifying Intermediate Latent Subprocesses). Let Og = {0.) denote the observed sub- (inl))EE(mmm...
Considdr observed subprocess Oj and its inferred observed parent-cause set Py C Og. For any O2 E Pg. let h be the largest JyE(m-n.....-- d-separates
processes, with the corresponding observed variable set O = (0 (1,.122....))
value such that the lagged variable set P. = =(O!") (o())nn(nn (n -o.-,,,..-llll... 1 JO,EPo
(n)) from the remaining variables O. \P)(P, ". Equivalently, f is the largest value such that:
rank (o vo( -Prrl-
This is equivalent to stating that the shortest directed path from O₂ to O₁ that does not pass through any other observed subprocess consists of h latent subprocesses.
Remark C.2. In Corollary C.I O and O2 may refer to the same subprocess in cases where Proposition 4.3 infers that O has self-loop. In such Corollary C.1 can be used to determine whether this self-loop represents direct seli-excitation mediated through intermediate latent subprocesses.
Proof. Let Og {0.) and O, {0(0)) (E(i))EE(n-e (E((n- Consider observed subprocess On and its (.....2....))
inferred parent-cause set Pg. For any O2 E Pg. assume the shortest directed path from O2 to O₂ consists of latent subprocesses. This implics that the lagged variables (0(²)) be( do not influence O("), while
the variables {O(1)) 1se mm..m do.
Thus, the variable set P. (0(2))
(al.)/..aa....-.............. ((O JosEPs
(ore-a))))ooooooo__o_o_.... (e( the minimal set that d-separates
(()) (m) from the remaining variables. By Lemma 4.2 this implies:
rank (2 ) PPl/
This completes the proof.
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D RANK FAITHFULNESS FOR THE HAWKES PROCESS
Assumption (Rank Faithfulness the Hawkes Process). probability distribution p is rank faithful to graph G every rank constraint on any sub cross-covariance matrix that holds in is entailed by every linear structural model (as defined in Eq. with respect to & and the excitation function (1)(5) = auy (t),()), Vi,j (I.....4).
The rank faithfulness assumption widely adopted causal discovery literature i..d. data (Spirees 2013 Huang et al. 2022). In our setting. concerns only the excitation function coefficients d;j. and prior studies have shown that violations of this assumption oceur only in degenerate cases of Lebesgue measure zero. Specifically, it fails only in pathological scenarios, such when multiple ay coefficients involving those of latent subprocesses are exactly equal across different subprocesses in manner that induces rank deficiency- situations that are highly unlikely to arise in practical applications.
To empirically assess the robustness of our method to potential violations of rank faithfulness, we con- duct sensitivity analysis where, for each synthetic graph, choose the exponential excitation function randomly selected edges, thereby artifi-
(s) agee -9 and deliberately assign identical 6,, values cially increasing the risk the violation of rank faithfulness. The results, reported Table in Appendix Q.3 demonstrate that our method remains robust even under such perturbations.
E ACCOUNTING FOR SELF-LOOPED OBSERVED SUBPROCESSES UNDER LATENT CONFOUNDER INFLUENCE
O2
0/"1
(b)
Figure 7: Illustration of self-Looped observed subprocesses under latent confounder influence. (a) Summary causal graph where O1, O2, O3, and O4 are observed subprocesses, and Ln latent confounder subprocess. All subprocesses have self-loops. (b) Corresponding window causal graph for (a), illustrating the discretized causal mechanisms among O₁, O2, and L1, with two effective lags.
Consider Fig. where Oz and O2 also have self-loops. As shown in Fig. 7b these self-loops introducc additional indirect effects, where the lagged latent variables {T. (e(nnmm...n propagate their influence the current variables " and O!!)) through the observed lagged variables (0.)) JE(n=m.....-)) (e((1.)
Fortunately, since these lagged variables are observed, they can he explicitly incorporated into the structural equations and, correspondingly, into the covariance matrix. Considering the window graph in Fig. 7b with
effective lagged variables, the structural equations for the observed variables {0(55) (e(()))ee))))))) m...... can be written as:
(ii(1.2)
I (x-1) 1)
(n)
(n)
(
ca
(n-1)
(0))
L (n (n-m)
601
n-11)
(())
E
601
(5)
(n) (00)
nn m-))
60z
(n-1) ((0)
(1 n-11)
6og
(n-m) ((0)
m)-m)
Cog
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Goyh So
w(s)ds
Bayh (m- w(s)ds
mm
1
...
1
0
0
(6)
dogt1
(s))ds
m dogga
(m 1)a (s))ds
0
0
...
...
0
m
0
It is straightforward to see that the rank of the coefficient matrix E is 2nn 1. Aecordingly, by including these observed lagged variables in the cross cova-covvrri matrix, obtain:
rank (oof))EE pelo ("())))""")) tof (of p m.......)) 2m + (C
(c(11.)
where 2m corresponds to the observed lagged variables of O₁ and O2. and corresponds to the latent con- founder subprocess L1. For formal proof. see Proposition 4.5 and Appendix K This result implies the presence of latent confounder subprocess T,1, such that the set [T1 01,02) forms the parent-cause set of O2) Conditioning on this renders {O1, O2] locally independent of O3 and O4.
F PROOF OF PROPOSITION 3.4
Prouf. According to Definition 3.3 set Pg1 is parent-cause set for N; E Ng and only every directed path from any node in Ni ViN.N,N | N, passes through at least one node Pgi. In the special case where N, has self-loop, N₂ itself is also included as parent cause. This implies that cach subprocess N, E Pgi must have a directed influence on N,, formally expressed as J 0t - s) dN, (8) 0, and zero otherwise.
On the other hand, as discussed in the last paragraph of Section 3.1 and in Didelez 2008). N; is said to locally independent
rest of the system given minimal set Poz Ng and only fü (t, s) (N,(s) > 0 for each N₂ E Pgz, and zero otherwise. Therefore,
Pg1 Pgz. N; is locally independent of Ng Pg given Pg if and only if Pg is the parent-cause for N; in Ng.. This completes the proof of Proposition 3.4
G PROOF OF THEOREM
steps. First, define the multivariate INAR sequence (Definition G.I and show that it admits a linear autoregressive model representation (Proposition G.3) Then, in Theorem G.5 we establish that this multivariate INAR counting process converges weakly to a multivariate Hawkes process as the bin size A O, with the correspondence between the parameters of both models made explicit. The details are follows:
Prouf. prove Theorem 4.1
we proceed
Step Definition of the Multivariate INAR model. We begin by introducing multivariate INAR model, adapted from Definition 20 the paper B. Hawkes forests (Kirchner 2017).
Definition G.1 (Multivariate integer-valued autoregressive model (Kirchner 2017). An multivariate integer- valued autoregressive time seriistmultivarritee INAR) is sequence of Va-valeed random variables X, [x(")) x (n) ,...,) x(m), JoEz+ with X (0) 0, defined as:
+c.") iE (e(,,..,.),nEE
where the reproduction coefficients
(A4) > with the subcritical matrix [Σ of Jussee...... and the (o ad Pois(0 are mutually independent and
immigration coefficients 0!!))
0. (n4) rid Poi)( and En
also independent of (n)
Remark G.2. Definition G. follows Definition 20 in paper B. Hawkes Forests in (Kirchner 2017, but with adapted notation match Theorem
4.1 Key correspondences include:
1, J. h.
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(1))
<(11 (Xn)nez + Xx,, X x - indices E Zz to match our Hawkes process formulation (Eq. D this purely notational and does not affect model semantics, as the indices are used to describe relative positions within the time series.
c Giih 03 .oo - -3 We also restrict
Step Linear autoregressive representation of the INAR model. The multivariate INAR sequenee admits an equivalent linear autoregressive representation, as shown in Proposition G33 corresponding to Propo- sition 3.1 in (Kirchner 2016a). The current variable X is expressed as weighted sum of all lagged variables
x - plus a constant term (()) and a stationary white-noise term (1).
