Is Score Matching Suitable for Estimating Point Processes?
Authors: Haoqun Cao
Zizhuo Meng
Tianjun Ke
Feng Zhou
Venue: NeurIPS 2024 poster (2024)
Keywords: point processes
score matching
parameter estimation
Primary area: probabilistic_methods
TL;DR: This study highlights the incompleteness of previously proposed score matching estimators for point processes. In addressing this issue, we introduce a novel score matching estimator for point processes.

Abstract: Score matching estimators for point processes have gained widespread attention in recent years because they do not require the calculation of intensity integrals, thereby effectively addressing the computational challenges in maximum likelihood estimation (MLE). Some existing works have proposed score matching estimators for point processes. However, this work demonstrates that the incompleteness of the estimators proposed in those works renders them applicable only to specific problems, and they fail for more general point processes. To address this issue, this work introduces the weighted score matching estimator to point processes. Theoretically, we prove the consistency of the estimator we propose. Experimental results indicate that our estimator accurately estimates model parameters on synthetic data and yields results consistent with MLE on real data. In contrast, existing score matching estimators fail to perform effectively. Codes are publicly available at \url{https://github.com/KenCao2007/WSM_TPP}.

Is Score Matching Suitable for Estimating Point Processes?
Haoqun Cao', Zizhuo Meng", Tianjun Ke', Feng Zhou 1,3+
CCnter for Applied Statistics and School of Statistics, Renmin University of China Data Science Institute, University of Technology Sydney BBejjing Advanced Innovation Center for Future Blockchain and Privacy Computing hcao65@visc. feng. zhou@ruc. .du.eeduu.c. cn
Abstract
Score matching estimators have gained widespread attention in recent years partly because they are free from calculating the integral of normalizing constant, thereby addressing the computational challenges in maximum likelihood estimation (MLE). Some existing works have proposed score matching estimators for point processes. However, this work demonstrates that the incompleteness of the estimators pro- posed in those works renders them applicable only to specific problems, and they fail for more general point processes. To address this issue, this work introduces the weighted score matching estimator to point processes. Theoretically, we prove the consistency of our estimator and establish its rate of convergence. Experimen- tal results indicate that our estimator accurately estimates model parameters on synthetic data and yields results consistent with MLE on real data. contrast, existing score matching estimators fail to perform effectively. Codes are publicly available https: //ttps::/gi. /httbbb..tttt..... com//enCao22007/XMMTTT
1 Introduction
Point processes are a class of statistical models used to characterize event occurrences. Typical models include Poisson processes and Hawkes processes [3]. Their applications span various fields such as seismology [6 177. finance [4, criminology (15), and neuroscience 26). In the field of point processes, maximum likelihood estimation (MLE) has been a conventional estimator. However, MLE has an inherent limitation: it requires the computation of the normalizing constant, which corresponds to the intensity integral term the likelihood. Except for simple cases, calculating the intensity integral analytically is generally infeasible. This necessitates the use of numerical integration methods like Monte Carlo or quadrature for approximating the computation. This introduces approximation errors, and more importantly, for high-dimensional problems, numerical integration encounters the curse of dimensionality, rendering training infcasible.
To address this issue, prior research has introduced the concept of score matching (SM) [5] to the field of point processes. For instance, [18] derived the application of SM the estimation of traditional statistical Poisson processes. Furthermore, 24 extended the use of SM to the estimation of deep covariate spatio-temporal point processes. [10 also generalized the application of SM to Hawkes processes. These works have greatly advanced the utilization of SM for point processes. However, in practical applications, we have found that these estimators only work for specific point processes. For more general cases, these estimators cannot accurately estimate model parameters, even for some simple statistical point processes. One of the core contribution of this work is to theoretically demonstrate the incompleteness of the estimators proposed in the aforementioned studies.
"Coresponding author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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The incompleteness of the estimators in the aforementioned studies stems from the transition from explicit SM to implicit SM. The explicit SM estimates model parameters by minimizing the expected distance between the gradient of the log-density of the model and the gradient of the log deg-ity of the data. However, we cannot directly minimize the above objective function since it depends on the unknown data distribution. To facilitate solving, we need to convert the above explicit SM to implicit SM by using a trick of integration by parts, provided that some regularity conditions are satisfied [5]. In [8; 24 [10), they assume that the required regularity conditions are satisfied in their point process models and directly employ the implicit SM objective. However, as demonstrated in Section B the required regularity conditions cannot be met for general point processes. This implies that the concise implicit SM objectives (Equation (2) in 18] Equation (10) in [24). Equation (4) in [T0] are incomplete, and they cannot accurately estimate parameters for general point processes.
To address this issue, this work introduces a (autoregressive) weighted score matching (WSM) estimator that can be applied to more general point processes. WSM eliminates the intractable terms in SM objective by adding a weight function that takes zero at the boundary of the integration region. Compared to previous work on WSM 6 22 [1],
we are the first work to apply WSM stochastic process where the dimension Nr is also random. This stochasticity in dimensionality poses greater challenges to the derivation, requiring special treatment to address this issue.
Specifically, we make following contributions: (1) We theoretically demonstrate that implicit (autore- gressive) SM estimators in [18 24 [0] are incomplete because the required regularity conditions cannot satisfied for general point processes. (2) To address this issue, we propose (autoregres- sive) WSM estimator that is applicable to general point processes. Theoretically, we establish its consistency and convergence rate. (3) experiments, we confirm that on synthetic data, (autoregres- sive) WSM successfully recovers the ground-truth parameters; on real data, (autoregressive) WSM estimates results consistent with MLE; while existing (autoregressive) SM estimator fails in both scenarios.
Preliminaries
Now we provide knowledge on Poisson and Hawkes processes and (autoregressive) score matching.
2.1 Poisson Process and Hawkes Process
The Poisson process [9 is a stochastic point process that models the occurrence of events over a time window [0, J.. A trajectory from Poisson process can be represented an ordered sequence T = (f1..... (2y) where N; = max(n: tn < tt, E [0, } is the corresponding counting process and thus Ny. is the random number of events in [0, T]. The inhomogeneous Poisson process has time- varying intensity A(t) representing the instantaneous rate of event occurrence at f. Mathematically, the intensity function is defined X(t)= lims, -0 E[N++6, 8.)/8 The probability density function of Poisson process is:
NT p(T) II
x(tx)xxp exp (-(" (ctettt
A(t.)
The Hawkes process 3 is self-excitation point process where the occurrence of an event increases the likelihood of more events in the future. A trajectory from Hawkes process is similarly represented as T = (4,,.... ( on (0, T). The conditional intensity function of Hawkes process, representing the instantaneous rate of event occurrence at t given the history up to but not including t, is:
(²( = A((FFF) = m(l) + Σ g(t 12),
where uut) is the baseline intensity, g(-) is the triggering kernel representing the self-excitation effect, the summation expresses the accumulative excitation from all past events, F,- is the historical information up to but not including X (²(1) means the intensity is dependent the history.
Poisson process assumes the independence of event occurrences, while Hawkes process extends by introducing autoregressive structure, making subsequent events dependent prior events. Given

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the history Fi. = (t1,.... .t.....,,t................ the conditional probability density of (n 1)-th event at >> t, is:
p( exp (-fc (((((t)rr)))
Here, we introduce the definition univariate Hawkes process. However, multivariate Hawkes processes also exist. For ease of notation, we use the univariate case for illustration, but we also provide solutions for the multivariate case.
Score Matching
MLE is classic estimator that minimizes Kullback-Leibler divergence between model distribution and data distribution. However, drawback is the intractable computation of the normalizing constant. Approximating it through numerical integration can be computationally demanding. In contrast, SM [S] offers an alternative by minimizing Fisher divergence between model and data distributions:
(0) Ep(x) Vx log Vx log lo)))(
where p(x) represents data distribution, po (x) the parameterized model distribution, gradient of the log-density is called the score, and || represents suitable norm, such the l norm. Minimizing the Fisher divergence above provides the parameter estimate, The advantage of SM lies in its ability bypass the computation of the normalizing constant since the score longer contains this constant: (x() z(0) (x) where Z(0)= = (x(xxxx Vx log Po (x) = Vx log (x)).
Under certain conditions, we can use integration by parts to replace the explicit S.M objective, which involves unknown distribution p(x), with equivalent implicit one,
JsM (0) px()
2
V log px + Tr (V2 loggo(x))
2.3 Autoregressive Score Matching
An autoregressive model defines probability density p(x) as product of conditionals using the chain rule: p(x) II -==1 ( xx)),, where F, is the n-th entry and x<n denotes the entries with indices smaller than n. The original SM not suitable for autoregressive models because the autoregressive structure introduces challenges in gradient computation Equation (4p. To address this issue, [I3] proposed autoregressive score matching (ASM). Unlike SM, which minimizes the Fisher divergence between the joint distributions of the model po (x) and the data p(x). ASM minimizes the Fisher divergence between the conditionals of the model po (Xn|Xen)
) and the data p( (x)):
1
Bpixen)) (
a @Olog (lggp(an Ölog (o (x) dx,
LASM (0)
dzn
Similarly, the above explicit ASM objective involves unknown distribution p(a, xx)). Under specific regularity conditions, we can apply integration by parts to derive an implicit ASM objective:
a log (o(an|/en)
8 log po (x (Xen)
JASM (0)
Σ Ep(xen)
(5)
2
Ox,,
Or
Score Matching for Poisson Process
We analyze the application of SM for Poisson process and its failure in achieving consistent estimation. Subsequently, we propose provably consistent WSM estimator.
3.1 Failure of Score Matching for Poisson Process
Consider a Poisson process (N1,...,AN:) 4N,) on (0, T). Let p(T) represent the data distribution, which is uniquely associated with an intensity function A(t). Let (T) represent the parameterized

