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REVIEW 5 major objections 5 minor 29 references

Likelihood-Free Estimation for Spatiotemporal Hawkes processes with missing data and application to predictive policing

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

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Pith's one-line read A Wasserstein GAN whose generator is an exact Hawkes simulator can recover the process parameters from crime data with heavy district-level under-reporting, matching 8 of 10 true hotspot cells where maximum likelihood on reported crimes…

desk verdict Plausible idea and honest application framing, but no WGAN parameter estimates are reported and the generator differentiability gap is unaddressed; needs major revision before the central claim is supported. read the letter →

arxiv 2502.07111 v1 pith:TF3EV5GD submitted 2025-02-10 cs.LG stat.APstat.ME

Pramit Das , Moulinath Banerjee , Yuekai Sun This is my paper · ORCID
classification cs.LGstat.APstat.ME MSC 60G5562M30
keywords predictivepolicingHawkesprocessesspatiotemporalpointlikelihood-freeestimationWassersteinGANmissingdatacrimeunder-reportinghotspotprediction
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

Crime data used for predictive policing records only reported crimes, and reporting rates vary sharply by district. Fitting a spatiotemporal Hawkes model to the reported events as if they were complete severely under-estimates the background rate and triggering parameters, which biases downstream hotspot forecasts. The paper proposes to bypass the intractable missing-data likelihood by training a Wasserstein GAN whose generator is an exact Hawkes simulator followed by district-specific thinning, with an LSTM critic comparing simulated and real event streams. On a Bogotá-based simulation, the method recovers parameters well enough that the estimated model's top-10 hotspot cells overlap the true model's by 80 percent, with average relative error around 10 percent, while maximum likelihood on reported data gives badly biased estimates. If the approach holds, it gives an interpretable, likelihood-free route to fitting crime models under differential under-reporting.

What carries the argument

The machinery is a Wasserstein GAN whose generator is not a neural network but the exact stochastic simulator of the Hawkes process followed by thinning. The simulator uses Ogata's thinning algorithm, chosen because its algebraic event-generation expressions can be placed in a deep-learning computational graph; the critic is an LSTM that scores real and generated event streams, and the 1-Lipschitz constraint is enforced with a gradient penalty. The WGAN objective approximates the Wasserstein distance between the reported-data distribution and the distribution generated by the candidate parameters, so minimizing it over $\theta$ is a minimum-distance estimator that never evaluates the missing-data likelihood. The same exact generator is reused for a goodness-of-fit check based on inter-arrival times of reported events.

What would settle it

Instrument the simulator's computational graph during training and compare the Adam update for $\theta$ with a finite-difference estimate of the WGAN loss gradient; if the gradient through the discrete sampling steps vanishes or disagrees, the claimed mechanism for updating $(\mu, \alpha, \beta, \sigma)$ is not in place.

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Extended reading notes

Core claim

The central claim is that WGAN-based likelihood-free estimation with an exact, interpretable generator reliably recovers the parameters of a spatiotemporal Hawkes process even when a substantial fraction of events is missing through location-dependent thinning. The paper demonstrates this by simulating Bogotá crime data from a Hawkes process whose background intensity is centered at a set of known locations, whose triggering kernel is a product of an exponential temporal decay and a Gaussian spatial kernel, and which is thinned by district-specific reporting rates; the WGAN is trained on 10,000 reported streams and the estimated model's expected cell counts are compared with the truth. The estimated parameters identify 8 of the 10 true hotspot cells and give a mean relative absolute error of 10.219 percent across the grid; across 27 nearby parameter settings the top-10 hotspot accuracy averages 74.8 percent. The contrast is with the EM-based maximum likelihood fit on reported data, which in the paper's table yields estimates of $\mu$, $\theta$, and $\omega$ far below their true values and therefore poor hotspot forecasts. This is what the paper is trying to establish: the missing-data likelihood does not have to be written down to do reliable inference, provided the generator mirrors the true data-generating mechanism.