Proposition G.3. Ler X. be a I-dimensional INAR sequence as in Definition G.1 with immigration coefficients
(0) 2 0, reproduction coefficients (6) 2 0, and x!!) = 0. Then
(n) = x!-) 0
o X (n- nEZt
(8)
defines a white-noise sequence, i.e., (1( is stationary, Blc(")] fl,..., n Zt. Moreover let the x noise matrices un u = |e(") | l) and repreduction-ccoffiiere matrices A =
[0 (I..)e we have:
diag Eni
n',
B B u
Orxt,
+ n'.
Remark G.4. Proposition G.3 is adapted from Proposition 3.1 of (Kirchner 2016a). which also appears
Proposition of the same paper in the author's 's doctoral thesis (Kirchner 2017). The original formulation uses full vector and matrix notation; here, we present each dimension separately for consistency with our notation. Moreover, we adapted notations in Remark G.2
Step 3: Convergence of the INAR to Hawkes process.
Finally, we show that the multivariate INAR The corresponding parameters of the INAR
process converges multivariate Hawkes process A and the Hawkes process are stated in the below theorem.
Theorem G.5 (Multivariate INAR converging to multivariate Hawkes process (Kirchner 2017)). Let Ng1 (N.) be a stationary multivariate Hawkes process with background intensities { and piecewise- continuous excitation functions {d))(s) Z O, As E (0, (0, For bin width A (0,5), let X. x.,, x(")) Ineat be multivariate INAR sequence with:
(X(*), Xx()) ....,
0 = - A.
0000 = Api.
dus(s)ss
(k-132
and (! 0. From the sequences X,, we define family of point processes Noa (NAHen where for each N
(² Σ (())
EE (0,
(10)
Then, Ng2 converges weakly to Ng1 in distribution, as A + 0.
Remark G.6. Theorem G.5 is simplified version of Theorem 25 Kirchner 2017). The original proof pro- ceeds via convergence of Hawkes forests (constructed via branching random walks), showing that the Hawkes process is a limit of INAR- approximating forests. The convergence of Hawkes process and INAR comes from the convergence of Hawkes forest and the approximating forest with corresponding parameters. We adapt it here with direct correspondence between Hawkes and INAR parameters, and restrict domains to f E oo) and E Z+ for consistency and clarification. Typically, Hawkes process results hold for both domains (Laub) ctt al, 2015) Remark 2), since variable t and n is used only to calibrate relative positions. Moreover, besides the notation changes in Remark G.2 we adopt: NF + No1. Np(a + Noz, the reproduction intensities
hi W + excitation function dijg.
Remark G.7. The constant 5 in the Theorem G.5 comes from the moment structure of the INAR sequence. For details, see Theorem (Kirchner 2016b) and Corollary 24 paper Hawkes forests Kirchner 2017).
In summary: The linear autoregressive representation of the multivariate INAR model is established Proposition G.3 based on the model definition provided in Detinition G.I The convergence of the multi- variate INAR process to the multivariate Hawkes process, along with the correspondence of their parameters, is presented in Theorem G.5 Together, these results validate the discrete-time linear formulation stated in This completes the proof.
Theorem 4.T
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H PROOF OF LEMMA 4.2
Proof. The proof of Lemma
2 is based on Proposition 2.2 and Theorem 2.4 from (Sulivantt al. 2010
which we restate here for completeness.
Proposition H.1 (Rank Characterization Conditional Independence (uulivant al. 2010 Let X (i, Σ) be a multivariate normal random vector, and let A. B. and C be disjoint subsets of indices. Then the conditional independence statement XA L X X holds if and only if the cross-co cross-covariance matrix BALC.BUC has rank [C].
Although this result was originally established for linear acyclic models with independent Gaussian noise, it relies solely on second-order properties (variance and covariance) the data and leverages path analysis rooted the independence of noise terms. Consequently, this result remains valid for linear models with arbitrary noise distributions, since the argument applies to any distribution with finite second moments.
Theorem H.2 (Conditional Independence in Directed Graphical Models (Sullivant et al. 2010)) In directed graph G. set C d-separates A and B if and only if the conditional independence statement XA X Xe holds for every distribution that is Markov with respect to G.
Combining the two results, we obtain following: For any linear acyclic causal model with disjoint variable sets A... B., and Cv,, the set C, d-separates A, and B, in the associated causal graph and only if:
(a.k(EA.c...Buuc = |Cel.
This equivalence confirms that the d-separation criterion the causal graph corresponds to a rank condition on the cross-covariance matrix U.A.UC.BBUC..
The window causal graph in PO-MHP is a directed acyclic graph with linear causal relations and serially uncorrelated white noise. Therefore, the above rank condition applies directly to the window causal graph in the PO-MHP framework. This completes the proof.
I PROOF OF PROPOSITION
Proef. For any subprocess O1, we prove the equivalence of the four statements step by step.
(1) + (2): If Pg is the parent-caut cause of O₂ in the summary graph, construction of the window causal graph. it equivalent to that the corresponding lagged variable set P. contains all direct parent variables of O(m)) This follows from the fact that, in the window graph, directed edges exist from the effective lagged variables of each parent subprocess to O(e), Moreover, by definition of the parent-cause set, Pg is minimal with this property.
(2) + (3): P. contains all parent variables of O(() in the window graph, by the Markov property of DAGs, P. d-separates ((")) from all other observed variahles O. (Po ((ff"))) Reversly, if P. d-separates (O(")))) by the Granger causality-events in the future 0(n) from all other observed variables in O (P.
cannot causally influence events in the past, P. should contain all parent variables of O(") in the window graph. Morcover, by definition of the parent-caut cause set, Pş minimal with this property.
(3) (4): By applying Lemma 4.2 the d-separation between Ofa) and the rest of the variables, conditioned on ... equivalent to the rank constraint:
rank (o oo!o!!")
|Pal-
(4) <> (1): Assume the rank condition holds for P... By Lemma 4.2 this implies that P,, d-separates of") from other observed variables the window graph. Translating back the summary graph, this implies that Pg is the minimal parent-cause set of O₁, as no smaller set can block all paths to O,.
Thus, all statements are equivalent. This completes the proof.
J PRELIMINARIES FOR PROOFS OF PROPOSITION 4.5 AND THEOREMS 4.7 AND 4.8
To establish this result, we rely the concepts of trek separation (t-separation) and d-separation introduced by (Sullivant al. 2010), which provide powerful tools for analyzing latent structures in linear causal models.
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Definition J.1 (Trek (Sullivant al. 2010 ). A trek in the DAG g from variable V, variable V, ordered pair of directed paths P2) where P1 has sink VG, Pz has sink V,, and both P1 and P2 have the same source V₁. The common source V22 is called the top of the trek, denoted top (1PP. 1, P2). Note that one or both of P1 and P2 may consist of single variable, that is, path with no edges. A trek (P1, P2) is simple if the only common variable among P1 and P2 is the common source top(P1. P2). We let TV,, V5) and SV, V3) denote the of all treks and all simple treks from Vi to Vj. respectively.
Definition J.2 (T-separation (Sullivant et al. 2010)). Let A.. B., CA. and CB be four subsets of total variable set V We say the ordered pair (CA. CB) L-separates A, from B. if. for every trek (ri: 72) from variable A to a variable in B., either 7i contains a variable Ca or 72 contains variable in CB.