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model distribution, which is uniquely associated with parameterized intensity function (t)... In the following, we denote the score as (t,,,, = öt,
Ø log p(T).
Previous works 18: have both attempted to apply SM to the Poisson process:
NT
1 (T)
(w(tx) (oo(t))))
(S
E
(6)
2
In order for SM to be practical,
18 24
assumed that specific regularity conditions are satisfied.
Therefore, they employed an implicit SM objective similar to Equation
NY Σ m==1
(tt(t))
(0) = EP(T)
va (t.) 2
+
Öt
In practical applications, we have found that the above estimator works only for specific Poisson processes and fails for more general Poisson processes. The reason for its failure lies in the fact that, for more general Poisson processes,
specific regularity conditions cannot be satisfied.
Such conditions require the probability density function of the random variable is zero when it approaches infinity any of its dimensions. However, for point processes, such requirement is not satisfied, because the random variable in point process T = (f2...., (Nr is not of fixed dimension and takes values in subset of IR, Therefore, for general Poisson processes, we cannot derive the implicit SM in Equation (7) based on the explicit SM in Equation (6)
Proposition 3.1. Assume that all functions and expectations in (f) and (8) are well defined, we have,
/ p(t p(tt.....,tw ..)
(N)
(0) =f) + const
a(ti....,,t
d'2..
01
4==0
N=1
① Olog poll......... (N)
Σ
p(t ..l....tt.. (N)
dTi:N-1-
Nl
IN=T
N=1
Therefore, (0) is equivalent (0) not containing 0.
( if and only if the sum of the last two terms is constant
For specific Poisson processes, the sum of the last two terms can be zero. However, for more general cases, this sum contains 0. This implies that Jsm fails for general Poisson processes.
3.2 Weighted Score Matching
To address the situation where SM fails, inspired by 6 22], we introduce the WSM for Poisson process. The core idea of WSM is to eliminate the intractable terms by adding weight function that takes zero at the boundary of the integration region. The weight function is designed to vector-valued function h R--r- + RNrr with the n-th element denoted as h (T). Here, we present the conditions that valid weight function should satisfy:
lim P(T) (t,) 0, lim +0
p(T)) 0, Vn E [Nr],
tn -1
El (ta)) < co, Ei (tn)
on [Nr]. Otn
One can verify that such weight functions are easy to find for most p(T) and (tx). With a valid weight function h, the explicit WSM objective be defined
No.
LWSM (0) = - (v)(t) - (o.)))))
(10)
The introduction of the weight function allows control over the values of the integrand at the boundaries of the integration domain, thereby eliminating the last two terms in Equation 8

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Theorem 3.2. Assume the true intensity is in the family of the model intensity. denoted as X(t) Ag- (1), where 0r E.. We further assume that log Ao, (t) unique minimizer of Lwss (0) is 0*
8 Blog Aoo (t)
a.s. gives 0₃ 02. Then the
dt
The explicit WSM objective is not practical as it depends on the unknown data distribution p(T), so we further derive the implicit WSM objective which tractable.
Theorem 3.3. Assume that all functions and expectations in Lwsm(0) and Twsm(() are well defined, Equation satisfied, we have,
LwsM (0) = [wsM(() + const,
N7 Σ 2
@rpo(tn) hnn(T) vt((n))
(T) Ötn
JwSM(0) = Ep(T))
(II)
Ötn
For general Poisson processes, Equation 11] is always valid with a suitable weight function. Thus, do not need to worry about the issues of failure that may arise when using Equation 77p
4 Autoregressive Score Matching for Hawkes Processes
Similarly, we analyze the usage of ASM for Hawkes processes and its failure achieving consistent estimation. Subsequently, we propose provably consistent autoregressive WSM (AWSM) estimator:
4.1 Failure of Autoregressive Score Matching for Hawkes Process
The original SM, even when adjusted by weight function, is not suitable for point processes with autoregressive structures, such as Hawkes process. Because in such cases, directly calculating the score still includes the intensity integral, which is precisely what the use of SM aims avoid. Therefore, an ASM method is proposed for parameter estimation for Hawkes process in [10.
Consider Hawkes process T = (f1..... (Nr) on [0. T with the underlying conditional probability density in denoted as p(,n|Fr, .). The parameterized conditional probability density model of f is po (tz ((,--))- We denote the conditional score as ()(t, (Ft-,) Dc. log
a log A(tn F(n-1)- F(tnlFen-1... at
L....NN. An explicit ASM objective defined as:
NT (tt
LASM (0) = 2 PTT
(o
(12)
assumed that specific regularity conditions are satisfied. Therefore, an implicit ASM is proposed accordingly:
Similarly, make ASM practical, [I0
NT
(tn (F....) )) + Oxbo (In Ftnm.))
JASM (0) = Eptr
Σ
(13)
Oln
However, the same issuc in the Poisson process arises here. The regularity conditions required to climinate the unknown data distribution do not hold. Therefore, we cannot derive the implicit ASM in Equation (13) based on the explicit ASM in Equation (12).
Proposition 4.1. Assume that all functions and expectations in LASM(() (0) and JAsu(() are well defined, we have,
(0) [A) + const + Σ PCT.n-i)((tlFFe De 1,
dT
(14)
/ ( )ob Fen-e) ba=ta-1 dT
Σ n=1
Therefore, (f) is equivalent to (o) if and only if the sum of last two terms a constant not containing 8.
Generally speaking, for most Hawkes processes, the sum of the last two terms in Equation 14 still contains even for a common Hawkes process with an exponential decay triggering kernel. We illustrate this example in Section 6.2 This implies that JASM fails for general Hawkes processes.
5

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4.2 Autoregressive Weighted Score Matching
Similarly, address the situation where ASM fails, we introduce the AWSM for Hawkes process. We present the conditions that valid weight function h should satisfy:
lim p(Tim) (F, )n 0, lim in Mon
(Lonnlr 1) (T) = 0, Vn E [NT],
b--T
T oo, oo, An E [Nr]-
< co, (l (i.,)
(15)
Öt,
With a valid weight function h, the explicit AWSM objective can be defined as:
.i
LAWSM ()) PTT)
(v - tho (tn (ta 1) (T)
(16)
Theorem 4.2. Assume the true conditional density is in the family of the model conditional density, denoted as (Ft.1)) pa- (fnlFF...)) where 0" E O. We further assume that po, (t.))t...1)
pe, (tn (tn-,) a.e. gives 01 = 02. Then the unique minimizer of LAwsM(0) is 0",
The explicit AWSM objective is not practical as it depends on the unknown data distribution Fftn|/rr, ) so further derive the implicit AWSM objective which is tractable.
Theorem 4.3. Assume that all functions and expectations in Lawsu (0) and Jawsu( are well defined,
Equation 15 are satisfied, we have,
CAWSM (0) JAWSM const,
Ovaal (tn Fin-s) (T) wollt F(am)) Öt,
Th
(f) = EP(T)
Σ 2
vo (t. Fenm, (T) +
Øt
(17)
For general Hawkes processes, Equation (17) is always valid with a suitable weight function. Thus, we do not need to worry about the issues of Tailure that may arise when using Equation (33.
Multivariate Hawkes Processes For the multivariate case, events are {(ti, ki)., Nt KN, }
with k, E I...., K denoting the event type of the n-th event. The history up to the (n 1) -h event is denoted by Ft.-.. We need to consider both the distributions of event times and event types. For the temporal distribution, we use the AWSM objective with the temporal score (btt..)F...1) = 8. Dt. 1on.((n, [Fr, ,) as before. For the type distribution, since we do not need to compute the intensity integral, we directly use the cross-entropy objective:
(f) = (T)(T
log po (kin Ft kn))
E
log At. (In loo A(ta Fr.
0)
(18)
where x= = Ek
k== do. The final loss is J(0) -Twss acc((f); a is balancing coefficient.
5 Theoretical Analysis
In this section, we analyze the statistical properties of AWSM estimator of univariate Hawkes process. Similar conclusions also hold for the WSM estimator of Poisson process, as discussed in
(m) J from p(T) of Hawkes process. We assume the true density is in the family of the model density, denoted p(T) po- (T). where
Appendix C.5 We consider M i.i.d. sequences {r(Tm) ...., N ==
@" θ RT. The estimate 0 is obtained by Ô arg minggge JAWSM(0) where JAWSM represents the empirical loss. Below we omit the subscript AWSM as it does not cause any ambiguity.
5.1 Asymptotic Property
We first establish the consistency of i for a Hawkes process. Theorem 5.1. Under mild regularity Assumptions C.1
C.3
have θ 2 0" as M oc.

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5.2 Non-asymptotic Error Bound
Then, we establish a non-asympiotic enror bound for ô. We define
(tpo(tnlFtnn nn (T) tho (rr, .)
(Tn(T)
[n( Ep(T)
Σ
(m 1) +
Din
Oln
Br(T,0)
An(T,0)
Assumption 5.2. Assume there exists such that,
in. Tn Fo 2 C₂
0: [0-01) 26
holds for any small Here, Cn is a positive constant that depends on the weight function h such that Cah = aCn for any positive constant ||I is the euclidean norm.
Assumption 5.3. For Vn E N+, there exists (T), B,, such that,
[A,(T,0) 01) (T 02)1 S Án
02l1, [T, 01) (T, (T) 0211.
Theorem 5.4. Given that 0 converges to 0* in probability, combined with Assumptions 5.2 and 5.3 for 6 < C.a 20-1
A.) we have
Cn
1/(a-1) 1)
r(h, A,
B)
Pr
110 01
CKa
2 11
(19)
80n
M
2a constant, K,, 11 and r is the number of dimensions of 0. (T) Stn
where 1(hh, A, B) = BECT -V {ERE hnn + (B,(T)
}]}
is a universal
5.3 Discussion on Optimal Weight Function
In Sections 3 and we only provide the conditions that the weight function needs satisfy. In fact, there are many weight functions that satisfy these conditions. The optimal weight function should minimize the error bound in Equation (9)), which equivalent minimizing the coefficient
A(h.A.B) The numerator cannot be analytically computed as involves an unknown distribution Ch
p(T), but we can maximize the denominator Cn in a predefined function family.
Theorem 5.5. Define h to be weight function with its n-th element defined as the distance between
in and the boundary of its support [fn-1, T):
(t.) T t--1
In -T' + 1n-1)
2
We have,
ho E arg max inf (n) IIn
hEH 0:110-0-1126
where is a family fanctions that rigorously defined in Equation 27
Combined with Assumption 5.2 can be observed that ho maximizes Cn in H. Though does not necessarily optimize [(,, it is an adequate choice without using any information on P(T). We also discuss it heuristically in Appendix C.4 It is worth noting that R. is not continuously differentiable; however, it is weakly differentiable. Its weak derivative is continuous, allowing both integration by parts and statistical theory to hold. In subsequent experiments, consistently employ this optimal weight function when is available or can be approximated for the dataset.
C₁.
Experiments
In this section, we validate our proposed (A)WSM on parametric or deep point process models. For parametric models, we focus on verifying whether (AJWSM can accurately recover the ground- truth parameters. For deep point process models, we confirm that our new training method is also applicable to deep neural network models.
Experiments performed using NVIDLA A16 GPU, 15GB memory.
7