Load-bearing premise

The method assumes that the code that simulates a Hawkes process can be differentiated with respect to the four parameters even though the simulation draws random event counts through rejection sampling; if gradients cannot flow through those discrete steps, the parameter update that drives the WGAN training is not well defined.

Editorial extensions

If this is right

  • If the central claim is correct, fitting Hawkes models for predictive policing no longer requires a tractable missing-data likelihood: a WGAN trained on reported streams can produce interpretable parameter estimates directly.
  • For the Bogotá simulation, the estimated parameters yield hotspot maps whose top-10 cells overlap the true model's by 80 percent, with mean relative cell-count error of about 10.2 percent.
  • Across 27 nearby choices of the true parameters, the top-10 hotspot accuracy averages 74.8 percent, suggesting the recovery is not tied to one hand-picked setting.
  • Because the generator is the exact Hawkes simulator rather than a black-box network, the fitted parameters remain interpretable and can be audited, which matters for accountability in policing applications.
  • Under a mild normalization condition on the reporting function, the Hawkes parameters are identifiable even when the reporting rates are unknown, so the same setup could in principle learn the missingness mechanism.

Reading between the lines

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

  • Beyond the paper: the claimed gradient flow through the exact simulator is the method's unstated load-bearing assumption; if the discrete rejection steps are not reparameterized, a practical variant would need a differentiable surrogate simulator or a gradient-free optimizer to make the same idea work.
  • Beyond the paper: because the reported goodness-of-fit statistic uses inter-arrival times of the temporal projection, it may be insensitive to errors in the spatial kernel shape; a diagnostic that separates background-rate recovery from triggering-kernel recovery would sharpen the claim.
  • Beyond the paper: the identifiability lemma suggests a testable extension in which the WGAN estimates the district-specific reporting rates jointly with the Hawkes parameters under a unit-integral constraint, rather than treating them as known inputs.
  • Beyond the paper: comparing against a missing-data EM or an approximate Bayesian computation baseline on the same simulation would quantify how much of the improvement comes from the WGAN formulation itself versus from using the correct generative model.
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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

5 major / 5 minor

Summary. The paper proposes a Wasserstein GAN (WGAN) based likelihood-free estimator for spatiotemporal Hawkes processes with location-dependent missingness, motivated by predictive policing data in Bogotá. The generator is an exact Hawkes-process simulator followed by district-specific thinning; the critic is an LSTM with gradient penalty; and final parameter estimates are selected from an initialization grid using a chi-square goodness-of-fit test on interarrival times. The empirical section applies the method to simulated crime data and reports hotspot-overlap and mean-absolute-error metrics, claiming in the Discussion that the method reliably recovers the underlying Hawkes parameters.

Significance. If fully supported, the method would address a genuine need, since the missing-data likelihood for this model is intractable and naive maximum likelihood estimation on reported data is badly biased. The simulation-based minimum-distance formulation is coherent in principle, and the paper correctly identifies the limitations of ignoring missingness. However, the central claim is not evidenced: no WGAN parameter estimates are reported, the proposed model-selection criterion cannot separate α and σ², and the generator update in Algorithm 1 is not well-defined for the exact simulator. The validation is also self-referential in that the same generative family is used for simulation and for evaluation, so the reported hotspot metrics do not compensate for the absence of direct parameter recovery results.