Theorem J.3 (Trek separation for directed graphical models (Sullivant al. 2010)). The sub-matrix ΣA. has rank less than equal to for all covariance matrices consistent with the graph G if and only if there exist subsety Ca. CB Ve with [Ca] + CC S such that (Ca, CB) I-separates A from B. Consequently,
rank(EA.B) & min + |Cal (Ca, Cs) I-separates A from B}
and equality holds for generic covariance matrices consistent with G.
Corollary J.4 (T-separation and D-separation (Sullivant et al. 2010). set d-separates A and B in 9 if and only if there is partition C = CA U Cg such that (Ca, Cp) I-separates A U C from B U C.
Therefore, when CA and CB are disjoint, the combined set CA CB also serves as d-separator between A and B. Moreover, since the window graph in the Hawkes process is a DAG with linear relations, the above results can be directly applied after suitable adaptation to the Hawkes process setting.
K PROOF OF PROPOSITION 4.5
Proof. We prove both directions of the equivalence.
(+) If such latent confounder L1 exists, the rank condition holds. Suppose there exists latent confounder L1 that is one common parent cause in the parent-cause set of (O,, O2). and that Lı together with {O,, O2} makes them locally independent of other (nonempty set) subprocesses.
Given that L and its paths Oj and O2 satisfy Definition 4. the contribution of I,1 to both O₁ and O2 in the window graph occurs through the number of latent intermediates, resulting in aligned contribution across time lags. this setup, influence of Lı will appear as a shared component across the observed variables f0 (1.2) (j.
Consider the window graph with considered effective lags. Following the logic of trek separation, in the window graph with m effective lagged variables, the minimal choke set CA that t-separates O("), ,0(") from the rest given by:
C. == (1 )e()))etmmmmmm-- U (0,0m) (m) (ol))eE(( pE(n-m....e-1)
(iE(1.2)
is equivalent that C. the minimal that d-separates (O("), " from the O 0.Mo of(").
Thus, by Theorem J.3 the generic rank of the cross-covariance matrix bounded above by |CA 2m + m 3m, where 2m comes from observed lagged variables of (O,, O2) and m comes from latent lagged variables of L1. However, due to the structure of the excitation function (s) = ws)w the latent subprocess L1 contributes effectively as single shared component across all its lagged variables, reducing the effective rank from m
To explain this, we first write the structural equations for the observed variables (0.) (0 as the linear regression those check points
I
ca (00)
(a)
(n-1) (00)
L (n-m)
601
(-11)
((0)
E
601
(II)
60z (00
(n)
m)
(n-1) ((0)
(1 n-1)
6og
(n-m) ((0)
m)
Cog
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Bog Io
Bogt
0
0
0
0
...
0
m
0
0
0
0
(12)
Aoyl To
0
1
...
1
0
0
0
1
0
m
0
0 ...
where a.111 denotes the product of the constant coefficients along the directed path, and Z." represents the sum of all integration combinations. For example, if L1 + L2 - O, and the intermediate latent subprocess Le has no self loop, then a.in = do1224221, and Im Σ (Ju- w(s)ds f/m--1 wss)ds) because the latent directed path acyelic and none self-loops. general, different length latent paths have different Ig_ which one reason that we re- quire latent paths with equal lengths. However, interesting, for exponential kernel auge(_0a) where w(s) e the element Ig = m (fe-1) w(s)ds [(---1) w(s)ds) = [ -mm --(mmm)))
intermediate latent subprocesses involved have
2 2(m--1)a cl-Ba) ds = 2 mm 1)A wis)dss Moreover, in the case of L1 + L2 + O2. IP should be O. This key reason that two paths must equal length for theoretical identifiability. Theoretically, if two paths don't tt have equal length, the two corresponding rows don't have equal number of leading Os so that they can not be linearly dependent and the rank deficiency can not happen.
is straightforward to see that the rank of the coefficient matrix is 2m because the two row corresponding
to o(") and O!r)) in E are linearly dependent (proportional to each other).
Furthermore, the cross-covariance matrix of {O(t) (e())de(nn-e (n-mm.....))) and O,11O(") of i.c., [EE(1.) be written F where E and F coefficient
2(o "(of")eeta PEte-mmm Ou\(of"),of" Heell))) matrix by regressing variables on those choke points. The rank(C rank(CaA.FT) has full column rank, be- cause F calculated from regressing all the rest variables Oz (-f( o on Ca and without blocking lagged variables, no shrinkage of rank occurs. Consequently, the rank of the cross-covariance matrix
rank (oo(!) P:o00)pt O, (ofo (of")) following theorem proofs also adopt similar way).
rank (ECACAF") rank(E) 2m (The
Thus, the total rank becomes:
rank 2m (from observed lags of O₂ and O2) + 1 (from L1) = 2m
(=) If the rank condition holds, there exists latent confounder L1 satisfying the claimed properties. Conversely, assume observed rank condition:
rank
(e(L))) o oooo of 2m +
By construction of the window graph (Eq, [ if there would be at most 2m, corresponding the observed lagged variables of O₂ and Og. The observed rank being strictly 2m + thus implies the presence an additional latent variable influencing both O₁ and O2-
latent confounder between On and O2, the rank
Due to the rank faithfulness assumption (Assumption J. such a rank elevation uniquely corresponds to a latent subprocess L1 acting as a parent cause of both O, and Oz. Furthermore, for the rank increment to be exactly one, the causal paths from L1 to O₃ and O₂ must satisfy the symmetric path situation (Definition 4.4): ie.. the paths only involve intermediate latent subprocesses of the same depth without self-loops, ensuring that the contribution of I,, introduces a single additional rank component shared by both O₂ and O₂ at the same temporal lag level.
Finally, by construction, conditioning on P/- = Li, O2) removes all causal influence from Lr, rendering (O1, O2) locally independent of the remaining observed subprocesses.
This completes the proof.
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L PROOF OF THEOREM 4.7
Proof. We prove both directions of the equivalence.
(+) If such parent-cause set P'g exists, the rank condition holds. set subprocesses such that:
Assume that Pg is the minimal
Pg is a subset of the parent-cause set of Ni.
Conditioning Sg == Pg U (D(NN) (De())D)))))))e) U USSIID((M
renders
N1 locally independent of all other (nonempty set) subprocesses in the system.
All possible observed surrogates of N. in Og have been identified.
For each L, Pg. the relationship between Le and its observed effects (De(NN), De(L.)) satisfies Defini- tion 4.4
this setup, the lagged variables De( and De(L.), as well as lagged and current variables of their observed siblings Sib(De(NV)) and (i) appear in both A and B.. The rank contribution from these observed variables is deterministically: [oal (De(N)) De(L.) 2,
"e pt (OP)) Yo (o,CSM(DLL(M) (SMrP.)L)) merr u (o (................-
1,Cr% The remaining part of A.. i.e. \ consists of the current variables (De(NN,) (e(L)) hueep
Given the symmetric path structure (Definition 4.4). each latent confounder L, E P/J contributes exactly one shared latent component, as the influence propagates through symmetric, acyclic paths. Due to the specific excitation function dey 0uy w(s). this results in precisely one rank contribution per latent subprocess, regardless of the number of lagged variables.
Thus, latent contribution adds exactly:
10 = [Liteerg = (el FLEPP
rank-one components.
Combining both observed and latent contributions, the total rank becomes:
|0.1 0 JO.ll + |O.a| (from (e( 11 |A,|
(-) The rank condition implies the claimed causal structure and local independence. Assume that P/ is the minimal set such that:
rank (CA.. B.) |A.| -
By the theory of trek separation (Theorem 1.3 such rank deficiency implies that information flow between A and B. must pass through a set of choke points, corresponding to the candidate parent causes P'g.
no latent confounders existed, or if P/gs were not part of the parent-cause set of N1, the rank would be exactly OO solely contributed by the lagged variables of observed surrogates and both the current and lagged variables of their siblings.