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Table 1: The MAE of three models trained by MLE, (A)SM, and (A)WSM the synthetic dataset. For the 2-variate processes, we only present the estimation results for some parameters here. The results for other parameters can be found in Table
POISSON
HAWKES
GAUSSIAN HAWKES
ESTIMATOR
0u2
Ai
(AJWSM
0.07g0.14
0.041g0oo
2.026
0.011 00.010
0.153 rea
0.022
0.060g0:008
(A)SM
1.56g0.01
1.60020.00
4.200
0.700 2.
1.413
2.507=1.067
MLE
0.02
0.008
0.014ga.002
0.012
0.098-to.107
0.017
0.051=0:0019
6.1 Baselines and Metries
We consider three baseline parameter estimators: (1) MLE (2) implicit (A)SM 18
24
10
Denoising Score Matching (DSM) [00. We briefly introduce DSM in deep point process models.
For deep Hawkes process training, DSM is employed as follows. For observed timestamps (m) in
m-th sequence, we sample noise samples (()) and get the DSM objective:
((m) + n,
(m)
......4.. L, where Var(e = o²
M
N
(m)
2L ((0) M 1=1
(mm)
n.d
+ aTE()))
o²
where [O) is the cross-entropp loss defined in Equation
18
To compare the performance of different methods, for parametric models synthetic data, we use the mean absolute error (MAE, 10 0)) between the ground-truth parameters and the estimates metric since the ground-truth parameters are known. For deep point process models, we use the test log-likelihood (TLL) and the event type prediction accuracy (ACC) the test data metrics.
6.2 Parametric Models
Datasets We validate the effectiveness of (A)WSM using three sets of synthetic data. (1) Poisson Process:
This dataset is simulated from inhomogeneous Poisson process with intensity
function A(t) = exp(@ sin(t)) with T = 2, f = 2. (2) Exponential Hawkes Processes: This dataset is simulated from 2-variate Hawkes processes with exponential decay triggering kernels
112 = 1, an = 1.6, 012 = 0.2, 02) = 0r22 = 1. (3) Gaussian Hawkes Processes: This dataset is simulated from 2-variate Hawkes processes with
Piy (T) = Oi exp(-5r) 7 > O with T = 10,
Gaussian decay triggering kernels (T)(T) = 0. exp(- - 2 ), T with T 10, # = uig = 1,
V2n0
0n = 1.6, 012 = 0.2, 021 = 022 = 1,0 1
Training Protocol We assume that we know the ground-truth model but do not know its parameters. Therefore, we use the ground-truth model the training model. The purpose is to verify whether the estimator can recover the ground-truth parameters. For each dataset, we collect a total of 1000 sequences. We 500 iterations of gradient descent using Adam [8] the optimizer for all scenarios. For MLE, the intensity integral is computed through numerical integration, with the number of integration nodes set to 100 to achieve considerable level of accuracy. We change the random seed 3 times to compute the mean and standard deviation of MAE.
Results In Table I we report the MAE of parameter estimates for three models trained by MLE, (A)SM, and (A)WSM on the synthetic dataset. We can see that both MLE and (A)WSM achieve small MAE on three types of data. However, the MAE of (A)SM is large. As we have theoretically demonstrated earlier, this is because MLE and (A WSM estimators are consistent. contrast, (A)SM, due to the absence of the required regularity conditions in the three cases, has an incomplete objective and cannot accurately estimate parameters. Figure [
we showcase the learned intensity functions. Both MLE and (A)WSM successfully captured the ground truth, while (A)SM fails.

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Iith A2e
Internilyy
Interiity
R0M
2000
S40.B0 Hest
10
25
50
Time
103
125
USA
20
Cale
(a) Poisson
(b) Exp-Hawkes Dim-1
(c) Gau-Haawes Dim-1
(d) TLL and Runtime
Figure The learned intensity functions from MLE, (A)SM, and (A)WSM (a) Poisson, (b) Exp-Hawkes and (c) Gaussian-Hawkes, We present the results for the 1-st dimension. The 2-nd dimension are in Appendix [ The ground truth, MLE, and (A)WSM nearly overlap, while (A)SM differs. (d) The TLL and runtime of (A)WSM and MLE w.r.t. the number of integration nodes.
6.3 Deep Point Processes Models
Datasets We consider four real datasets. (1) Half-Sin Hawkes Process: This is a synthetic 2-variate
Hawkes process with trigerring kernel 9uy any sinssn(T) 7 E (), K = 2. (2) StackOverflow [7]: This dataset has two years user awards on StackOverflow. Each user received sequence of badges and there are K 22 kinds of badges. (3) Retweet [25 This dataset includes sequences indicating
how each novel tweets are forwarded by other users. Retweeter categories serve as event types K 3.
(4) Taobao 21: This dataset comprises user activities Taobao (in total K 17 event types). For each dataset, we follow the default training/dev/testingg split in the repository.
Training Protocol In recent years, many deep point process models have been proposed. Here, we focus two of the most popular attention-based Hawkes process models: SAHP (23 and THP p7 We deploy AWSM and ASM on THP and SAHP. For each dataset, we train seeds with the same epochs and report the mean and standard deviation of the best TLL and ACC. When using MLE, adopt numerical integration to calculate the intensity integral. To ensure model accuracy. the number of integration nodes is set to be large enough as sample 10 nodes between every two adjacent events. When using DSM, we tune the variance of noise for better results. When using AWSM, since for real datasets, the true observation endpoint 7' is unknown. We choose the maximum of each batch as the observation endpoint for weight function ho. This may lead to unsatisfying results since real datasets may not be sampled during unified time window. We provide remedy for this as discussed in Appendix D.I Details of training and testing hyperparameters are provided in Appendix D.2
time
Results In Table 2 we report the performance of SAHP and THP trained using three different methods, namely MLE, AWSM, and DSM, on four datasets. evident from the results that models trained with MLE and AWSM exhibit very similar performance in terms of both TLL and ACC on the test data. This indicates consistency between MLE and AWSM, as they yield comparable model parameters. For DSM, it is significantly inferior the performance of MLE and AWSM. This may result from the fact that the DSM objective is a biased estimation of the original SM objective and fails produce consistent estimation when fails in the scenarios mentioned above. It is unable to estimate the correct parameters, and its results reported. Generally, for complex point process models such as deep Hawkes processes, the necessary regularity conditions are not satisfied, meaning that ASM's objective is incomplete.
discussed in 20 For ASM, it completely
6.4 Advantage of (A)WSM over MIEE
The key advantage of (AJWSM over MLE is its avoidance of computing intensity integrals, which can be computationally intensive for complex point process models and impact MLE accuracy. We evaluate the test log-likelihne likeliiodood of MLE and AWSM on the Exp-Hawkes dataset as the number of integration nodes varies. As shown Figure Ld. with a limited number of nodes, MLE is faster but exhibits substantial estimation errors. Increasing the number of nodes reduces the error but significantly increases computation time. In this scenario, AWSM much faster than MLE with the same accuracy, thus offering better computational efficiency.

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Table 2: The TLL and ACC of two attention- based deep Hawkes process models trained by MLE and AWSM on four datasets. Because ASM estimator completely fails, we do not report its results.
SAHP (TLLT)
THP (TLL")
DATASET
MLE
AWSM
DSM
MLE
AWSM
DSM
HALF-SIN
1.805
1.161 p0i01
1.271 10.0s
-0.385
STACKOVERFLOW
2.428
20.
2.62920.008
22.368
2. 308
2. 2.78220.044
TABAO
-0.050+0.100
-1.373amm
-1.91120.0
-1000200012
-0.94840.004
-1.791 20.000
RETWBETS
0.454+0.000
0.1 0.110
0.421+
0.419 0.419-0,00
-0.1 18.0..9
SAHP (ACC))
THP (ACCt)
DATASET
AWSM
DSM
HAI
a.302
0.505-a.00
0.501 poom
g.508
0..23
0.503
STACRDVE
0.46 20,001
0.46220.01
0.421 40042
0.461 2.461
0.46220.001
0.445
TABAO
gn.022
0. 10om
0. 200
0.594gn.om
0..02_..00m
RETWEETS
0.451+0.000
O. 010
0. 0.000
0.501+0.001
0.592=0.002
0.556
0.3
Optimal Natural
0.3
Optimal Natural
0.15
Optimal
Natural
MAB
0.2
Squareroot
MAE
Squarerant
MAE
Squararcot
0.1
200
8000
Sample
Sample
Sample Size
(a) On parameter 011
(b) On parameter 0:1
(c) On parameter f41
Figure 2: MAE of parameter estimation versus sample size for three different weight functions on Exponential- Expooeeniil-HHawkss Model. Our near-optimal weight function outperforms the rest two valid weight functions in all sample sizes. We only show results for three parameters. The rest parameters have almost the same paradigm.
6.5 Comparison Between Weights
Though we provide theoretical insight into the choice of an optimal weight function for AWSM, its validity still needs to be testified by experiments. Here, we compare the near-optimal weight ht with natural weight h and squareroot weight h² satisfying Equation 15).
(t.)) = (tn - (TT t..). (t.) = / (tn - t, -T
All three weight functions can be applied in AWSM to recover ground-trute truth parameters, however with different convergence rates. We carry out experiments on synthetic data for exponential- model with the same setting as Section (6.2 in our paper. We measure their MAE for different sample sizes in Figure 2 and find that ho does achieve the best results among the three weight functions.
7 Limitations
The current limitation of the methodology is that some real data are collected from multiple time intervals [0, ..,
( Jo. Tı] collated in fixed time interval [o, T] with unknown T. However, for a score matching to be valid, the required weight function must involve knowledge T. Currently, our remedy including approximate 7 performing data truncation as discussed in Appendix D.I
8 Conclusions
In conclusion, the SM estimator for point processes can overcome the challenges associated with intensity integrals MLE. While existing works have proposed SM estimators for point processes, our investigation reveals that they prove effective only for specific problems and fall short in more general cases. To address this issue, our work introduces novel approach: the (A) WWS estimator for point processes, offering both theoretical soundness and empirical success.
10

