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REVIEW 3 major objections 5 minor 32 references

Neural Networks for Parameter Estimation of the Discretely Observed Hawkes Process

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A neural network trained on simulated interval counts can estimate Hawkes-process parameters about as accurately as costly likelihood-based methods, but in far less time.

desk verdict A practical amortized inference method for interval-censored Hawkes processes with strong exponential-kernel results, but the non-exponential claim needs more than a Gamma-kernel simulation. read the letter →

arxiv 2506.01258 v3 pith:6XYMRI76 submitted 2025-06-02 stat.ME

Jason J. Lambe , Feng Chen , Tom Stindl , Tsz-Kit Jeffrey Kwan This is my paper
classification stat.ME MSC 62M0962M4560G5562F40
keywords Hawkesprocessinterval-censoreddataneuralnetworkestimationsummarystatisticuniformimputationnegativebinomialautoregressionbootstrapuncertaintyquantificationamortisedinference
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that a fully connected neural network, trained on simulated data, can estimate the parameters of a Hawkes process from interval-censored counts with accuracy close to the best existing likelihood-based method but at a fraction of the computational cost. The key is a cheap summary statistic: maximum-likelihood estimates from event times uniformly sprinkled into each interval, augmented by negative-binomial autoregressive coefficients for the non-exponential case. Simulation experiments show low bias, comparable mean-squared error to the particle-based benchmark, and bootstrap-based standard errors and bias correction that produce near-nominal coverage. Applications to weekly measles and Salmonella counts, including a seasonal background rate, illustrate the method in practice. If the paper is right, expensive likelihood calculations for this intractable setting can be replaced by a one-time training investment.

What carries the argument

The load-bearing object is the summary statistic $s(n_{1:K}) = (\hat{\theta}_{\mathrm{imp}}, \hat{\gamma}_{0:p}, \hat{\delta})$. For the first component, event times are imputed uniformly inside each censoring interval and a (possibly misspecified) exponential Hawkes model is fit by maximum likelihood, giving a fast, minimum-dimension input that is sensitive to the background rate, branching ratio, and mean offspring delay. The second component, a negative-binomial autoregression with $p$ lags applied to the counts, carries information about the shape and scale of a non-exponential offspring density, which the imputation estimate alone cannot identify. The neural network learns the inverse mapping from this summary to the true parameter vector, and a parametric bootstrap, applied by simulating new paths from the network estimate and re-running the network, produces standard errors and bias correction without any additional training.

What would settle it

For a non-exponential offspring density such as a Weibull or log-normal kernel, simulate interval-censored paths with known parameters, compute the paper's summary statistic, and train the network following the paper's recipe; the central claim would fail if the shape and scale estimates remain biased at aggregation levels where the Gamma kernel performed well.

Watch

Extended reading notes

Core claim

The central discovery is that a deliberately misspecified but fast summary statistic is enough for a neural network to recover the parameters of a discretely observed Hawkes process. Instead of reconstructing latent event times carefully, the authors sprinkle them uniformly into each observation interval, fit an exponential Hawkes model by maximum likelihood, and append the coefficients and dispersion parameter from a negative-binomial autoregression on the counts. This fixed-dimensional input lets a small fully connected network learn the inverse map from data to parameters; the same network then serves as an amortised estimator for any new dataset. The paper reports that this estimator has empirical bias comparable to the particle Markov chain Monte Carlo benchmark, with similar or slightly smaller standard errors, and it extends naturally to unequal censoring intervals and time-varying background rates.

Load-bearing premise

The central assumption is that the cheap summary statistic, built from the MLE of a misspecified exponential Hawkes model on uniformly imputed event times plus negative-binomial autoregressive coefficients, is informative enough for the network to recover the true offspring-density parameters; the paper supports this with simulations only for a Gamma kernel and reports that the shape estimate depends on the hand-chosen lag order.