Since all possible observed surrogates of N; in Oş have been identified, the extra deficiency of rank (i.c., (Oya)) thus directly implies the existence latent subprocesses contributing shared rank-one components. By the rank faithfulness, this observed rank pattern is only consistent with the existence of latent subprocesses (Ix) L,eg that act as confounders between De(Ni) and their respective observed effects, and these latent subprocesses are members of the parent-cause set of Ni.
For the rank deficit latent subprocess to be exactly one, the contribution from each latent subprocess must propagate through symmetric acyclic paths, consistent with Definition 44 ensuring single rank-one compo- nent contribution per latent subprocess. Moreover, the inclusion of the observed surrogates and their siblings ensures that no alternative paths can explain the dependency patterns. Thus, P/ must be the subset of parent causes, satisfying the conditional local independence of Ni given Sg-
Therefore, the rank condition is both necessary and sufficient to identify P as the subset of parent causes of Ni. considering both observed and latent subprocesses. This completes the proof.
M PROOF OF THEOREM 4.8
Proof. We prove both directions of the equivalence.
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(+) If such latent confounder L1 exists, the rank condition holds. Assume there exists latent confounder subprocess such that:
Li is common parent cause of {Ni, Na).
Conditioning on Pg= == L, U N1, N2 U Sib(De(Na))e((u.)) renders (N,, Na) locally independent of the
rest (nonempty set) of the system NG VPG.
All possible observed surrogates of NN) in Og have been identified.
and its observed effects (De(Ni ), De(Na)) satisfy Definition
By the Definition 4.4 the causal influence from L1 {De(NN); De(NN)) is symmetric and only propagates through the same number of intermediate latent subprocesses without self-loops. Under this condition, the contributions of Lı to the observed surrogates (De(Ni), De(Na)) appear a rank-one component across the lagged variables of these subprocesses, aligned in time.
Thus, in the window graph, the latent influence from L1 will introduce exactly one additional rank compo- nent across the observed variable set A, beyond the rank contribution from the observed lagged variables themselves.
Formally, following the arguments for Proposition 4.5 the rank of EA..B, is determined by the minimal set of choke points that t-separate A. from B, in the window graph. Given the assumed structure:
The lagged variables of (De(NN), De(Na)) and both the current and lagged variables of their observed sihlings, denoted as O,1 = (De(NN (N Fee
(O,CSib(P((M)))ss ()SI((((())))))))))))) appear in (n- U (
(e (E(nnm.....
both A., and B., contributing deterministically |O.1] to the rank.
The influence from La propagates symmetrically both De(N)) and De()N) through acyclic paths com-
posed exclusively latent subprocesses, per Definition 4.4 As result, due to the excitation function (s) a.z a.jw(s), ws)). the total rank contribution from L1 is exactly one.
Therefore, the total rank becomes:
rank (Ea. B.) |0.i| + |A
(-) If the rank condition holds, such a latent confounder L, must exist. Now assume the ob- served rank condition:
rank (E,,. 1
Wc know that parent-cause sets of all inferred latent confounder processes in Neg remain unidentified even after applying Theorem 4.7 In the absence of any new latent confounder, the maximum possible rank would be [O corresponding solely the contributions of the lagged variables of (De(Ni), De([N)) and both the current lagged variables of their observed siblings. The observed rank being exactly |O.1 + A. implies the existence of an additional latent source influencing both N1, N2 and their observed surrogates.
Due to the rank faithfulness, this increment must be attributed to a unique latent subprocess L1 that acts as a confounder for N1 and N2. Moreover, the fact that the rank increment only one implies that the paths from L, N1, N2 must satisfy the symmetric and acyclic conditions in Definition 4.4 ensuring that the influence of L1 is captured as a rank-one shared component at the observed surrogates level.
Moreover, the inclusion of the observed surrogates and their siblings ensures that all other possible paths and confounding structures are blocked, enforcing P/ = L1 U (N1, N22 J U (N) local independence and all possible observed surrogates of {N1, N2) } in Og have been identified.
in ensuring
Thus, the rank pattern is both necessary and sufficient imply the existence of I1 and the claimed causal and conditional independence structure. This completes the proof.
N PROOF OF THEOREM 5.1
Proof. We prove the theorem by considering the two cases separately: (i) the system contains no latent subpro- cesses, and (ii) the system contains latent subprocesses.
Case (i): No latent subprocesses. In this case, the system consists solely of observed suhprocesses Og. Since there are no latent confounders, Phase alone is sufficient for identifiabilit. This follows directly from Proposition [.3] which ensures that for each observed subprocess, its parent-cause set can uniquely identified by checking the rank condition of the relevant cross- cross-covariance matrices. Thus. the entire causal graph can be identified solely through Phase I.
Case (ii): Presence of latent subprocesses. In the general case where latent subprocesses exist, the algorithm relies on the synergy between Phase and Phase
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Phase I iteratively identifies the parent-cause for each subprocess (including both observed and previ- ously discovered latent subprocesses) whose parent-cause set fully contained in current set known subprocesses. By Proposition|4.3jan Theorem (4.] this identification is guaranteed when no unknown latent confounders intervene or when latent confounders are already represented by their observed surrogates. Phase II handles the discovery of new latent confounder subprocesses by systematically applying Propo- sition 4.5 and Theorem 4.8 The identifiability is guaranteed under the condition that all latent confounder subprocesses and their associated observed effects satisfy Definition 4.4 This condition uniquely ensures that each latent confounder subprocess contributes a unique, identifiable rank pattern in the cross-cos coos-icovr matrix of its observed surrogates and their siblings, enabling its detection through the rank conditions estab- lished in the theorems. The latent cofounder subprocesses that do not satisfy Definition 4.4 remain unidenti- fied.
Termination and completeness. The algorithm alternates between Phase and Phase II. Since each iter- ation either identifies new parent-caus, set or discovers a new latent subprocess, and given the finite number of subprocesses (including latent ones), the algorithm must eventually terminate.
By construction:
All observed subprocesses will eventually have their parent-caupe sets identified through Phase 1.
All latent subprocesses satisfying Definition 4.4 will be identified through Phase II and incorporated into the active set for further investigation.
The recursive application the identification theorems cnsures that no causal relationships (either between observed, latent, or between observed and latent) will remain unidentified under the conditions.
If Definition 4.4 fails for any latent, the algorithm terminates without fabricating that latent any edges it would entail, thereby returning only the identifiable portion of the causal graph (sound abstention).
Thus, under excitation function dis(s) asyw(s) and rank faithfulness, the entire causal graph consisting of both observed subprocesses and latent confounders can identified. This completes the proof.
O DETAILS OF IDENTIFICATION ALGORITHM
O.1 PHASE
The detailed algorithm for Phase I is in Algorithm
Algorithm 2 Identifying Causal Relations
Input: Partial causal graph G, Active subprocess set Ag, Observed subprocess set Og Output: Partial causal graph G, Active subprocess set Ag
1: repeat
2: Select subprocess N₁ from Ag.
3: for Len = to [Ag U Ogl do
4: repeat
5: Select subset Pdd C Ag U Og such that [P/] Len.
6: if (Ag J Og. N2 satisfies Proposition 4.3 and Theorem 4.7 then
7:
Ag AA Agg..N (N. and update G.