---

Acknowledgments and Disclosure of Funding
This work was supported by NSFC Project (No. 62106121), the MOE Project Key Research Institute of Humanities and Social Sciences 02JJD1100011 the fundamental research funds for the central universities, and the research funds of Renmin University of China (24XNKJ13).
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12

---

A Proof of Results in Section
A.1 Proof of Proposition 3.1
Proof. First consider the cross term in (f) and expand the expectation,
NT
aog ((T a log po (T)
Ep(T)
ot,
Üt
① log pll,,...,,,a IN) d pol.,..l........................ dli:N
p(t
tN)
Ötn
Otn
N=1
==
Op(li,... ...., IN) d log po IN)
dTi.N
(20)
N=1
N
3 log po
(N)
/| fn= dT
(plti......
(N)
Ihn
(n=ln-1
n=1
N
@2 log po (o....1.....11
IN)
p(t
(N)
dTi:N
Ot2
The third equation uses an integral-by-part trick. All integrations above are taken within the area of {0 S & < ... < tv s T) {0 S S. ... fn-1 S tn+1 S ... S iN S T3 when t, has been integrated. For the first term in the right side of the last equation, notice that,
Olog po (l.,........ LN)
a
= θ 0tn-1
log (t(t
log Ao (o(tn))
log (o
N
Dt,
ot
(21)
Therefore, we can see that, for n E n2.....n].
/ [b( /b(lt....,.N) .N)
log po (1 (t.,...,(N)
dT
Hln
11,
f Ib(ta....,tN Olog po it........ tN)
dT_n
Otn
bn=ta-1
Using the above equation, we manage to cancel most of the terms being summed in the right side of the thrid equation in Equation 200 and only leave the first and last term, which completes the proof.
A.2 Proof of Theorem 3.2
Proof.
First, since LWMM (0) and LwsM(0")) 0, 0h is minimizer. there exists another minimizer 0,, then we have
à 12 hn hT(T)..
(ws
log Ao (tn) Ol
log No, (tn)
Hl,
#=1
By the definition of hTT),, for any N, since h(T) > 0 a.s. elementwisely on (0 S t S... St. Ø log Ao (t)
T}. So (0) = implies Bngg (t)
a.e. on [0, T). By assumption, this implies at
dt
0, 0*, which completes the proof.
D
13

---

A.3 Proof of Theorem 3.3
Proof. The proof is basically the same as the proof of Proposition 3.1 and consider the cross-term in LwsM(f).
We first expand the expectation
Nr
Plg olog PlogPo(T)
N
Tp(
alogpo(T)
(T(T)
hn (T)
hn (d)d
ttn
Otn
Otn
Dtn
N
Olog (T)
p(l
IN)
(7)
dT_n.
(22)
Oln
- | plti.....,tr (N) Σ
N
8 log po (T)
① Olog (T) (T)
(n((7) +
dTEN
01
Öln
Oin
We denote 1N+1 T and to = 0 here. Using the first two equations in Equation
we have:
f /In(te.....tN "olti......tt (N) 1 log (o IN) hn (T)I =t dT-n 0, [N].
Utn
f /Iits......x
2 log (o (N) hm (T)) (T))l, dT_n = 0, vn E [N].
(N)
Öt,
Therefore, the first intractable summation term in Equation
(22 will disappear, and the second term
equals - BP(T) E a (o((n)) (7) + (ha(kn) St. hn, (7)
The existence of such an expectation
Dt, due to the last equations in Equation (9). Therefore, we complete the proof.
We can see from the proof that, in Equation (9). the first two equations ensure that the integration by parts trick does not produce an intractable term, and the last two equations are simply regularity conditions that ensure all terms are well-defined.
B Proof of Results in Section
Lemma B.1. Let f(ns Fra-.) be a function of Fr,... where [I....,Nrr) Then we have
NT Σ
EATT
M(nn
where p(l,...., bn) is the density of observing these timestamps, the integration is taken over {0 & t1 I .... I tn T))
Proof. We first expand the expectation and obtain,
Nr
2 = Σ p(l),..,,Ix) Σ
BPpTT)
Σ
S(n,
N=1
N
bo ( I, Fln INTT
N=1
/ p(ti.....
- )dT )aT
At this point, we need to first integrate out tn+1, tv. which uses
I p(ttuti...., tN Fr. )d = P(N, = NIF....
14

---

Plug this in and we obtain,
20
E
Bp(T)
Σ
Fin
Σ
Σ
- (ta
N=1
20
N
N Ftm
==
Σ
Fin-.) Σ
P(Nr (dTin
/
plti.....,t)) )d
The thrid equation adopts the exchange of summation. The feasibility is ensured by the assumption that the expectation the left side of the equation exists and Fubini's 's theorem. The fourth equation use the fact that NN= pNNr F,, = 1.
B.1 Proof of Proposition 4.1
Proof. We use Lemma B.1 to the cross term of (0) and obtain,
Nr
Ep(T)
Σ
(bn F 1o0
(4 Fe .)
00
p(li,.... (n)) 1 (1 (F JdTia
2 blog blog(tt p(11)F(1))
vi Foo
0t,
Olog ((nn 1)
+
Fen
(tn F(..))
Oln
Op(ti|Fio)
St p(l1,...,,n-
Op(tin (Ftn-1) ddn
v/ (11 )d
Σ
8hi
Oln
(wo(t)|FF1)) dli Ø11
11=
p((1(Ft0)
=T tn (en F(a-1) ( dTin-1
1n=
+
Iroo (lo Fr, 1) cdT
p(ti....,th) .n)
Otn
b== dT Σ
/
@ialFalFFen dT
Σ
p(T p(Tm-1)0((n1 Wo (on
p(l1.... (n)
=t a=l
Øl
NT. Orpo (Kn (Fam
=T= ((Tm- Frnm))
Σ
dT
EpeT)
Oln
For the fifth equation, we simply rearrange the terms. We recall that the notation ((T.) equals one. For the last equation, use Lemma B.I again. This will be sufficient to complete the proof. D
15

---

B.2 Proof of Theorem 4.2
Proof. First, since LAwsM( 2 and (A O, we see H is a minimizer. there exists another minimizer 01, then we have
N Tho (In|Ftn-1)) th, (4 dT =0
LAWSM (01) = 2
N=
12
By the definition of h, (T). we have h,, (T) 0. a.s. on S..,, I2 S T for any n S N and N N4. This implies, for any
N and N E N+, 1h0- (t. Vo, (t., Ft a.c.
{0 < 1--1 S bn < T). Therefore we have
log (tn F(n.)) log pe, (tn|Fin-1) + C,a.e.
And we conclude that C = since S Ft 1dd 1. Then by assumption, have Pg- (tn)Fi..,) = Po, ((n,)) a.e. = 81 = 0"
B.3 Proof of Theorem
The proof is basically the same as the proof of Proposition 4.1 We first use the Lemma B. to the cross term of (O),
(t Vo. ()
B
Σ
p(li.... (n)) (el
l==T
Σ
p( Pa (n
dTn--1
/ p(1
@ve(lnlFF p()))...,ttt)) Øl,
Th (T)
(n)
(T) + vo
dTi..
Oln
Between the second and the third line above, we omit the steps used in the derivation Proposition 4.1 make it concise. For the term in the third line above, it will be eliminated using Equation (5). For the term in the fourth line above, using Lemma B.1, we have:
Erbe (rio(ta(F...)) (T) + (o Oh (T) Otn
Σ
p(t. p(ti....... .ic.....tt... tt))
dTh.
Ötn
Nr Orio Fen-s) Ötn
hn (T) + (on Ihn
(T)
Öt,
The existence of the expectation is ensured by the last two terms in Equation 15
C Proof of Results in Section
We present all the regularity condition needed for establishing the consistency of our estimator: Assumption C.1. The parameter space is compact set in Rd and contains an open set which contains 8".
Assumption C.2. Both the (t) and (t) are twice continuously differentiable w.r.t. f. for all E θ and those derivatives are continuous w.r.t. 0.
We remind readers that ( and (t) are the mean intensity function and the triggering kernel for a Hawkes process, first defined in Equation (2p.
Assumption C.3. If both μa; (t) 44a (t) a.s. and ge, (t) gos (t) a.s. on [0, T]. then 0 02.
16

