Neural Diffusion Intensity Models for Point Process Data

Xinlong Du; Harsha Honnappa; Vinayak Rao

cs.LGFeb 27, 2026

Neural Diffusion Intensity Models for Point Process Data

Authors: Xinlong Du, Harsha Honnappa, Vinayak Rao

Organizations: Edwardson School of Industrial Engineering, Purdue University · Department of Statistics, Purdue University

Abstract

Cox processes model overdispersed point process data via a latent stochastic intensity, but both nonparametric estimation of the intensity model and posterior inference over intensity paths are typically intractable, relying on expensive MCMC methods. We introduce Neural Diffusion Intensity Models, a variational framework for Cox processes driven by neural SDEs. Our key theoretical result, based on enlargement of filtrations, shows that conditioning on point process observations preserves the diffusion structure of the latent intensity with an explicit drift correction. This guarantees the variational family contains the true posterior, so that ELBO maximization coincides with maximum likelihood estimation under sufficient model capacity. We design an amortized encoder architecture that maps variable-length event sequences to posterior intensity paths by simulating the drift-corrected SDE, replacing repeated MCMC runs with a single forward pass. Experiments on synthetic and real-world data demonstrate accurate recovery of latent intensity dynamics and posterior paths, with orders-of-magnitude speedups over MCMC-based methods.

Figures & tables

Figure 1: The variance curve exceeds the mean curve, indicating strong overdispersion of the arrival point process, in the US Bank data from ( SEE Lab, Technion, 2024 ) .

Figure 2: Comparing learned prior drift $b_{\theta}(Z_{t},t)$ (purple) to the ground truth drift $\tilde{b}(Z_{t},t)=0.3(80-Z_{t})$ (yellow).

Figure 3: The learned data samples (red) compared to the true data samples (blue).

Panel of Figure 4 — Figure 4: The learned posterior sample paths (red) are generated using the learned $(b_{\theta},u_{\beta})$ pair, and the baseline “true” posterior sample paths are generated using extensive MCMC simulations (blue).

Figure 5: Train vs. test Wasserstein distance between amortized posterior samples and high-fidelity MCMC posterior samples as a function of the training sample size $n$ . A train-test gap indicates overfitting of the amortized correction $u_{\beta}(\cdots,X)$ , which vanishes as $n$ goes beyond $16$ .

True drift $\tilde{b}$	EM ( $\ell^{2}$ loss)	VI ( $\ell^{2}$ loss)
$0.3(80-Z_{t})$	6.159	5.792
$0.3(80-Z_{t})-5t$	6.721	4.989

Table 1: Comparison of the learned prior quality in terms of

\ell^{2}

deviation of the intensity paths.

Figure 6: The learned data samples (red) compared to the true data samples (blue).

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 7 — Figure 7: Overdispersion affects downstream inferences.

Figure 8: Amortized variational inference architecture for Neural Diffusion Intensity Models.

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Bayesian inference for latent point process intensities often relies on per-dataset sampling or optimization, making repeated inference computationally expensive. We propose a neural-network-based amortized variational framework that learns a reusable inference rule mapping a variable number of replicated point patterns to a distribution over their shared latent intensity. Trained from event times using the point process likelihood, the model learns an amortized inference rule without ground-truth intensity supervision. At test time, it performs inference of the latent distribution in a forward pass. On synthetic data, additional realizations improve intensity recovery and yield narrower credible intervals with broadly stable coverage. Our method is tested on bicycle-sharing data and achieves the highest mean held-out log likelihood among the evaluated methods while requiring substantially less inference time than the Log-Gaussian Cox Process baseline.

Abstract

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