Return to line 9: end if
8:
10: until All subsets of Ag U with size Len selected.
end for
12:
until Ag is not updated or |Agl < 1.
return: G. Ag
O.2 PHASE II
The detailed algorithm for Phase II is in Algorithm
P COMPUTATIONAL COMPLEXITY OF THE ALGORITHM
In this section, we analyze the computational complexity of our two- woo-phase iterative algorithm, which alternates between: (1) inferring causal relationships among discovered subprocesses and (2) identifying new latent sub- processes. Let denote the number of processes in the active process Ag and denote the number of
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exponential Published as a conference paper at ICLR 2026
Algorithm 3
Input: Partial causal graph G, Active subprocess Ag. Observed subprocess set Og
Output: Partial causal graph G. Active subprocess set Ag
1: Initialize cluster set C == 0 and the group size Len = 2.
2: repeat
3: Select subset 3'g from Ag such that [Vg] Len.
4: if (Ag L Og. V/g) satisfies Proposition 4.5 and Theorem 4.8 then
5: Add V'g into C.
6: end if
7: until All subset of As with size Len selected.
8: Merge all the overlapping sets in C.
9: for each merged set C2 E C do
10: Introduce a new latent subprocess Lj.
11: Ag- = Ag U Leg Ci, and update G.
12: end for
13: return: G. Ag
subprocesses in the augmented process set Tg Ag U Og at the start of each phase. Assume each test is an oracle
Phase Inferring Causal Relationships
For each component process N1 E Ag, we evaluate subsets of Tg starting from subsets size up to size of Tg. stopping when the test result is positive. In the worst case, for each N1, we need to evaluate all subsets of Tg. which requires E (*) tests. For one subprocess Nı E Ag. if its parent-cause set is found, Ag is updated. After that, the algorithm will restart go all the subprocesses in Aş make sure parent- worst case. the algorithm find parent-cause set for the last
cause set subprocesses Ag can be found. component process Ag each time. The complexity of Phase upper bounded by: O (n!) Ex= (x)).
Phase I:: Identifying New Latent Subprocesses
In this phase, test all subsets of Ag of size 2. Since there are (*) such subsets, the complexity of Phase II is upper bounded by: O ((2))
Overall Complexity
The total complexity of the algorithm depends on the number of (both observed and latent) subprocesses and the structural density of the causal graph, as these factors determine the number of iterations required for the algorithm run. Combining the two phases, for each iteration, overall complexity approximately upper bounded by: O (n! Ex= (R) + (2))
In practical scenarios, the structural density of the causal graph and sparsity of dependencies may reduce the number of required iterations and tests, leading to improved efficiency compared to this worst-case analysis.
Q MORE DETAILS OF EXPERIMENTS
Q.1 SYNTHETIC DATA GENERATION AND IMPLEMENTATION
We evaluate our method on two types of synthetic data: event sequences generated by the Hawkes process in Eq. [ and discrete-time time data generated directly from the discrete-time model in Eq. (2)
Hawkes Process Data:
We generate event sequences using the tick library (Bacry al. 2017). an effi- cient framework for simulating multivariate Hawkes processes. The excitation function
kernel (s) e -3s where fixed Cuy for edges involving latent subprocesses are sampled uni- formly from [0.90, 0.99], and duy for other edges including observed sclf-loops are sampled uniformly from (0.30, 0.50], because the causal influences attenuate along latent paths. And in Case 1, sampling asy from [0.30, 0.50] avoids the cycle between N₂ and N3 becoming nonstationary. Moreover, to ensure stationarity and avoid explosive hehavior, verify the spectral radius of the integrated excitation matrix after generating aly. To discretize the sequences our method, we select the time hins length 0. and consider 600 effec- tive lag time bins as discretized lagged variables for the calculation sub-covariance matrix. The sample size corresponds to the number of discrete data points.
Discrete-Time Series Data: To assess method under ideal discrete-time conditions (i.c., exactly satisfying Thcorem 4.1 we generate data directly from Eq. @2) The excitation function is exponential kernel
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(s) agg The coefficients au and decay parameter 3 above. Similar the Hawkes data, we verify the spectral radius ensure stationarity. The noise terms are drawn from independent Gaussian distributions. We set the number of effective lagged variables to 200. The sample size corresponds to the number of discrete data points.
Preprocessing and Rank Deficiency Testing: For each trial, we standardize the discretized data to ensure fair comparison. To test for rank deficiency, use canonical correlation analysis (CCA) (w Anderson 1974), following the procedure in (Huang al. (2022), We use the grid search to find the best rank fest threshold. We also conduct empirical sensitivity analysis for test threshold. The result is in Appendix QQ.
Data Usage for Baselines: For Hawkes process-based methods (SHP QQao et al. 2023). THP Ca al. 2022 and NPHC (Achab et al. 2018)) we use the raw Hawkes process data produced by the tick library. For rank- based methods designed for i.i.d. data with linear relations (Hier. Rank (Huang et al. 2022) and RLCD (Dong
et al. ) and time series baseline (LPCMCI (Gerhardus & Runge 2020), we use the discretized Hawkes process data.
We run all the experiments on personal CPU machine.
Q.2 EVALUATION METRICS
We evaluate the accuracy of causal structure recovery using the standard Fl-score, which combines precision and recall.
Causal relationships among both latent and observed subprocesses represented by an adjacency matrix, where cach entry either I 0, indicating the presence or absence of a directed edge, respectively. Specifically, Adia(t, (t. j) = denotes a directed edge from the j-th subprocess to the i-th subprocess, while Adj = indicates no such edge.
We measure the similarity between the estimated and ground-truth adjacency matrices using the Fi-score. First, compute precision, defined
true positives
precision total inferred positives"
which represents the proportion of correctly inferred edges among all predicted edges. Next, we calculate recall, defined as
true positives
recall total ground-truth positives"
which captures the proportion of correctly inferred edges relative to the true causal edges. The F1-score, given by
Fl-score = 2. precision x recall
precision + rocall'
harmonizes precision and recall to provide a balanced measure of structural recovery.
Practical Considerations
practice, the indices latent subprocesses the estimated (summary) graph may not correspond those in the ground truth. To address this, following Huang et al. Huang et al. 2022). we permute and divide the latent subpracess indices in the estimated graph and select the result that minimizes the difference from the true graph. When the number of estimated latent subprocesses is smaller than the true number after division, we add isolated latent nodes balance the comparison. Conversely, if the estimate exceeds the number, we merge and select subset that best matches the true latent subprocesses.
Additionally, since our inferred summary graph simplifies the underlying causal structure, by omitting interme- diate latent subprocesses and redundant edges as formalized in our thcorems and Definition 4.4 we adjust the ground-truth adjacency matrix to this idcalized representation before comparison. This ensures fair evaluation of causal discovery.
[.4
For baselines designed for i.i.d. data with linear relations (i.e., Hier. Rank (Huang et al.) 2022 and RLCD (Dong et al. 2.23)), their output graphs capture relationships among discretized variables, rather than subpro- cesses. To enable fair comparison, we regard an edge N₁ + N2 as correctly identified if more than half of the considered variables associated with N, have inferred edges to those of N₂.
Q.3 ADDITIONAL SYNTHETIC EXPERIMENT RESULTS
Comparisons on Cases 5 and 6 Fig. shows the F1-score comparisons for Cases 5 and 6, which corre- spond to intricate latent confounder structures illustrated Fig and Fig These cases involve interactions
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between latent confounders. The results indicate that method maintains strong performance even under these challenging causal configurations.
--- Ours (Discrete)
Ours (Hawkass
SHP
NPHC
THP
Hier. Rank
RLCD
LPCMCI
1.0
04
02
0.0
30000
S0000
80000
30000
50300
80000
Samplcs
Samples
Figure 8: F1-score comparisons the remaining causal graphs (Cases and 6), involving latent confounder interactions. Case 5 and Case correspond to the causal structures in Figs. 3c and 3d respectively.