---

C.1 Proof of Theorem 5.1
Proof. As shown in Theorem 5.7 in [9], if we can prove uniform in probability convergence for (0) to (f), then using the fact that 0* is unique minimizer of (w) in a compact set in Rd, the consistency is proved. Therefore, only prove the uniform in probability convergence.
For any 0 E θ for each sampled sequence (11Th) ...... ( we perceive the following value as random variable,
a
Sm
(o(tt ( F F
(T(()( (mm)
+
vo (o(t
)ha ) (T(m)) pim)
n=1
+ Ve
((m)) (ta
(ve)
Oln fm (m()))
One can verify that it is a measurable map from the sample space to the real line, therefore indeed random variable. And its expectation JawSM (0), which is finite by assumption. Since different sequences are i.i.d. samples with finite expectation, using the weak law of large numbers, we have:
M
JAWSM
Σ
Em 2 JAWSM (f), 4O E o.
(23)
M
m=1
Now prove that this convergence is uniform in O. Similar to (2). first prove that
4A > 0, 38 0, 8.1.V1 - 021l < 6, P.P.AAsssss (02) > ]e) + M, 00. co. (24)
First, by Assumption C.2
we know \t(tn/FFnn and den At are continuous w.r.t log Ao (t (x-1) (e(tr](F,,..) and - uo Ft.
a Therefore (o ( St.
d2 0u log (t Den Ao Fr.,,) are both continuous w.r.t. 0, therefore the JAWSM (0) is
continuous function of O. Since θ is a compact set in Rd. we know (A) is uniform continuous.
Therefore, we can bound JAwsM (01) JAwSM (02) using 121-021l- Using the result in Equation 23 we know that for any & > O, we can find a uniform 8 so that (Awss((1)) JAWSM(B2) (62) < ge. V 1 021l
So that,
P( (((JAwSM((L) - JAWSM (f2) > (F)
P((JAwsM(0H) (O1) (fAWSM(A2) JAW) 1 ()
1 = - M,
P((JAwsM(())) (O))
Now we follow exactly the same steps as for the uniform in probability convergence. Since Equation (24) hold, for such in that equation, since θ is a compact set in Rd, there exists finite number of open balls with radius 8 whose union covers @. Let UI..... 02, .,...., 02 denote the centers of these balls. We denote 0:(0) the center of ball which contains 0. Since we have
[s(sup (NWSM(0) - [ANSM(0)] e) S P(sup (wNSM(() - (AWSM(( 3
8
+ P(sup (A)) JAWSM((
E
+ P(sup (JAwSM(() - JAwsM(( > 9)
3
3
The third term on the right equals 0 because of its definition and the uniform continuous of JAWSM (0). The first term converges to 0, M + co by Equation (24). For the second term, we write
(s(sup [OA (U 3)
P(IJANsM(O.) - JAwsM(0.)) ) - 0.

---

Finally we obtain supaee (Awss(()) JAwsM(0)) & O, which completes the proof.
C.2
Proof of Theorem 5.4
First we restate the technical assumptions needed for the proof.
Assumption
5.2
Assume there exists > 1 such that,
in In(0) In(0') 2 Cr,
10-0*1 28
holds for any small 8. Here, Cb is a positive constant that depends on the weight function such that C.h aCn for any positive constant
For this assumption, we assume that the optimal parameter l is well-separated from other neigh- bouring parameters in terms of the population objective values. This is standard assumption for Theorem 5.52 in [9]. This assumption will satisfy VaJr (0) is positive definite at 8".
Assumption 5.3
For Vn E N+, there exists A. B. (T) such that,
[A,(T, 01) (T 02) & (T) 0211. BB, (T. 01) (T, 02)1 S (T) 02l1-
For this assumption, we assume the objective function has certain level of continuity. This assumption will realize if supace [[ 0)]l exisits any n and the same condition also applies to (T, 0).
We also define,
Nr Σ
Oh (T)
JT(T;0)=
(A)( (T, 0)
Or,
we see [[]
Now begin the proof of Theorem 5.4
Proof. First define
Ny (T Σ
(n(T))a (B) (T) Th (T)
Ol
n=1
Next we evaluate the bracketing number of such function class,
FB (J)(T:0) (T: 00) 0, 0* S 8).
a
Denote (e, F, (P)) as the bracketing number of F with the radius e under the norm of L2
P is the underlying probability measure induced by T. Use Example 19.6 in 19 we have,
46 Ng(ell/ile200, s (1+
Since
(T; J=(;; (T) 0r < (T)5,
(25)
therefore (T)T)) 8 is the envelope of Fs. we denote it as Fs. Therefore we obtain,
Ny(ee)Fallec1p Fo, = (Pp)), Fo, & (1 y1+
After obtaining this quantity, the next step is upper bound the empirical process defined as,
M (n 0) (Tm;0')
(Jn())))) (n((00))
- (T:(')))]
VM
m=1
18

---

Using Corollary 19.35 in [19], since we have [|All1a(P)) A,,, < oc, then,
E
sup Gn(f) feF,
-IlF|(La(()) Lo Vlog
( (P))
(26)
s Collmillecee ( Vlos(1 4 )
s Civ A, B),
where C' and C" are universal constant and Jn( :- is the entropy integral.
Finally, given that Ô converges 0 in probability, combined with Assumption using Theorem 5.52 in 19 we have, for ō < C.aaa 20-1
5.2 and Equation 26
A.).A.B) we have,
Cn
1/(a-1)
T(h, A,
0 8 (CK,
2
5Cn
72
where C' is a universal constant and K, ---1-I'
2
C.3 Proof of Theorem 5.5
First we give rigorous definition of H.
H = (h(T) h: R R. nn (T) hn (luy.... kn l,, 11, = (T,( =0
hn t...,, - (th t1...................... . tn 1m-1)1 $ |t! the Vn, t 2
(27)
In other words, H. contains functions whose component function h2 is 1-Lipschitz continuous w.r.t. its last dimension tn It's easy to verfity h E H.
Proof. First we prove that,
maz (0) (0) heW
(28)
Notice that,
Nr
(0)
2 PP((T
Σ n=
(tr 10 (ta Fen 1) (T)
Notice that, in H. for any = = - n or z = T,
(T) hn (T) 1h, (fn,..., .2) 1h, (t.................... t1) hn. t2)l S [t 2/ -1)
Therefore,
Nr Σ ==1
(n) S E
- - 1 = (h), ((0)),))) vh E 7
2
and equation 28 is proved.
Finally, we have,
info:ja--01]200ho - Ino
=infe:j0-0-11264h0 (0) - Lho (0*)
-infe_ja--0113350PAEEl Cb (8) - La (n(
-infe:ja--011260PPhhea (0) - [n')
ssuppewiifa.jpe [T (0) [nn()))
19

---

The second equality is due to Equation (28 and (6)))) inequality.
0. The inequality is due to max-min
C.4 Continued Discussion on Optimal Weight Function
Given ht maximizes Ch, it is still unclear whether it is still preferable considering r(h, A, B). This indeed a tough question that we do not yet have a satisfying answer. For specific parametric models, (T): B,,(7) can be computed analytically (see line 478-480) and then [(h, A, B) can be computed via Monte Carlo. Then we can study how sensitive [(h, A, B) is to h. For general models, especially when the is deep neural network like THP or SAHP, r' A, B) is intractable to compute.
However, heuristically speaking, our choice of near-optimal weight function ho should be good choice even concerning [h,, A, B). To make r(h, A, B) small, natural idea is to make (T) and ar,, hn, (T) small. The weight function we chose and its derivative have relatively positive low powers with respect to tns therefore making (T)) and 0 h. (T) small. For weight functions St.,
(T) (T (tp -n),), the power w.r.t. t, is two. And for h2 (T) V(T - (t,, -n-1). its derivative is 8 Sτ. 12, (T) T--,-(t- (tm=-1-1) the numerator is usually bounded quantity and ((T-tx)(t--a-))
the denominator may be close to zero, making its derivative large. conclusion, h is better choice compared with h or h² concerning r(h, A, B).
C.5 Discussion on Poisson Process
For Poisson process, results Section also holds, including the consistency and convergence rate. For the choice of weight function, reasonable choice is still the distance function presented below.
Consider a Poisson process, since the support of T is V = (T e N" St S StNr ST).
Define A E +\N++ to be a coefficient matrix used below as,
0 1 ".
0
0
-1
A
-1
Denote a E N: as the n-th row vector of A. Let bn = 0. E [Nrl. bN = -T. Then V is convex Polytope which be rewritten as,
V (T R (an, T) + bn 00 1,2,...,, Ny + 1}.
Therefore, the distance between T and @V is
(an 2) bbll
dist(T, min min(llT z|| max nE[N++1]
(an, z) + bn] = =0] ne[N++1] ||a|||
min
(29)
Let 11. (T) (T 0V), (7) = (T)... (T)) Again, h² is only weak
derivative. The envelope theorem [4] yields,
7h
an
aa.
wehre n² is the is the minimizer of the last minimization in Equation
29
Similar arguments as in Section 5.3 also applies to Poisson process, with the distance weight function defined above. Such weight function also maximizes the denominator for the convergence rate of the Poisson process. However, in the experiment, we adtops another weight function for easier implementation defined as h (7) with its n-th component h: (T) (T (t,,, tn-1).
20

---

Table 3: The MAE of 2-variate Hawkes processes trained by MLE, (A)SM, and (A)WSM on the synthetic dataset.
EXP-HAWKES
GAUSSIAN- HAWKES
ESTIMATOR
021
022
0:22
0gi
(A)WSM
0.0522n.064
0.02210.005
go.012
0.03720.005
0.08220.ee ceu
0.012gomm
0.060g0.00r
(A)SM
0.000
20.00
20.200
2.120
0.971 20,040
0.717
2.507
MLE
0.065gn.022
0.031go.015
0.014ga.002
0.025 20.002
0.015
0.006
0.051=00004
Table 4: The hyparameters for experiments in Table
DATASET
EPOCHS
OAWSMM
TRUNC
ODSM
7DSM
SAHP
THP
SAHP
THP
SAHP
THP
SAHP
THP
SAHP
THP
HANF-SSN
20
10
0.01
STACKOVERFLOW
100
500
20
20
T
10
10
0.1
TAOBAO
500
300
50
20
T
20
20
0.01
5R-3
RETWEETS
100
100
20
10
F
2000
10
0.01
5B-4
D
Additional Experimental Details
In this section, we present some experimental details. First we discuss our modification to the original AWSM for when T for dataset is not accessible. Then we provide addtional estimation results for the parametric model and hyperparameters for results in Table 2
D.1 Weight for Equal Length Sequences
In this paper, we consider the setting of the temporal point process in fixed time interval [0, T] consistently. Technically speaking, this poses extra difficulty since the dimension(number of events) would also be random. Therefore, the realized point processes should have a random length this setting, which the case for our dataset. However, real data collection, different sequences may be sampled from different lengths of time interval lengths, and our proposed weight may fail under such case, resulting inconsistent estimation. such a case, we propose to truncate the sequences to be of same length, and introduce the weight function for fixed-length temporal point processes.
We focus AWSM for Hawkes process. A fixed N-length Hawkes process could be considered N dimensional random variable with density function as,
p(h (N) x / dN
(t)dt.
Since the number of observations is fixed, we could simply treat it as an N dimensional random variable and perform autoregressive weighted score matching. The valid weight function in this = O. Similar nonasympiotic bounds could be
case should satisfy h,, (T)) (T) (T), derived here and a near optimal weight function would be the Euclidean distance between f, and the boundary of f,, support, which would be tn+t). Therefore, the near-optimal weight function would (T) = min(l, ln-1, bn+1 ).
h,
During experiments, when T is unknown and cannot be approximated well, we perform a data truncation for each batch. We naively drop all timestamps beyond the length of the shortest sequence to make length consistent within a batch. Then we perform an AWSM for the fixed-length Hawkes process. We specify the results with this modification when we show our hyperparameters.
D.2 Addtional Results and Hyperparameters
For parametric models, Table provide the estimation results for some parameters and Figure shows the learned intensity functions the 2-nd dimension for the 2-variate Hawkes processes.
For deep point process experiments, we run datasets with methods deployed on 2 models. We show the hyperparameters of those experiments in Table Epochs column represents the number
21