Editorial extensions

If this is right

  • For an exponential Hawkes process observed as interval counts, the neural-network estimator matches the accuracy of the particle Markov chain Monte Carlo benchmark while cutting estimation time from minutes to milliseconds.
  • For non-exponential excitation kernels, adding negative-binomial autoregressive lags identifies the shape and scale of the offspring density; larger lag counts remove most of the bias in the shape estimate.
  • Bootstrap bias correction and standard errors work well enough that approximate 95% confidence intervals achieve near-nominal coverage across simulation settings.
  • The same summary-statistic construction accommodates unequal censoring intervals and time-varying background rates, a capability many competing likelihood methods lack.
  • Because training cost is paid once, repeated inferences on many datasets become near instantaneous, which is the defining advantage of amortised inference.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The imputation-plus-autoregression recipe may transfer to other point-process models whose likelihood is intractable under incomplete observation, such as renewal Hawkes processes, since it avoids hand-building proposal distributions for latent event times.
  • The paper validates the non-exponential summary statistic only on a Gamma kernel; a natural stress test is to try Weibull or log-normal offspring densities, where shape and scale are not matched by the exponential imputation model in the same way.
  • The bootstrap bias-correction step could double as a diagnostic: if the bootstrap median is far from the point estimate, the summary statistic is probably not capturing enough information, and subtracting the bias would conceal rather than fix the problem.
  • Because the shape estimate is sensitive to the hand-chosen lag order $p$, users would benefit from a data-driven rule for choosing $p$, such as selecting the smallest lag at which the bootstrap median stabilises.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a likelihood-free, neural-network-based method for estimating the parameters of a Hawkes process from interval-censored (count) data. The method constructs a summary statistic from a uniform imputation MLE of a misspecified exponential Hawkes process, augmented with negative binomial autoregression (NBAR) coefficients for non-exponential excitation kernels. A fully connected feed-forward neural network is trained on simulated data to predict the model parameters, with bootstrap procedures for bias correction and standard-error estimation. The method is evaluated on exponential-kernel simulations (benchmarked against PMMH), Gamma-kernel simulations, time-varying background rate scenarios, and two infectious disease applications (measles in Tokyo and Salmonella in New South Wales).

Significance. If validated, the method offers a practical amortised-inference solution to an intractable likelihood problem, and the exponential-kernel experimental design is a clear strength: Table 1 reports 3,000 test paths, bias-corrected estimates, bootstrap standard errors, and coverage probabilities near nominal, all benchmarked against PMMH. The bias-correction procedure is simple and appears effective. The central unsupported claim is that the method generalises to non-exponential excitation kernels: the only evidence is for Gamma kernels, the summary-statistic identifiability is asserted heuristically, and Table 3 shows the shape estimate depends strongly on the hand-chosen lag order p. The paper would be substantially strengthened by an out-of-family simulation (e.g., Weibull or log-normal kernel) and by adding a likelihood-based comparator to the non-exponential experiments.

major comments (3)
  1. [§3.1.2 and Table 3] The summary statistic s(n1:K) = (θimp, γ̂0:p, δ̂) is claimed to be sensitive to the offspring waiting-time distribution, but no identifiability or sufficiency argument is given. In the Gamma-kernel experiment with α=1.5, β=1.0, and Δ=0.1, the NN estimate of α is 1.843 at p=3 but 1.549 at p=48, a relative change of 23% that is large compared with the reported standard errors. The recommendation to 'increase p until estimates stabilise' is post hoc and does not establish that the summary statistic can identify the offspring shape and scale. The abstract's general claim about non-exponential kernels is not yet supported; an out-of-family test (e.g., Weibull or log-normal kernel) and ideally a theoretical justification of informativeness are needed.
  2. [Table 2 and §4.2] The non-exponential experiments report no competing approximate-likelihood method. The parity claim in the abstract ('comparable to the best extant approximate likelihood estimators in terms of mean-squared error') is therefore not demonstrated outside the exponential-kernel setting. The only benchmark shown for Gamma kernels is the MLE from continuous observation, which is not a meaningful comparison for discretely observed data. Adding a PMMH (or other) benchmark on the same test paths for at least one non-exponential configuration would substantiate the claim.
  3. [Abstract and §2.2] The abstract states 'We implement NN quantile estimation for fast uncertainty quantification', but the methods described in Section 2.2 (and used throughout the paper) only provide bootstrap-based standard errors and bias correction. No quantile estimation procedure is described or evaluated. Either the abstract should be corrected to describe the bootstrap-only approach, or the quantile method should be presented and tested.
minor comments (5)
  1. [Abstract / §4] The abstract claims comparison 'in terms of mean-squared error', but the reported results show biases and standard errors rather than an explicit MSE comparison; an MSE table or figure would make the claim easier to verify.
  2. [§3.1.1] The paper fixes the random seed when performing uniform imputation to make the summary statistic deterministic. It would be useful to discuss the sensitivity of the results to this seed choice, especially because the imputation estimate is a random quantity that becomes a fixed input to the NN.
  3. [Table 1] For β at Δ=0.1 and Δ=1.0, the NN coverage probabilities are 0.984 and 0.980, respectively, which may indicate slightly conservative bootstrap standard errors; a brief comment on the calibration of the bootstrap would be helpful.
  4. [§5.1] In the paragraph preceding Table 6, 'PMMMH' appears to be a typo for 'PMMH'.
  5. [§4.3] The practical recommendations for choosing p are phrased loosely ('ceasing to increase once estimates stabilise'); a more concrete rule, such as using the largest p for which the NBAR coefficients remain statistically meaningful, would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the NN estimates are genuine supervised predictions on held-out simulated data, and the summary statistic is an input for error correction rather than a fitted target.

