Neural Diffusion Intensity Models for Point Process Data
| Authors | Xinlong Du et al. |
| Year | 2026 |
| Field | Machine Learning |
| arXiv | 2602.24083 |
| Download | |
| Categories | cs.LG, stat.ML |
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.
Engineering Breakdown
Plain English
This paper solves a fundamental problem in modeling point process data (like event sequences or arrival times): Cox processes use hidden stochastic intensity functions to capture overdispersion, but inferring these intensities and computing posteriors has been computationally intractable, requiring expensive MCMC sampling. The authors introduce Neural Diffusion Intensity Models, which use neural stochastic differential equations (SDEs) inside a variational inference framework to make this tractable. Their key theoretical breakthrough proves that conditioning on point process observations preserves the diffusion structure of the latent intensity with an explicit drift correction—meaning the variational family provably contains the true posterior, so maximizing the ELBO gives you maximum likelihood estimation if your model is expressive enough. This replaces MCMC with scalable amortized neural inference.
Core Technical Contribution
The core contribution is a variational framework that reformulates Cox process inference as neural SDE learning with provable posterior coverage. Using filtration theory, the authors show that when you condition a diffusion process on point observations, the posterior intensity still follows a diffusion with a computable drift term—a non-obvious result that ensures the variational approximation can match the true posterior exactly under sufficient model capacity. This is fundamentally different from prior approaches that either use MCMC (slow), Laplace approximations (loose bounds), or Gaussian processes (limited scalability). The amortized encoder architecture learns to map observed point patterns directly to posterior intensity paths without sampling, making inference fast and scalable to long sequences.
How It Works
The system takes a sequence of observed event times as input and produces a posterior distribution over the latent intensity function. The amortized encoder (a neural network) processes the event sequence to output parameters of a variational posterior—specifically, the drift and diffusion coefficients of a posterior SDE. The key technical mechanism leverages the enlargement of filtrations theorem: if the intensity follows dI_t = μ(I_t, t)dt + σ(I_t, t)dW_t, then conditioning on observed events changes only the drift to μ + (drift correction term), leaving the diffusion structure unchanged. A neural SDE is then integrated forward using these corrected parameters to produce intensity samples. The ELBO combines a reconstruction term (log-likelihood of events given intensity) and a KL divergence term between the posterior and prior SDE processes. Training maximizes this ELBO, and theory guarantees this equals maximum likelihood estimation when the neural network is sufficiently expressive.
Production Impact
For production systems handling event data (e.g., user activity sequences, system logs, sensor events), this replaces weeks of MCMC sampling with millisecond-scale neural inference. If you're currently fitting Cox processes via MCMC on datasets with thousands of events, switching to Neural Diffusion Intensity Models would reduce inference latency by orders of magnitude while providing principled uncertainty quantification through the posterior SDE samples. The amortized architecture means after training once on your data distribution, you can instantly infer posteriors for new event sequences without retraining. The practical trade-off is that you need sufficient neural network capacity—the theoretical guarantee only holds if the network is expressive enough—and you must choose appropriate SDE parameterizations (drift and diffusion forms) a priori. Integration into existing pipelines is moderate: replace MCMC samplers with forward passes through the encoder and SDE integrator, requiring only changes to the inference module rather than data processing or feature engineering.
Limitations and When Not to Use This
The paper assumes you can specify the prior intensity SDE form (drift and diffusion functions) correctly; misspecification here will violate the posterior coverage guarantees. The theoretical result requires sufficient model capacity, but in practice there's no simple way to verify this, so you may converge on a poor local optimum if the network is too small. The approach is designed for moderate-length sequences where numerical SDE integration is tractable; extremely long sequences or high-dimensional intensity processes (e.g., multi-type point processes with many event classes) will face scalability challenges not addressed here. The paper likely lacks experiments on real-world datasets with known ground truth—validation relies heavily on simulation studies—so robustness to realistic data distribution shift is unclear. Follow-up work should address multi-type/marked point processes, adaptive SDE parameterization, and comparison with modern neural alternatives like neural ODEs or transformer-based sequence models for point processes.
Research Context
This work builds directly on Cox processes from the 1950s (foundational for overdispersed event data) and modern variational inference theory, but its key novelty is applying filtration enlargement results—a classical theorem in probability theory—to prove posterior coverage for neural SDEs in the point process setting. It extends recent work on neural SDEs and neural point processes (which use neural networks to learn intensity directly) by combining them with formal variational bounds. The paper benchmarks against traditional Cox process inference (MCMC) and likely compares to recent neural point process baselines like Neural Hawkes processes or Transformer Hawkes models on synthetic datasets. This opens a research direction toward combining filtration theory with modern deep generative models, suggesting similar results might apply to other partially-observed latent diffusion systems (e.g., stochastic volatility in finance, latent disease dynamics in epidemiology).
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