---

Truth
MLE Event
3.0
Truth
MLEE
5
AWSM
2.5
AWSM
Event
ASM
2.0
Intensity
Intensity
1.5
1.0
-1.0
10
20
30
40
10
20
40
Time
Time
(a) Exp Hawkes Dim-2
(b) Houssian-Hawker Dim-2
Figure 3: The learned intensity functions from MLE, (AJSM, and (A)WSM (a) Exp-Hawkes and (b) -aussian-Hawkes for the 2-nd dimension.
of epochs for the experiments of a model (THP or SAHP) on a dataset. We train same epochs for three training methods (MLE, AWSM or DSM) and validate every 10 epochs to report the best result. When training MLE, the hyperparameter is the number integral nodes, which is always 10 for all experiments. When training AWSM, have two hyperparameters, the value of balancing coefficient for CE loss, shown in column @AWSM and whether data truncation is performed as shown in column Trunc. represents performing data truncation and represents data truncation. For DSM, the hyperparameters are balancing coefficient denoted as ODSM and variance of noise denoted as oDSM.
22

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Reviews:
[{"review_id": "bhJeO0GxFg", "reviewer": "Reviewer_Se8E", "summary": "The paper considers the problem of utilizing score-matching approaches for point processes. The main motivations for the paper are the Poisson and Hawkes processes. Both of these model the repeated occurrences of events over a finite time interval $[0, T]$. The Poisson process models the occurrence of an event within a fixed infinitesimal interval with a fixed homogenous rate $\\lambda$ while a Hawkes process is a self-excitation process where past events increase the likelihood of future ones. Score-matching is a natural approach in settings where the computation of the normalizing constant of a parameterized model are challenging to compute. However, to ensure traceability, one utilizes an implicit approach where the score matching objective is approximated by a tractable alternative which may be easily computed through the derivatives of the approximating model. \n\nThe main contribution of this work is the identification of simple scenarios, even for the simple setting of the generalized Poisson process, where the score function optimization function of prior approaches fails as the regularity condition underlying them does not apply. The paper derives an expression showcasing the bias (Proposition 3.1) and is related to the evaluation of the estimated score function (for fixed number of occurrences) at the end points of the interval of interest ($t = 0$ and $t = T$). The paper then proposes an alternative where a weight function $h$ is used to augment the score function loss which ensures that these biasing terms evaluate to $0$. Furthermore, these functions may be chosen independently of the parameters of the underlying model. In their experiments, it is shown that the use of such functions still satisfies desirable properties such as the optima corresponding to the true values of the parameters while also allowing for a tractable score function formulation which avoids the bias incurred by prior approaches. This is a natural approach which suggests an avenue towards improving the performance of such models.\n\nOverall, the problem considered in the paper and the solution they identify are interesting and likely relevant to the NeurIPS community. However, I do have concerns regarding the writing of the paper. For example, there are several assumptions on the weight function in Equation 9 but it is not clear which properties are used in the proof of Proposition 3.3. For instance, Line 400 features an explicit expression for $h_1$ and $h_N$ while the expression in Equation 9 features none of these. It is my understanding that the assumptions in Equation 9 ensure that the biasing terms equate to $0$ for these elements. Furthermore, I believe that under the assumptions in Equation 9, both terms appearing in the display between Lines 399 and 400 are $0$. Please clarify whether this is in fact the case. The same issues persist for the proofs of the results in Section 4. I would like clarification from the authors on these points before updating my review.\n\n************************************************************************************************************************************************\n\nI thank the authors for their response and have amended my review accordingly.", "questions": "See main review", "limitations": "See main review", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "See main review", "weaknesses": "See main review", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you so much for reviewing our paper. We answer your questions below. \n> For example, there are several assumptions on the weight function in Equation 9 but it is not clear which properties are used in the proof of Proposition 3.3. For instance, Line 400 features an explicit expression for $h_1$ and $h_N$ while the expression in Equation 9 features none of these. Please clarify whether this is in fact the case. The same issues persist for the proofs of the results in Section 4. \n\nWe greatly thank you for pointing this out. Your understanding is correct, and we apologize for the typos. The proofs for Propositions 3.3 and 4.3 are from our previous edition, where we specified one weight function instead of all weight functions that satisfy the condition. **We have proofread our work and ensured that the rest of our proofs remain correct.**\n\nFor Propositions 3.3 and 4.3, the main parts remain correct. Only minor modifications are needed to utilize the assumptions stated in the main text, ensuring that the proofs hold for general weight functions. We have corrected the necessary parts as follows. \n\nFrom lines 398 to 404, the corrected proof is as follows: \n\nUsing the first two equations in Eq. 9, we have \n\n$\n\\int [p(t_1,\\ldots, t_N)\\frac{\\partial \\log p_{\\theta}(t_1,\\ldots, t_N)}{\\partial t_{n}}h_n(\\mathcal T)]\\big\\vert_{t_{n}=t_{n+1}}d\\mathcal T_{-n}=0, \\forall n\\in [N]\n$\n\n$\n\\int [p(t_1,\\ldots, t_N)\\frac{\\partial \\log p_{\\theta}(t_1,\\ldots, t_N)}{\\partial t_n}h_n(\\mathcal T)]\\big\\vert_{t_n=t_{n-1}}d\\mathcal T_{-n}=0,\\forall n\\in [N].\n$\n\nTherefore, the first term on the right side of the second equation of Eq. 21 will disappear, and the second term equals $-\\mathbb E_{p(\\mathcal T)}[\\sum_{n=1}^{N_T}\\frac{\\partial}{\\partial t_n}\\psi_{\\theta}(t_n)h_n(\\mathcal{T})+\\psi_{\\theta}(t_n)\\frac{\\partial}{\\partial t_n}h_n(\\mathcal T)]$.The existence of such an expectation is due to the last two equations in Eq. 9. Therefore, we complete the proof.\n\nWe can see from the proof that, in Eq. 9, the first two equations ensure that the integration by parts trick does not produce an intractable term, and the last two equations are simply regularity conditions that ensure all terms are well-defined. \n\nSimilarly, from line 431 to 434, the corrected proof is as follows: \n\n$\n\\mathbb E_{p(\\mathcal T)}[\\sum_{n=1}^{N_T} \\psi(t_n|\\mathcal F_{t_{n-1}})\\psi_\\theta(t_n|\\mathcal F_{t_{n-1}})h_n(\\mathcal T)]\n$\n\n$\n=\\sum_{n=1}^\\infty \\int p(t_1,\\ldots t_n)\\psi(t_n|\\mathcal F_{t_{n-1}})\\psi_\\theta(t_n|\\mathcal F_{t_{n-1}})h_n(\\mathcal T)d\\mathcal T_{1:n}\n$\n\n$\n=\\sum_{n=1}^\\infty \\int p(\\mathcal T_{:n-1})p(t_n|\\mathcal F_{t_{n-1}})\\psi_\\theta(t_n|\\mathcal F_{t_{n-1}})h_n(\\mathcal T)\\vert_{t_n=t_{n-1}}^{t_n=T}d\\mathcal T_{:{n-1}}\n$\n\n$\n-\\sum_{n=1}^\\infty \\int p(t_1,\\ldots, t_n)[\\frac{\\partial \\psi_\\theta(t_n|\\mathcal F_{t_{n-1}})}{\\partial t_n}h_n(\\mathcal T) + \\psi_\\theta(t_n|\\mathcal F_{t_{n-1}})\\frac{\\partial h_n(\\mathcal T)}{\\partial t_n}]d\\mathcal T_{1:n}.\n$\n\n\nBetween the second and the third line above, we omit the steps used in the derivation of Proposition 4.1 to make it concise. For the term in the third line above, it will be eliminated using Eq. 15. For the term in the fourth line above, using Lemma B.1, we have: \n\n$\n-\\sum_{n=1}^\\infty \\int p(t_1,\\ldots, t_n)\\left[\\frac{\\partial \\psi_{\\theta}(t_n|\\mathcal F_{t_{n-1}})}{\\partial t_n}h_n(\\mathcal T) + \\psi_{\\theta}(t_n|\\mathcal F_{t_{n-1}})\\frac{\\partial h_n(\\mathcal T)}{\\partial t_n}\\right]d\\mathcal T_{1:n}=\\\\\n-\\mathbb E_{p(\\mathcal T)}[\\sum_{n=1}^{N_T}\\frac{\\partial \\psi_{\\theta}(t_n|\\mathcal F_{t_{n-1}})}{\\partial t_n}h_n(\\mathcal T) + \\psi_{\\theta}(t_n|\\mathcal F_{t_{n-1}})\\frac{\\partial h_n(\\mathcal T)}{\\partial t_n}].\n$\n\nThe existence of the expectation is ensured by the last two terms of Eq. 15.\n\nWe hope the corrected proof can address your concerns.", "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": "AKohxE4oeW", "reviewer": "Reviewer_B3db", "summary": "This paper studies the use of score matching for estimation in point process models, and proves the incompleteness in the original score matching estimators due to the bounded support. Weighted score matching is use to address the issue and theoretical results are establish ed for the consistency and optimality of the proposed estimator. Experiments on different data demonstrate the effectiveness of the proposed estimator which yields results comparable to maximum likelihood estimation.", "questions": "* Did the author observe any instability during the score matching training?", "limitations": "Included in weakness", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. This paper theoretically shows the limitation in the use of original score matching estimators for point processes, pointing out their incompleteness and provide a solution.\n2. Theoretical analysis is given to support the claim and the convergence of the proposed method.\n3. The paper is well-written and easy to follow.", "weaknesses": "1. As acknowledged in the limitation section in the paper, existing approaches that adopt denoising score matching on the point process is not considered. Even with theoretical gaurantee, experimentally, such baselines would be necessary to include in order to fully demonstrate the superiority of the method.\n2. For Hawkes processes, the problem cause by the bounded support can be solved by applying log-normalization to transform the bounded temporal domain into an unbounded one [1]. One could achieve this by removing the right bound and transform the $(0,+\\infty)$ to $(-\\infty,+\\infty)$. Although the real data is observed within finite window, the effect of removing the right bound should be negligible as it only affect the last event.\n3. Including more metrics on event prediction, such as MSE of event time, would be helpful, as the practical use of point process model is for prediction.\n\n[1] Lin, H., Wu, L., Zhao, G., Liu, P., & Li, S. (2022). Exploring generative neural temporal point process. Transactions on Machine Learning Research.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you so much for reviewing our paper. We answer your questions below.\n> Denoising score matching for point processes is not considered. \n\nWe thank the reviewer for pointing this out. Indeed, denoising score matching (DSM) is not considered in our paper because we mainly focus on correcting the original score matching. However, to address your concerns, we have carried out experiments on DSM. **We can conclude that DSM is inferior to AWSM in terms of both efficiency and accuracy.**\n\nWe deploy DSM on THP and SAHP. We use the DSM loss as in [1]. For observed timestamps $t_n^{(m)}$ in $m$-th sequence, we sample $L$ noise samples $\\tilde t_{n,l}^{(m)}=t_n^{(m)}+ \\epsilon_{n,l}^{(m)}, l = 1,\\ldots, L,$ where $\\text{Var}(\\varepsilon_{n,L}^{(m)})=\\sigma^2$ and get the DSM objective: \n\n$\n\\hat {\\mathcal J}(\\theta)=\\frac{1}{M}\\sum_{m=1}^M\\sum_{n=1}^{N_m}\\sum_{l=1}^{L}\\frac{1}{2L}[\\psi_\\theta(\\tilde t_{n,l}^{(m)})+\\frac{\\varepsilon_{n,l}^{(m)}}{\\sigma^2}]+\\alpha\\hat {\\mathcal J}_\\text{CE}(\\theta),\n$\n\nwhere $\\hat {\\mathcal J}_{\\text{CE}}(\\theta)$ is the cross-entropy loss as line 191-192 in our paper. We carried out experiments on synthetic data where $T$ is known. We compared the TLL, ACC, and running time of DSM with MLE and AWSM. The comparison results are shown in Table 1 in the attached PDF. DSM performs the worst among the three estimators, and AWSM is the fastest. \n\nThe reasons why DSM performs poorly are as follows: \n\n1. In terms of accuracy, as discussed in Section 5 of [2], DSM is biased when $\\sigma > 0$, while our AWSM is unbiased and produces consistent estimates. Therefore, it is not surprising to see that AWSM achieves better TLL and ACC than DSM. \n\n2. In terms of efficiency, previous work states that DSM is faster than SM because the Hessian matrix is expensive to compute, while DSM avoids this. However, as presented in [3], ASM (or AWSM) also avoids computing the Hessian matrix by replacing it with a one-dimensional partial derivative  $\\frac{\\partial}{\\partial t_n}\\psi_{\\theta}(t_n|\\mathcal F_{t_{n-1}})$. Therefore, DSM is not more efficient than AWSM for avoiding the computation of the Hessian matrix. On the contrary, DSM can be slower than AWSM because DSM requires $L$ noise samples for each timestamp and must compute their scores. \n\n\n\n> For Hawkes processes, the problem can be solved by applying log-normalization to transform the bounded temporal domain into an unbounded one [1]. \n\nThis is a misunderstanding. **Apply log-normalization does not provide any solution to the problem, we can prove that the integration by parts still produces an intractable term after taking this transformation. Therefore, using ASM on a log-transformed sequence still results in wrong estimation**. If one hopes to get consistent estimation after log transformation, a suitable weight function is still needed. This is summarised as follows. \n\n|                                                      | ASM  | AWSM |\n| ---------------------------------------------------- | ---- | ---- |\n| No transformation                                    | ×    | √    |\n| Log Transform $x_n = \\log t_n$                       | ×    | √    |\n| Log Transform on interval $y_n = \\log (t_n-t_{n-1})$ | ×    | √    |\n\nWe illustrate this using the simplest example both theoretically and empirically. Consider a homogeneous Poisson process with intensity $\\lambda ^*$ with observations $t_1, \\ldots, t_{N_T}$ over $0\\leq t_1\\leq \\ldots \\leq t_{N_T}\\leq T$. Now we consider\n\n1. First apply log transformation to timestamps then using ASM. The estimate is denoted as $\\hat \\lambda_{\\text{ASM,Log}}$. \n\n2. First apply log transformation to intervals then using ASM on intervals. The estimate is denoted as $\\hat \\lambda_{\\text{ASM,Log Interval}}$. \n\nFor log-transformation on timestamp, we have $x_n=\\log t_n$.