Sensitivity to Time Discretization Interval We evaluate the sensitivity of method the choice of the discretization interval A with decay parameter 3 1 in the exponential excitation function dhy (s) asye a.ye-Pr As shown in Table 1 when A is set to 0.01 or 0.05, our method achieves consistently high F1-scores across all cases, confirming that the discretized representation sufficiently preserves the temporal dynamics of the underlying Hawkes process. Even A 0.1. performance remains stable. However, when A increases to 0.3, we observe a sharp drop in performance, highlighting that overly coarse discretization leads to significant loss of temporal resolution, impairing the estimation of causal structures. The result shows the need choose small bin width A relative the typical support of the excitation function (Kirchner 2016a).
Table I: Performance of our method under varying A values using 80k Hawkes process samples generated by the tick library with decay parameter 3 = in the exponential excitation function. Case 1-3 correspond to Figs. [b] 2a and 3a
Performance remains stable and high when respectively. Results are averaged over ten runs. the loss of fine-grained temporal information.
0.1, but degrades significantly at A = 0.3 due to
Precision
Recall
F F1-Score
A
Case
Case 2
Case 3
Case 1
Case 2
Case 3
Case 1
Case 2
Case 3
0,01
0.98
0.91
0.84
0.92
0.93
0.83
0.93
0.92
0.84
0.05
1.00
0.96
0.83
0.84
0.98
0.82
0.90
0.97
0.82
0.10
1.00
0.91
0.86
0.87
0.93
0.83
0.93
0.92
0.84
0.30
0.50
0.55
0.50
0.17
0.63
0.33
0.25
0.59
0.40
Sensitivity to Rank-Test Threshold evaluate sensitivity of our method
threshold used
rank test (i.c., the cutoff deciding rank deficiency). We vary 7 E {0.01, 0.05, 0.10, 0.200 and assess three representative cases. Each experiment uses 30k Hawkes samples generated by the t. i lihrary under exponential excitation function (s) a:je- with 9 and time interval A 0.1; results averaged
runs. As shown in Table in the fully observed setting (Case 1) precision remains 1.00 while recall decreases as т increases, whereas in latent settings (Cases 2-3) a moderately larger threshold improves precision because of the attenuation of causal influences through the latent subprocesses. Overall, a threshold of 0010 provides a good balance across different scenarios.
Robustness to Violations of Rank Faithfulness To test robustness under violations of rank faithful- ness, we randomly select two edges in each synthetic graph and assign them identical coefficients ogy for the exponential excitation function (s) = agge- in every run, with 3 = and time interval A = 0.1. This manipulation introduces potential linear dependencies in the cross-covariance matrix, which could challenge rank-based methods. As presented Table despite the induced degeneracy, our method maintains strong per- formance, especially as the sample size increases. These results suggest that our approach is robust to moderate violations of rank faithfulness in practical scenarios.
Sensitivity to smaller sample size We also examine robustness under reduced data availability by evalu- ating all methods using only 5,000 samples Case (Fig. [b], where the underlying graph is fully observed
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Table 2: Sensitivity to the rank-test threshold 5. Each entry is averaged over ten runs on 30k sam- ples generated with an exponential kernel (B = 1). Case 1-3 correspond to Figs. Tb 2a and 3a
respectively. Overall, threshold of 0.10 provides a good balance across different scenarios.
Precision
Recall
F1-Score
Threshold
Case 1
Case 2
Case 3
Case
Case 2
Case
Case 1
Case 2
Case 3
0.01
1.00
0.42
0.57
0.80
0.53
0.50
0.88
0.47
0.53
0.05
1.00
0.62
0.62
0,64
0.73
0,54
0.77
0.67
0.57
0.10
1.00
0.66
0.72
0.60
0.75
0.65
0.74
0.71
0.68
0.20
1.00
0.76
0.68
0.47
0.85
0.63
0.62
0.80
0.65
Table 3: Performance of our method when, in each run, two edges in each graph are randomly assigned identical coefficients Ogj for the exponential excitation function, increasing the risk of rank deficiency. Hawkes process samples are generated by the tick library. Case 1-3 correspond to Figs. lb 2a and 3a respectively. Results are averaged over ten runs. Despite these perturbations, our method maintains strong performance, demonstrating robustness to such violations.
Precision
Recall
F1-Score
#Samples
Case 1
Case
Case 3
Case
Case 2
Case 3
Case
Case 2
Case 3
30k
0.87
0.60
0.72
0.87
0.75
0.71
0.87
0.67
0.71
50k
0.92
0.83
0.76
0.84
0.82
0.73
0.87
0.82
0.74
80k
0.95
0.84
0.83
0.90
0.83
0.80
0.92
0.83
0.81
and contains no latent confounders. All bascline methods, except Hicr.Rank, which assumes no direct edges among observed subprocesses, were included. The results (mean + standard deviation) in Table 4 show that both variants our method maintain competitive Precision and Fl-scores under this low-sample case. These findings indicate that our approach remains comparatively robust even when only limited observational data are available.
Table 4: Performance of all methods on Case (fully observed, no latent confounders) using only 5k samples. Values are mean scores across ten runs, with standard deviation shown in parentheses.
Ours (Diserete)
Ours (Hawkes)
SHP
NPHC
THP
RLCD
LPCMCI
Precision
0.95 (0.10)
0.97 (0.10)
1.00 (0.00)
0.53 (0.27)
0.68 (0.27)
0.40 (0.36)
0.92 (0.13)
Recall
0.62 (0.13)
0.57 (0.13)
0.93 (3.13)
0.36 (0. 10)
0.56 (0.30)
0.13 (0.10)
0.53 (0.10)
Fl-score
0.74 (0.11)
0.71 (0.12)
0.96 (0.08)
0.42 (0.09)
0.60 (0.28)
0.19 (0.13)
0.67 (0.10)
Sensitivity to violations of the symmetric path condition We further evaluate the robustness of the proposed method when the symmetric acyclic path condition Definition[4.4]p violated. Starting from Case in Fig. 2a we modify the path between L, and O₂ by introducing additional intermediate latent subprocesses: (i) inserting one latent variable (Case A), and (ii) inserting two latent variables (Case B). These modifications make the path from L1 to O₂ longer than the paths to the remaining observed subprocesses, thereby brcaking the symmetry required for rank deficiency. 5 the resulting precision, recall, and Fl-scores (mean standard deviation). While such violations theoretically prevent O from heing identifiahle as child Ta the empirical performance shows only mild degradation. This is partly because exponential kernels preserve proportionality across products of segmented integrals, and small differences in leading zeros induced hy path length differences may be statistically weak in finite samples. These results suggest that the proposed method can remain practically robust under mild deviations from the symmetric path structure.
Evaluation on Larger and More Complex Causal Graph
We further evaluate our method on a larger causal graph with 14 subprocesses, as shown in Fig. 1 Table repports the Fl-scores averaged over ten runs. Despite the increased complexity, our method successfully recovers the underlying causal structure with high accuracy.
We profile runtime three representative synthetic graphs (Cases 1- 3) and two real-world settings. All runs were executed on AMD EPYC 9454 CPU. The first real-world
Scalability and Runtime Profiling
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Table 5: Performance under violations of the symmetric path condition in Definition 4.4 (Case 2 in Fig. 2ah.
Values are mean + standard deviation.