major comments (5)
  1. [Algorithm 1, step 17; Section 3] The generator update uses Adam on ∇θ (1/L) Σ f_w(g_θ(z^(j))), but the exact Ogata-thinning simulator involves acceptance/rejection of candidate times and a random number of events, so the mapping from θ to the generated sample is piecewise constant and not differentiable with respect to θ. The paper's assertion in Section 3 that the simulation expressions are 'clear and concise algebraic expressions' suitable for a computational graph does not address the discrete decisions or the variable-length output; padding and masking do not restore differentiability. Without a reparameterization, a REINFORCE-type estimator, or a smoothing device, step 17 is not well-defined.
  2. [Section 3.1.1] The interarrival-time goodness-of-fit criterion cannot identify α and σ² separately. The temporal projection derived in the same section reduces the spatiotemporal Hawkes process to a temporal Hawkes process with mean μ and triggering kernel B g₁(t−t_i), where B = 2πσ². Any parameter pair (α, σ²) with the same product Bα yields the same temporal interarrival distribution, so the chi-square selection between initialization points cannot distinguish such pairs. This invalidates the model selection step and the claim of reliable parameter recovery.
  3. [Section 4] No WGAN parameter estimates are reported. Table 1 reports EM estimates on reported data; Section 4 reports only hotspot overlap (80% for the base case, 74.8% average over robustness runs) and a cell-level MAE. The Discussion's claim that 'the WGAN-based method reliably recovers the underlying parameters of the Hawkes process' is therefore unsupported by any displayed comparison between estimated and true θ.
  4. [Algorithm 1, lines 10-13] The critic loss in line 11 uses f_{w_{t-1}}(\tilde S_j) − f_{w_{t-1}}(S_θ^(j)) + λ(‖∇_{\tilde S_j} f_{w_{t-1}}(\tilde S_j)‖₂ − 1)², where \tilde S_j is the interpolated sample. The standard WGAN-GP critic loss is f(fake) − f(real) + penalty (or its negative, depending on convention); replacing the real term by the interpolated term is not a legitimate variant, so the discriminator update is incorrect as written.
  5. [Supplement 6.2] The identifiability lemma's proof is invalid. Equation (6) assumes that taking t ↓ t1 yields only the triggering term from event 1, but the Hawkes intensity at that limit also contains the background intensity μ_i(x,y) and contributions from all events before t1. Consequently, the cancellation used to derive α_i/μ_i and to compare σ_i and σ*_i is not justified, and the claimed identifiability under an unknown distortion function is not established.
minor comments (5)
  1. [Throughout] There are many typographical and spelling errors, including 'backgroud', 'Jenson-Shannon', 'Kantarovich', and 'e^{-β!}' in the Supplement; these should be corrected.
  2. [Section 3.1.1] The chi-square binning is described only as 'equal-width intervals'; the number of bins and the handling of empty bins are not specified, so the selection results are not reproducible.
  3. [Section 4] The reference to 'the grid map in Figure 12' should be to an existing figure number, and the hotspot list contains a formatting typo with an extra parenthesis.
  4. [Figure 3] The figure caption says 'Comparison of true vs. estimated crime hotspots' but the plots show expected cell counts; the axes and color scale are not labeled, which makes the comparison difficult to interpret.
  5. [Supplement 6.2] In the definition of g_i, the spatial variance σ^{*2}_1 is used for both i=1 and i=2; it should presumably be σ^{*2}_i. Additionally, the proof's equation (7) requires the distortion functions to be positive, which is stated, but the division step should be justified more carefully.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the estimation is a simulation-based minimum-distance procedure whose objective is not equivalent to its inputs.

full rationale

The paper's central claim is that a WGAN trained with an exact spatiotemporal Hawkes simulator plus a missingness mechanism can recover the Hawkes parameters in a likelihood-free way. This is a minimum-distance estimation scheme: the generator produces samples from P_theta after thinning, and the critic estimates the Wasserstein distance between those samples and the observed reported-crime streams. The estimated parameters are then evaluated against the known simulation truth in a separate Monte Carlo validation step. This derivation does not define the target in terms of the estimator, nor does it fit a parameter to a subset of data and then relabel the fit as a prediction. The model family used for simulation is the same as the family used for validation, which limits external validity but is not circular: the WGAN objective and the reported hotspot accuracy are computed from the actual parameterized distributions, not from the training loss itself. The paper does cite prior work by other authors for the crime simulation setup and the EM baseline, but those citations are not load-bearing self-citations and do not supply the main result. The identifiability lemma in Supplement 6.2 is used only to discuss identifiability under unknown reporting rates, not to justify the estimation procedure. Concerns about gradient differentiability of the discrete Hawkes simulator are substantive correctness risks, but they are not circularity: a broken or undefined gradient update would make the algorithm ill-posed, not self-referential. Overall, the derivation chain is self-contained in the sense that the estimator is not constructed from the quantities it claims to predict.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the generative model being correct, on the WGAN objective behaving as a minimum distance estimator, and on the simulator being differentiable. The paper does not validate these assumptions directly. The model selection criterion is not invariant to reparameterization of the trigger kernel, which is the most concrete structural gap.