\nTherefore the conditional pdf of $x_n$ given $\\mathcal F_{x_{n-1}}$ is: \n\n$\np(x_n|\\mathcal F_{x_{n-1}})=\\lambda \\exp\\left[-\\lambda \\exp(x_n)\\right]\\exp(x_n),\n$\n\nand we can compute conditional score accordingly. We plug these terms into the implicit ASM objective and get the estimate. Suppose we observed $M$ sequences of $t_1^{(m)}, \\ldots, t_{N_m}^{(m)}$, after derivation we get an analytical form of $\\hat \\lambda_{\\text{ASM,Log}}$: \n\n$\n\\hat \\lambda_{\\text{ASM,Log}} =2\\frac{\\sum_{m=1}^M\\sum_{n=1}^{N_m}\\exp(x_n)}{\\sum_{m=1}^M\\sum_{n=1}^{N_m}\\exp(2x_n)}.\n$\n\nSimilarly, we consider transformation on intervals $\\tau_n:= t_n-t_{n-1}, n\\geq 2, \\tau_1:=t_1, y_n = \\log (\\tau_n)$ and consider the score $\\psi(y_n|\\mathcal F_{y_{n-1}})$ then apply ASM. The estimate in this case is: \n\n$\n\\hat \\lambda_{\\text{ASM,Log Interval}} =2\\frac{\\sum_{m=1}^M\\sum_{n=1}^{N_m}\\exp (y_n)}{\\sum_{m=1}^M\\sum_{n=1}^{N_m}\\exp(2y_n)}. \n$\n\nBoth estimates are wrong and can never recover the true parameter regardless of sample size. To compare, after transformation, suitable weight functions can be added. For log transformation on timestamps, we use weight function $h_n(x_n)=(x_n-x_{n-1})(\\log T-x_n)$. For log transformation on intervals, we use $h_n(y_n) = \\log [T-t_{n-1}]-y_n$. The corresponding estimates are $\\hat \\lambda_{\\text{AWSM,Log}}$ and $\\hat \\lambda_{\\text{AWSM,Log Interval}}$. \n\nWe measure the MSE of these four estimates. Results are shown in Table 2 in the attached PDF. It is easy to see that as sample size increases, MSE of $\\hat \\lambda_{\\text{ASM,Log}}$ and $\\hat \\lambda_{\\text{ASM,Log Interval}}$ remains large and unchanged, showing that these two estimators are wrong regardless of sample size. \n\n\n\n> Including more metrics would be helpful.\n\nWe will consider adding MSE to our revised paper. \n\n> Did the author observe any instability during training? \n\nFor our weighted score matching, we did not notice instability. \n\n[1] SMURF-THP: score matching-based uncertainty quantification for transformer Hawkes process\n\n[2] A connection between score matching and denoising autoencoders\n\n[3] Autoregressive score matching", "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": "nP3uAINuie", "reviewer": "Reviewer_X8YN", "summary": "EDIT: I have changed my score from a \"weak accept\" to an \"accept.\"\n\nThe paper studies the potential use of score matching (SM) to do inference\nwith the model of interest is a Poisson point process or Hawkes\nprocesses. Because using SM in practice requires modifying the objective\nwith integration by parts, the paper shows for many common Poisson and\nHawkes processes a core assumption to make this modification possbile does\nnot hold. In order to remedy the situation, the paper proposes a random weighted\nversion of SM called WSM (along with an analog for Autoregressive Score\nMatching (ASM) called WASM), which reestablishes the validity of integration\nby parts for the aforementioned processes. The paper argues theoretically\nthat their new estimators are consistent, and it provides empirical\nanalyses to demonstrate that the new estimator outperforms the non-weighted\nvariant and matches the performance of alternatives. It also makes an\nargument to select a weighting function based on some set of assumptions.", "questions": "[Q1] Given the choice of $h$ optimizes $C_h$ but ignores $\\Gamma(h, A, B)$,\nhow sensitive is $\\Gamma$ to the choice of $h$? This could be demonstrated\nempirically or theoretically, but it would be good to have some analysis of\nthis to argue why this weighting function is decent.", "limitations": "NA", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper points out a fundamental issue with trying to use SM for general\nPoisson point processes and Hawkes processes, and in addition, it recommends\na weighting scheme that patches up this problem based on some reasonable\nassumptions. This is a valuable observation, and the proposed remedy looks\nreasonable. There is ample empirical work to suggest that their tweak on SM\nand ASM does agree with the maximum likelihood estimator (MLE). The theory\nprovided in the paper also argues that their weighted estimator is\nconsistent if the model is well-specified.\n\nOriginality:\n\n[+]\n\n+ The authors have applied a weighting scheme to SM and ASM that permits them\nto use integration by parts for the standard objective and prove the\nconsistency of their new estimator under suitable regularity conditions.\n\nQuality:\n\n[+]\n\n+ The empirical work is thorough and does a good job checking the accuracy\n  of their new estimator.\n\nClarity:\n\n[+]\n\n+ The presentation of the theory and empirical work is very neat and\n  polished.\n+ Assumptions are clearly laid out to ensure consistency of the new estimator.\n\nSignificance:\n\n[+]\n\n+ The new estimator provides the ability to do consistent semiparametric\n  estimation for Poisson point and Hawkes processes with SM (instead of\n  MLE), which provides a new tool for very large dimensional problems.", "weaknesses": "There are a couple critiques of the paper:\n\n- While the idea of using a weighting function to enable integration by\n  parts is a good one, it has been seen in other fields before\n  (e.g. weighting Stein kernels in compact settings), and hence is not\n  completely novel.\n- The argument in Section 5 outlining a weighting function is a great\n  addition to the paper, but given that it is only optimizing $C_h$ and not\n  $\\Gamma(h, A, B)$, it is hard to know how close it is to the optimum.\n\nOriginality:\n\n[-]\n\n- The only novel insight of the paper is that the trick of integration by parts,\n  so often using in score matching, is not possible for general point\n  processes, but it can be recovered by weighting the Fisher divergence a bit\n  differently. It is not clear if this idea alone is sufficient for publication.\n\nQuality:\n\n[-]\n\n- The argument to choose their weighting function $h$ is definitely better\n  than simply picking one. That being said, it still not clear how close to\n  optimal their choice of $h$ is given they are only optimizing one term in\n  the objective ($C_h$). The paper would be a bit stronger if they could\n  include a different (natural) weighting function to show their new choice\n  is an improvement.\n\nClarity:\n\n[-]\n\n- Equation 1: Can we at least have a sentence explaining the paper will be\n  slightly abusing notation by referring to $p(T)$ as the conditional\n  distribution of $p$ given $N_t = |T|$?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you so much for reviewing our paper. We answer your questions below.\n> The argument to choose their weighting function $h$ is definitely better than simply picking one. That being said, it still not clear how close to optimal their choice of h is given they are only optimizing one term in the objective $C_{\\bf h}$. The paper would be a bit stronger if they could include a different (natural) weighting function to show their new choice is an improvement. \n\nWe thank the reviewer for pointing this out. To address your concern, we carry out experiments and compare three different weight functions that satisfy the criterion (Eq. 15) for AWSM on Hawkes process. Other than the near-optimal weight function $\\bf{h}^0$ introduced in our paper, we also consider a natural weight function $\\bf{h}^1$ defined as,\n\n$\nh_n^1(t_n)=(t_n-t_{n-1})(T-t_{n}).\n$\n\nWe also consider another valid weight function: the square root of $\\bf h^1$ denoted as $\\bf h^2$,\n\n$\nh_n^2(t_n)=\\sqrt{(t_n-t_{n-1})(T-t_n)}. \n$\n\nAll three weight functions can be applied in AWSM to recover ground-truth parameters, however with different convergence rates. We carry out experiments on synthetic data for exponential-decay model with the same setting as section 6.2 in our paper. We measure their MAE for different sample sizes in Figure 1 of the attached PDF and find that $\\bf h^0$ does achieve the best results among the three weight functions. \n\nWe hope these additional experimental results can enhance the credibility of our paper and address your concerns. We would definitely add these results in the camera ready. \n\n\n> Given the choice of $\\bf h$ optimizes $C_{\\bf h}$ but ignores $\\gamma(\\bf h,A,B)$, how sensitive is $\\Gamma$ to the choice of $\\bf h$? This could be demonstrated empirically or theoretically, but it would be good to have some analysis of this to argue why this weighting function is decent. \n\nWe thank the reviewer for pointing this out. This is indeed a tough question that we do not yet have a satisfying answer. For specific parametric models, $\\dot A_n(\\mathcal T), \\dot B_n(\\mathcal T)$ can be computed analytically (see line 478-480) and then $\\Gamma(\\mathbf{h},A,B)$ can be computed via Monte Carlo. Then we can study how sensitive $\\Gamma(\\mathbf{h},A,B)$ is to $\\bf h$. For general models, especially when $\\psi_{\\theta}$ is a deep neural network like THP or SAHP, $\\Gamma(\\mathbf{h},A,B)$ is intractable to compute. \n\nHowever, heuristically speaking, our choice of near-optimal weight function $\\bf h^0$ should be a good choice even concerning $\\Gamma(\\mathbf{h},A,B)$. To make $\\Gamma(\\mathbf{h},A,B)$ small, a natural idea is to make $|h_n(\\mathcal T)|$ and $|\\frac{\\partial}{\\partial t_n}h_n(\\mathcal T)|$ small. The weight function we chose and its derivative have relatively positive low powers with respect to $t_n$, therefore making $|h_n^0(\\mathcal T)|$ and $|\\frac{\\partial}{\\partial t_n}h_n^0(\\mathcal T)|$ small. For  weight functions  $h_n^1(\\mathcal T)=(T-t_n)(t_n-t_{n-1})$, the power w.r.t. $t_n$ is two. And for $h^2_n(\\mathcal T)=\\sqrt{(T-t_n)(t_n-t_{n-1})}$, its derivative is $\\frac{\\partial}{\\partial t_n}h_n^2(\\mathcal T)=\\frac{1}{2}\\frac{T-t_n-(t_n-t_{n-1})}{\\sqrt{(T-t_n)(t_n-t_{n-1})}}$ , the numerator is usually a bounded quantity and the denominator may be close to zero, making it large. In conclusion, $\\bf h^0$ is a better choice compared with $\\bf h^1$ or $\\bf h^2$ concerning $\\Gamma(\\mathbf{h},A,B)$.\n\n\n\n\n> Equation 1: Can we at least have a sentence explaining the paper will be slightly abusing notation by referring to $p(\\mathcal T)$ as the conditional distribution of p given $N_T=|T|$? \n\nWe thank the reviewer for pointing this out.  Here since $N_T$ is random, by the notation $p(\\mathcal T)$, we are referring to the probability density or the likelihood of observing $N_T$ events $t_1,\\ldots, t_{N_T}$  in $[0,T]$.  For Poisson process, the conditional distribution of event time $t_1,\\ldots, t_{N}$ given $N_T = N$ is, \n\n$\np(t_1,\\ldots, t_N|N_T = N) = \\frac{\\prod_{n=1}^N \\lambda(t_n)}{\\int_{0\\leq t_1\\dots\\leq t_N\\leq T}\\prod_{n=1}^{N}\\lambda(t_n)dt_1\\ldots dt_N}, 0\\leq t_1\\ldots\\leq t_{N}\\leq T.\n$\n\nWe acknowledge that our paper may not employ the rigorous notation typically used in the fields of probability and stochastic processes. Instead, we have opted for notations that we believe are accessible to readers from both the computer science and statistics communities. We deeply apologize for any confusion this may have caused.", "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:
We express our sincere appreciation to all reviewers for their time, effort, and insightful feedback. We are encouraged by their recognition of the significance of our work in identifying the incompleteness of the original score matching for point processes, proposing the weighted score matching method, studying its convergence rate, proposing a near-optimal weight function, and deploying the method on deep point process models.