full rationale

After walking the derivation chain, I find no circular step that can be exhibited as an equation-level reduction. The estimator is a supervised neural network trained on simulated pairs (theta^(m), s(n^(m))); the target is the posterior mean E_pi[theta | s], and on test paths the outputs are genuine predictions. The uniform-imputation MLE theta_imp is an input summary statistic, explicitly described as a naive estimate that the NN corrects ("the NN can be equivalently interpreted as an error correction tool"), not as the output by construction. The NBAR components are additional inputs, not fitted to the test target. The data-dependent prior for nu centered at r_hat_T is a modeling choice; it constrains but does not define the estimator, and the simulation coverage/error tables are held-out empirical checks. The PMMH benchmark and SMC proposal are from the same research group, but they are cited as published comparisons and are not used to derive the NN estimator; no uniqueness theorem or self-citation is invoked to force the choice. Section 4.3's sensitivity to p and the heuristic identifiability of the non-exponential kernel are correctness/identifiability limitations, not circularity, since the claimed reduction is absent.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim rests on four axioms: standard Hawkes ergodicity, the posterior-mean interpretation of NN estimators from prior literature, the unproved identifiability of the summary statistic for non-exponential offspring kernels, and standard bootstrap theory. No new physical or statistical entities are introduced. The free parameters are analyst-chosen hyperparameters and model choices, not fitted constants, but they affect the estimator's performance.

free parameters (5)
  • ISN prior location mu_beta for offspring scale beta = 5 in simulations, 4 in the measles application
    Hand-chosen to center the training distribution of the exponential kernel scale; affects which parameter regions the NN sees.
  • ISN prior scale sigma_beta = 3 in simulations, 2.5 in the measles application
    Hand-chosen spread of the offspring scale prior; wider priors are recommended but this value is not derived.
  • Background-rate prior noise sigma_nu = 0.25
    Hand-chosen standard deviation around the data-dependent centering r_hat_T; controls how strongly the prior fixes the ratio nu/(1-eta).
  • NBAR lag order p = 1/Delta in simulations, 10 for Salmonella
    Selected heuristically; Section 4.3 shows it materially affects the estimated offspring shape parameter alpha.
  • Spline knot locations for seasonal background = weeks 0, 2.5, 5, 38, 52
    Chosen post hoc to match the empirical minimum and maximum of the data's weekly median counts; this is a model selection choice that influences the fit.
assumptions (4)
  • domain assumption The Hawkes process is stationary with branching ratio eta<1, so N(T)/T converges almost surely to nu0/(1-eta0).
    Used in Section 3.3 to center the prior for nu on the observed average count rate; standard for stable Hawkes processes.
  • domain assumption A neural network trained with mean-squared error loss approximates the posterior mean E[theta|s(n)] when the training sample is drawn from the prior and the model is correctly specified.
    Invoked in Section 2.2 via Sainsbury-Dale et al. (2024); this is the theoretical justification for using the NN output as a point estimate.
  • ad hoc to paper The uniform imputation estimate from a misspecified exponential Hawkes process, together with NBAR coefficients, identifies the parameters of the true (possibly non-exponential) offspring density.
    This is the core identifiability assumption of the summary statistic; it is validated only empirically in Sections 4.2 and 4.3 and is not proved.
  • domain assumption The parametric bootstrap distribution of the NN estimator approximates its sampling distribution, so the median and standard deviation of bootstrap estimates can be used for bias correction and standard errors.
    Standard bootstrap theory; used in Section 2.2 and throughout the simulation study.