free parameters (2)
  • Initialization grid for theta = unspecified
    The paper uses multiple starting values for theta (Section 3.1) and selects the best estimate via a goodness-of-fit criterion, but the grid itself is never described, so the reported hotspot results depend on an unstated modeling choice.
  • Chi-square binning for goodness-of-fit = equal-width bins, number unspecified
    The goodness-of-fit criterion in Section 3.1.1 compares histograms of interarrival times using bins of equal width, but the number of bins is not specified. This choice affects which candidate estimate is selected.
assumptions (4)
  • standard math The WGAN minimax objective with gradient penalty approximates the Wasserstein distance, and optimizing it over exact simulators recovers the true distribution P_theta.
    The method relies on WGAN theory from Arjovsky et al. 2017 and Gulrajani et al. 2017, including Kantorovich duality and the gradient penalty approximation.
  • domain assumption Crime data are generated by a spatiotemporal Hawkes process with Gaussian background centers and an exponential-Gaussian triggering kernel, followed by district-specific binomial thinning.
    This is the generative model taken from Akpinar et al. 2021a and used throughout the simulation study in Sections 1 and 4.
  • ad hoc to paper The interarrival-time distribution is sufficient to rank candidate parameter estimates in the goodness-of-fit selection.
    Section 3.1.1 selects theta by comparing interarrival times, but the temporal projection's triggering kernel depends on the product alpha times sigma squared, so alpha and sigma squared are not separately identified by this criterion.
  • ad hoc to paper The exact Hawkes simulator is differentiable with respect to theta through the simulation algorithm.
    The Adam update for theta in Algorithm 1 requires gradients of the discriminator output with respect to the generated event coordinates. The paper asserts algebraic expressions allow a computational graph, but does not explain how gradients flow through Ogata thinning or random event counts.

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Cite this review

Pith. "Pith review of Likelihood-Free Estimation for Spatiotemporal Hawkes processes with missing data and application to predictive policing." pith.science (2026). https://pith.science/paper/TF3EV5GD

@misc{pith2026250207111,
  author       = {Pith},
  title        = {Pith review of: Likelihood-Free Estimation for Spatiotemporal Hawkes processes with missing data and application to predictive policing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TF3EV5GD}},
  note         = {Machine review of arXiv:2502.07111}
}
read the original abstract

With the growing use of AI technology, many police departments use forecasting software to predict probable crime hotspots and allocate patrolling resources effectively for crime prevention. The clustered nature of crime data makes self-exciting Hawkes processes a popular modeling choice. However, one significant challenge in fitting such models is the inherent missingness in crime data due to non-reporting, which can bias the estimated parameters of the predictive model, leading to inaccurate downstream hotspot forecasts, often resulting in over or under-policing in various communities, especially the vulnerable ones. Our work introduces a Wasserstein Generative Adversarial Networks (WGAN) driven likelihood-free approach to account for unreported crimes in Spatiotemporal Hawkes models. We demonstrate through empirical analysis how this methodology improves the accuracy of parametric estimation in the presence of data missingness, leading to more reliable and efficient policing strategies.

Figures

Figures reproduced from arXiv: 2502.07111 by the authors.

Figure 1
Figure 1. Bogota map and crime statistics: figure from Akpinar et al. [2021a] [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Bogota district-wise comparison of expected crimes [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Comparison of true vs. estimated crime hotspots for the Bogota crime data [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Architecture of RNN from Christopher Olah’s blog [PITH_FULL_IMAGE:figures/full_fig_p030_4.png]
Figure 5
Figure 5. Figure 5: Structure of LSTM from Christopher Olah’s blog [PITH_FULL_IMAGE:figures/full_fig_p031_5.png]

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Reference graph

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