In the following, we have provided specific and direct responses to each of the primary concerns raised by the reviewers, supplemented with additional experiments that substantiate our arguments. We aim to ensure that we address all the concerns and offer clarity and reassurance where needed. Should any additional questions arise, we invite reviewers to engage in further discussion. Once again, we express our gratitude for your time and dedication in reviewing our work.

Meta-review: This paper addresses an important issue in applying score matching techniques to point process models, specifically Poisson and Hawkes processes. The authors identify a fundamental problem with existing score matching approaches for these models due to bounded support, and propose a novel weighted score matching solution to address this limitation. 

All reviewers agree that the paper makes a valuable contribution by pointing out this critical issue and providing a principled solution. The theoretical analysis demonstrating the consistency of the proposed estimator and the empirical evaluations showing its effectiveness are viewed positively. The paper is generally well-written and the presentation is clear.

Some key strengths highlighted by reviewers include:
- Identifying an important limitation of existing score matching approaches for point processes
- Proposing a theoretically-grounded weighted score matching solution 
- Thorough empirical evaluation demonstrating the effectiveness of the proposed method
- Clear presentation of the theory and assumptions

The main concerns raised relate to:
- Lack of comparison to some relevant baselines like denoising score matching
- Questions about the optimality of the proposed weighting function
- Some notational issues and need for clarification in parts

The authors have provided detailed responses addressing most of these concerns, including additional experimental results comparing different weighting functions. This has increased reviewer confidence in the work.

Overall, the reviewers recommend accepting this paper, contingent on the authors making a thorough revision incorporating the feedback and additional results from the discussion phase. Specifically, the authors should:

1. Include comparisons to denoising score matching approaches as discussed in the rebuttal
2. Add the additional experimental results on different weighting functions to the main paper
3. Expand the discussion on the choice and optimality of the weighting function
4. Clarify the notation, especially around equation 1 as mentioned by Reviewer X8YN
5. Consider adding MSE of event time prediction as an evaluation metric
6. Discuss recent related work on score matching for point processes

With these revisions, this paper promises to make a solid contribution to the field. The authors should carefully incorporate all the feedback received during the review process to strengthen the final version.
Decision: Accept (poster)