Precision
Recall
FI Fl-score
Case A
0.91 (0.14)
0.93 (0.11) 11) 0.90 (0.12)
0.92 (0.13)
Case B
0.89 15)
0.89 (0.13)
O2
O2
O
Figure 9: Illustration of larger causal graph consisting of 14 subprocesses, used to evaluate scala- bility and robustness.
setting follows our main paper: a five-alarm subgraph device_id = 8 The second merges all devices into single multivariate event sequence with all 18 alarms gauge scaling with graph size. We observe that Case 1 fastest latent confounders are present and Phase suffices. Case 2 introduces latent confounders, requiring both phases the first iteration and increasing runtime. Case 3 slowest among synthetic cases because the latent confounder is itself caused by an observed subprocess, triggering an additional iteration to identify its observed parent. For real data, merging all devices markedly increases runtime the sequence spans all 18 alarms and may deviate from a homogeneous Hawkes mechanism. phase-wise complexity breakdown is provided in Appendix P which offers further insight into the scalability the algorithm.
with one latent Alarm_ich (mmmmmii>))) from
Q.4 ANALYSIS OF REAL- WORLD DATASET RESULTS
evaluate our method on real-world cellular network dataset (Qiao al. 2023), which includes expert- validated ground-truth causal relationships. The dataset comprises 18 distinct alarm types and ~35,000 recorded alarm events collected over eight months from operational telccommunication network. This benchmark has been widely used in prior work (e.g.. the PCIC 2021 causal discovery track and (Qiao et al. 2023),, where performance many methods available and top Fl-scores are reported up to = 0.6.
our evaluation, we focus on subgraph involving five alarm types (A1 aelTarridds-ae and 7) from device jid 8, where Alarm am id-7 is manually excluded and treated as a latent subprocess. Both Alarm id-1 and Alarm id=3 are observed effects of this latent subprocess, providing ability infer latent confounders. The ground-truth causal subgraph is shown in Figure I0 Compared with inferred causal graph, the ground truth contains an additional edge from AL arm_id= to Alarm.id= How- ever, as noted Definition 4.4 causal edges between observed effects of a latent confounder are permissible in our framework.
opportunity to assess our method's
During inference, using Proposition 4 and Theorem 4.7
correctly identify Alarm Alarm_id==0,7 2, 3 as the the parent causes of Alarm_id=0. The parent-
parent causes of Alarm_id=2, and A. Allrmmids=1,> cause sets of Alarm.id=1 and Alar Alarm_id=3 cannot be fully explained by the observed subprocesses alone. This necessitates the existence of latent confounder influencing both, leading to the successful identification of Alarm_id=7 as a latent subprocess.
Table 6: Performance of method on the larger causal graph in Fig. 9 using Hawkes process data generated by the tick library. Results are averaged over ten runs. The method consistently recovers the causal structure with improving accuracy sample size increases.
Sample Size
Precision
Recall
F1-score
30k
0.65
0.52
0.58
50k
0.71
0.58
0.64
80k
0.80
0.71
0.75
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Table 7: Runtime across synthetic and real-world settings.
Graph Type
Runtime (s)
Case 1
227.80
Case 2
1036.01
Case 3
2603.95
Real Dataset aallammids=-0 --- 3, device id = 8)
1364.71
Real Dataset (all devices merged; 18 alarms)
20914.29
Alammus 1t
Aammjis
Alamjii
Figure 10: Ground-truth causal subgraph from the metropolitan cellular network dataset. 7 is treated as latent subprocess.
Alarm id-7
Baselines and Schemes. We also quantitatively compare method with representative Hawkes-based approaches (SHP (Qiao al. (203), THP (Cai al. 2022). NPHC (Achab et al. (0188). two rank-based latent-variable methods for i.i.d. data (Hier. Rank (Huang et al. 2022), RLCD (Dong al. (2023) ), and a time-series method designed for exogenous latents (LPCMCT (Gerhardus & Runge 2020). ). For fairness, all baselines evaluated on the same sub-dataset (A lA rm (ds-0==3 and from device id=8) with performance averaged ten using parameter grid search.
method,
Results on the Sub-dataset. Our method achieves the best F1-score when the data conforms single multivariate Hawkes process (per-device setting). reports the average F1-scores.
Table 8: F1-scores on the cellular network sub-dataset (Alarm_ids=0) and 7, device _id 8) where Ala rm_id77 manually excluded and treated as a latent subprocess; averages over 10 runs.
Algorithm
F1-score
SHP
0.49
THP
0.48
NPHC
0.42
Hier. Rank
0.00
RLCD
0.39
LPCMCI
0.43
Ours
0.76
Merged-devices analysis. For completeness, we also merge events from all 55 devices into a single mul- tivariate sequence with all 18 alarm types and analyze with our method. This setting violates the assumption that samples share the same generative mechanism (devices can be heterogeneous), and yields much lower Fl-score (0.17). This illustrates why per-device analysis is more compatible with assumptions, whereas merged-device data can confound structure learning.
The dataset records 34,838 alarm events from metropolitan cellular network Qiao 2023) covering 18 alarm types and devices. Each record contains: ct al.
Dataset description.
Alarm ID: one of 18 alarm types,
Device ID: one of 55 devices.
Start Timestamp: time when the alarm triggered,
End Timestamp: time when the alarm resolved.
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For causal analysis, events alarm type and the start timestamp the event time, yielding temporally ordered sequence suitable for inference.
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Reviews:
[{"review_id": "m6W9hcJpsE", "reviewer": "Reviewer_ZQpB", "summary": "This paper studies the problem of causal discovery in multivariate Hawkes processes (MHPs) with latent confounders. The idea is to represent a MHP with a specific form of excitation functions as a linear autoregressive model over discretised variables. Afterwards, the authors introduce a set of conditions under which the causal structure is identifiable using rank tests on covariance matrices of observed discretised variables.", "questions": "Please see above comments.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper addresses an important and relevant problem, i.e., causal discovery in multivariate Hawkes processes (MHPs) with hidden confounders. \nThe theoretical contributions provide valuable insights into causal discovery without assuming causal sufficiency in MHPs and represent an important step toward advancing research in this area.", "weaknesses": "The main result builds on representing an MHP as a linear autoregressive model through discretization. However, according to Theorem 4.1, this result holds only when the discretization parameter (\\Delta) tends to zero. In practice, for small but finite (\\Delta), this leads to model mismatch, which can also be observed in the sensitivity analysis with respect to (\\Delta) in Table 1.\n\nMoreover, all identifiability results are derived under the assumption that the linear representation holds, i.e., as \\Delta -> 0. However, no guidance is provided on how to choose (\\Delta) in practice to ensure consistent results.\n\nThe identifiability results further rely on an additional assumption that the excitation functions take the form (a_{i,j}w(s)), for example, the exponential decay function a_{i,j}\\exp(-\\beta s). While this is a common assumption in the MHP literature, it is often extended to cases where the decay rate \\beta is also an unknown, node-specific parameter, i.e., (a_{i,j}\\exp(-\\beta_i s)). Although this may appear to be a minor modification, it is non-trivial to see how the results of this work extend to such more general excitation functions.\n\nThe proposed algorithm has exponential complexity, which limits its scalability. As discussed above, its performance is also sensitive to the choice of \\Delta. Furthermore, the method relies on rank tests, which typically require large amounts of observational data. This raises a question regarding Figure 4: assuming the experimental setting is favorable to all baseline methods as well as the proposed approach (i.e., with no latent confounders), how would these methods perform with substantially fewer observations, e.g., significantly less than 30,000?", "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": "", "extra_scores": {}}, {"review_id": "FiYfQ3YaV0", "reviewer": "Reviewer_hpp2", "summary": "The paper studies structure learning (causal discovery) for partially observed multivariate Hawkes processes (PO-MHP) and provides the first principled framework that identifies latent sub-processes and recovers causal structure in continuous-time event sequences without prior knowledge.\n\nThe authors make a key theoretical contribution (Theorem 4.1) by showing that a continuous-time multivariate Hawkes process can be represented by a discrete-time linear causal model when the event-count data is appropriately binned. They further prove that the low-rank constraints on the cross-covariance matrices induced by the linear representation can be used to (1) detect the presence of latent confounder subprocesses, and (2) identify parent–cause sets and causal edges under explicit path-based conditions (Definitions 4.4, Propositions 4.3/4.5, Theorems 4.7/4.8).