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Cite this review

Pith. "Pith review of Neural Networks for Parameter Estimation of the Discretely Observed Hawkes Process." pith.science (2026). https://pith.science/paper/6XYMRI76

@misc{pith2026250601258,
  author       = {Pith},
  title        = {Pith review of: Neural Networks for Parameter Estimation of the Discretely Observed Hawkes Process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6XYMRI76}},
  note         = {Machine review of arXiv:2506.01258}
}
read the original abstract

When the sample path of a Hawkes process is observed discretely, such that only the total event counts in disjoint time intervals are known, the likelihood function becomes intractable. To overcome the challenge of likelihood-based inference in this setting, we propose to use a likelihood-free approach that uses simulated data to train a fully connected neural network (NN) to estimate the parameters of the Hawkes process from a summary statistic of the count data. A naive imputation estimate of the parameters forms the basis for our summary statistic, which is fast to generate and requires minimal expert knowledge to design. The resulting NN estimator is comparable to the best extant approximate likelihood estimators in terms of mean-squared error but requires significantly less computational time. We implement NN quantile estimation for fast uncertainty quantification. The proposed estimation procedure is applied to weekly count data for two infectious diseases, with a time-varying background rate used to capture seasonal fluctuations in infection risk.

Figures

Figures reproduced from arXiv: 2506.01258 by the authors.

Figure 1
Figure 1. Training samples with T = 400, σν = 0.25 and ˆrT = 5.225. This allows the NN to focus on (ν, η) pairs that agree with the observed data. The value σν reflects uncertainty around the estimate ˆrT of ν0/(1 − η0) [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Comparison of ISN to Gamma priors, each with mean of approximately 5 and [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Salmonella Typhimurium Cases in NSW, Jan 2009 - Dec 2017 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Salmonella Typhimurium Cases in NSW, 2009 - 2017, compared to results [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: Salmonella Typhimurium Cases in NSW, 2009 - 2017, compared to results [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Density histograms of bias corrected NN estimates for the case of ∆ = 0 [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: Density histograms of bias corrected NN estimates for the case of ∆ = 0 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]

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Works this paper leans on

32 extracted references · 31 canonical work pages

  1. [1]

    Andrieu, C., Doucet, A., and Holenstein, R. (2010). Particle Markov Chain Monte Carlo Methods . Journal of the Royal Statistical Society Series B: Statistical Methodology , 72(3):269--342

    work page 2010

  2. [2]

    COVID-19 statistics to move from daily to weekly reports across Australia , health ministers say

    Australian Broadcasting Corporation (2022). COVID-19 statistics to move from daily to weekly reports across Australia , health ministers say. ABC News Article Webpage https://www.abc.net.au/news/2022-09-07/covid-19-statistics-to-be-reported-weekly-health-ministers-say/101414964

    work page arXiv 2022

  3. [3]

    National Notifiable Diseases Surveillance System ( NNDSS ) public dataset -- salmonella

    Australian Government Department of Health, Disability and Ageing (2024). National Notifiable Diseases Surveillance System ( NNDSS ) public dataset -- salmonella. NNDSS Public Dataset Webpage https://www.health.gov.au/resources/publications/national-notifiable-diseases-surveillance-system-nndss-public-dataset-salmonella?language=en

    work page 2024

  4. [4]

    Bonnet, A., Dion-Blanc , C., Gindraud, F., and Lemler, S. (2022). Neuronal network inference and membrane potential model using multivariate Hawkes processes. Journal of Neuroscience Methods , 372:109550

    work page 2022

  5. [5]

    About Salmonella Infection

    CDC (2024). About Salmonella Infection . CDC Salmonella Information Webpage https://www.cdc.gov/salmonella/about/index.html

    work page 2024

  6. [6]

    J., and Stindl, T

    Chen, F., Kwan, T.-K. J., and Stindl, T. (2025). Estimating the H awkes process from a discretely observed sample path. Journal of Computational and Graphical Statistics , pages 1--13

    work page 2025

  7. [7]

    and Lang, G

    Cheysson, F. and Lang, G. (2022). Spectral estimation of Hawkes processes from count data. The Annals of Statistics , 50(3)

    work page 2022

  8. [8]

    S., Schramm, L

    Chornoboy, E. S., Schramm, L. P., and Karr, A. F. (1988). Maximum likelihood identification of neural point process systems. Biological Cybernetics , 59(4-5):265--275

    work page 1988

Show all 32 references
  1. [9]

    and Yoshida, N

    Clinet, S. and Yoshida, N. (2017). Statistical inference for ergodic point processes and application to Limit Order Book . Stochastic Processes and their Applications , 127(6):1800--1839

    2017

  2. [10]

    Creel, M. (2017). Neural nets for indirect inference. Econometrics and Statistics , 2:36--49

    2017

  3. [11]

    Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika , 57(1):97--109

    1970

  4. [12]

    Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika , 58(1):83--90

    1971

  5. [13]

    Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. Journal of Applied Probability , 11(3):493--503

    1974

  6. [14]

    Jiang, B., Wu, T.-Y., Zheng, C., and Wong, W. H. (2017). Learning Summary Statistic for Approximate Bayesian Computation Via Deep Neural Network . Statistica Sinica , 27(4):1595--1618

    2017

  7. [15]

    and Cook, S

    Jun, M. and Cook, S. (2024). Flexible multivariate spatiotemporal Hawkes process models of terrorism. The Annals of Applied Statistics , 18(2):1378--1403

    2024

  8. [16]

    Kirchner, M. (2017). An estimation procedure for the Hawkes process. Quantitative Finance , 17(4):571--595

    2017

  9. [17]

    J., Chen, F., Stindl, T., and Kwan, T.-K

    Lambe, J. J., Chen, F., Stindl, T., and Kwan, T.-K. J. (2025). Fitting multivariate Hawkes processes to interval count data with an application to terrorist activity modelling -- a particle Markov chain Monte Carlo approach. https://doi.org/10.48550/arXiv.2503.18351 https://do...

    2025 doi arXiv

  10. [18]

    Lenzi, A., Bessac, J., Rudi, J., and Stein, M. L. (2023). Neural networks for parameter estimation in intractable models. Computational Statistics & Data Analysis , 185:107762

    2023

  11. [19]

    W., Rosenbluth, M

    Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953). Equation of State Calculations by Fast Computing Machines . The Journal of Chemical Physics , 21(6):1087--1092

    1953

  12. [20]

    Salmonellosis (excluding S

    NSW Health (2021). Salmonellosis (excluding S . Typhi and Paratyphi Infection ). NSW Health Salmonella Information Webpage https://www.health.nsw.gov.au:443/Infectious/controlguideline/Pages/salmonellosis.aspx

    2021

  13. [21]

    Ogata, Y. (1978). The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Annals of the Institute of Statistical Mathematics , 30(2):243--261

    1978

  14. [22]

    Ogata, Y. (1988). Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes . Journal of the American Statistical Association , 83(401):9--27

    1988

  15. [23]

    Ozaki, T. (1979). Maximum likelihood estimation of Hawkes ' self-exciting point processes. Annals of the Institute of Statistical Mathematics , 31(1):145--155

    1979

  16. [24]

    Salmonella infection - including symptoms, treatment and prevention

    SA Health (2023). Salmonella infection - including symptoms, treatment and prevention . SA Health Salmonella Information Webpage https://www.sahealth.sa.gov.au/wps/wcm/connect/Public+Content/SA+Health+Internet/Conditions/Infectious+diseases/Salmonella+infection/Salmonella+infe...

    2023

  17. [25]

    Sainsbury-Dale , M., Zammit-Mangion , A., and Huser, R. (2024). Likelihood- Free Parameter Estimation with Neural Bayes Estimators . The American Statistician , 78(1):1--14

    2024

  18. [26]

    Schneider, P. J. and Weber, T. A. (2023). Estimation of self-exciting point processes from time-censored data. Physical Review E , 108(1):015303

    2023

  19. [27]

    Shanker, M., Hu, M., and Hung, M. (1996). Effect of data standardization on neural network training. Omega , 24(4):385--397

    1996

  20. [28]

    Shlomovich, L., Cohen, E. A. K., and Adams, N. (2022a). A parameter estimation method for multivariate binned Hawkes processes. Statistics and Computing , 32(6):98

    2022

  21. [29]

    Shlomovich, L., Cohen, E. A. K., Adams, N., and Patel, L. (2022b). Parameter Estimation of Binned Hawkes Processes . Journal of Computational and Graphical Statistics , 31(4):990--1000

    2022

  22. [30]

    and Chen, F

    Stindl, T. and Chen, F. (2018). Likelihood based inference for the multivariate renewal Hawkes process. Computational Statistics & Data Analysis , 123:131--145

    2018

  23. [31]

    Zaheer, M., Kottur, S., Ravanbhakhsh, S., P \'o czos, B., Salakhutdinov, R., and Smola, A. J. (2017). Deep Sets . In Proceedings of the 31st International Conference on Neural Information Processing Systems , NIPS '17, pages 3394--3404, Red Hook, NY, USA. Curran Associates Inc

    2017

  24. [32]

    Zammit-Mangion , A., Sainsbury-Dale , M., and Huser, R. (2024). Neural Methods for Amortized Inference . Annual Review of Statistics and Its Application

    2024

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Reviewed August 7, 2026 · model on record in the stance chip above.