\n\nBased on this theoretical foundation, the paper proposes a novel two-phase iterative algorithm where Phase I identifies causal relationships among the currently known (observed and inferred) subprocesses and Phase I discovers new latent confounders via rank tests. The authors also prove that this method guarantees the identifiability of the causal graph. Experiments on both synthetic and real-world datasets show that the proposed method effectively recovers the ground-truth causal graphs, outperforming existing baselines, especially in settings with complex latent structures.", "questions": "1. How sensitive is the performance of your algorithm to the choice of the discretization interval Δ? Is there a principled way to select an optimal Δ, or is it purely an empirical choice? How does data sparsity affect this choice?\n\n2. Some additional experiments could be added. For example, what if some latent confounders are removed (which violates the condition in Definition 4.4)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. One of the paper's main strengths is its strong theoretical foundation. Theorem 4.1 is a powerful result, which innovatively establishes a connection between continuous-time Hawkes processes and a discrete-time linear autoregressive representation. Moreover, the *Definition 4.4 + Proposition 4.5 + Theorems 4.7/4.8* that links symmetric path structures to observable rank deficiencies is also original and enables finding latent confounders without prior information of the existence or number of latent subprocesses.\n\n2. The paper addresses a critical challenge, where many previous causal discovery algorithms assume that all relevant variables are observed. This paper instead studies a more realistic and difficult scenario under partial observability, where they propose a novel framework to uncover causal structure with unknown latent subprocesses. It is scientifically important.\n\n3. The proposed two-phase iterative algorithm is a direct and elegant consequence of the theoretical results. The experiments, while concise, are well-designed to validate the paper's core claims. Specifically, the synthetic experiments include multiple graph families, sample sizes, and sensitivity checks.", "weaknesses": "1. Strong structural assumptions. Definition 4.4 formalizes the Symmetric Acyclic Path Situation (the observed effects being connected to the latent via paths of equal length and acyclic intermediate latents), which is a somewhat special topology. However, in complex systems intermediate latents can have varying path lengths or additional cross-links, which would break the condition and make that latent unidentifiable by the method.\n\n2. While the paper is theoretically rigorous, it is also extremely dense. The written could be polished with a motivating real world example in Figure 1, then expand the theoretical proof based on this step-by-step real world example. Moreover, some intuitions could be explained before show the theorems and proofs. For example, the transition from 4.2.1 to 4.2.2 could give some intuitions.\n\n3. Only one real world dataset is limited. This small dataset (evaluation on a five-alarm subgraph) can not support the model effectiveness on large and noise real-world systems. Meanwhile, the reported results on Tables 1-4 do not show variance.", "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": "", "extra_scores": {}}, {"review_id": "O1cuoBY1tg", "reviewer": "Reviewer_gH4E", "summary": "This paper addresses the problem of causal structure discovery in multivariate Hawkes processes under partial observability. Authors consider Hawkes processes with elaborate latent structure, and derive the conditions under which the latent variables and the relationship between them is identifiable. Similar results exist in the context of linear autoregressive processes and as acknowledged by the authors, they inspire the latent structure discovery results in this manuscript (Theorem 4.7 and 4.8). By transforming the Hawkes process inference problem into a discrete-time linear autoregressive formulation (Theorem 4.1), the authors establish the results for Hawkes processes. \n\nThe paper proposes a two-phase iterative algorithm that alternates between (i) discovering causal relations among existing subprocesses and (ii) inferring new latent subprocesses based on rank constraints of cross-covariance matrices. Necessary and sufficient conditions are derived for identifiability, including the introduction of path-based conditions (Definition 3.4) ensuring one-to-one correspondence between latent confounder structures and observable rank deficiencies. Empirical results on synthetic and real-world data show that the proposed method successfully recovers causal structures even when latent confounders exist.", "questions": "Please address the weaknesses mentioned above.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "Compared to the previous NeurIPS 2025 submission that I had reviewed, the manuscript has substantially improved in clarity of the claims and structure of the paper. Most importantly, the iterative algorithm is now explicitly defined, assumptions and identifiability conditions are more carefully motivated, and the connection to prior work—including an additional LPCMCI baseline in experiments, rank-based latent structure discovery methods, and INAR processes—has been expanded.", "weaknesses": "1. Novelty of Theorem 4.1 can still be debated. While the authors’ expanded discussion distinguishes their formulation from prior binning-based estimation approaches, the contribution could still benefit from more explicit formal comparison (e.g., showing in what sense their linear representation differs from INAR-based or EM-based formulations beyond the absence of likelihood modeling).\n\n2. Motivation benefits from more discussion. In much of the classical literature on the broader causal discovery problem, the structure within them is not discussed, as the latent confounder are often treated as root nodes affecting the observables. The work is indeed interesting in a theoretical sense, yet, I'd like to question the motivation for latent structure discover: Since these variables are not observed, it is hard to imagine interventions on them, so why is it of interest to practitioners to identify the structure of the latent variables?\n\n3. Assumptions and their implications. The identifiability results depend on assumptions about the Hawkes process and the structure of latent confounder. In which practical scenarios are these assumptions justifiable? On the other hand, in misspecified cases, how poorly is the latent structure discovered? A short illustrative example, even synthetic, where the assumption are valid and one where they fail would greatly benefit the reader.\n\n4. Missing acknowledgement/comparison with recent work on causal discovery in Hawkes processes via compression schemes, e.g., [1,2]\n\n[1] Hlaváčková-Schindler, K., Melnykova, A., & Tubikanec, I. (2024). “Granger causal inference in multivariate Hawkes processes by minimum message length.” JMLR 25(133): 1–26\n\n[2] Jalaldoust, A., Hlaváčková-Schindler, K., & Plant, C. (2022). “Causal Discovery in Hawkes Processes by Minimum Description Length.” AAAI 36(6): 6978–87.", "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": "", "extra_scores": {}}, {"review_id": "H476E4k1ML", "reviewer": "Reviewer_VH6W", "summary": "The paper proposes a causal discovery method for identifying latent subprocesses and their causal structures in partially observed multivariate Hawkes processes. The authors introduce a causal model for partially observed multivariate Hawkes processes to represent continuous-time event sequences. Based on this model, they leverage rank constraints on the covariance matrix to identify causal influences without prior knowledge of the existence or number of latent subprocesses.", "questions": "What parameters were used for the method proposed in the real-world data experiments?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The authors discretize the Hawkes process, transforming the multivariate Hawkes process causal model into a linear autoregressive model, and theoretically prove this conclusion.\n2. It proposed a method for identifying latent subprocesses and causal structures solely by leveraging rank constraints on second-order statistics.", "weaknesses": "1. In Proposition 4.5, should it be that the rank constraint is both necessary and sufficient for the corresponding local independence in the graph, under the structure defined in Definition 4.4 and the data generated accordingly?\n2. The results mentioned in the original SHP paper differ significantly from those presented in this paper. Additionally, could results for different time intervals be provided, similar to the approach in the SHP paper?\n3. When the paper transforms the model into a linear autoregressive model, does it require the noise to be constrained as Gaussian?\n4. The proposed two-phase iterative algorithm suffers from severe scalability limitations, which restrict its practical applicability.\n\n\nTypo:\n\nThere is an extra } at line 300.", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Oral)