Summary
This paper tackles how to estimate event rates that change depending on outside conditions, even when those rates can get very low or drop to zero. The authors develop a new way to estimate these event rates without needing the usual assumption that rates stay above zero everywhere. They use deep neural networks, which can learn complex underlying patterns in the data, and show that this leads to better accuracy and faster learning compared to traditional methods. Their techniques also adapt to cases where events happen on simpler geometric shapes within complex spaces. They tested their method on simulated data and real brain activity data, finding it performs better than common alternatives.
What this means in practice
- •For neural data analysts: Model neural spiking activity from stimulus and covariates with higher accuracy and better prediction on unseen data.
- •For event-driven system engineers: Estimate variable event rates dependent on external conditions without assuming minimum event frequencies, improving system monitoring.
Abstract
We consider nonparametric maximum likelihood estimation of the intensity function of a Poisson process, allowing the intensity to depend on observed covariates. Such models describe how event rates vary with external conditions, including conditions under which the intensity becomes arbitrarily small or vanishes. However, risk analysis based on the Kullback-Leibler (KL) divergence typically requires the true intensity to be bounded away from zero, a restrictive assumption in applications. We develop a unified approach that avoids this assumption and derive an oracle inequality for the covariate-averaged Hellinger distance between the true intensity and the nonparametric maximum likelihood estimator (NPMLE). As an application of the oracle inequality, we establish convergence rates for NPMLEs based on deep neural networks (DNNs). Our results show that, as in nonparametric regression, the ability of DNNs to capture underlying structure leads to improved convergence rates. Specifically, DNN-based NPMLEs mitigate the curse of dimensionality when the true intensity has a compositional structure and automatically adapt to a low-dimensional Riemannian manifold structure of the event space. For intensity functions with a compositional structure, DNN-based NPMLEs achieve nearly minimax-optimal convergence rates without explicitly incorporating this structure into the network architecture. Simulations demonstrate that DNN-based estimators outperform kernel estimators in most settings, while manifold kernels remain competitive in some designs. Finally, we analyze stimulus-evoked neural spiking using data from the Allen Brain Observatory Visual Coding Neuropixels project. DNN-based NPMLEs provide better out-of-sample performance than a conditional kernel estimator, as evaluated by held-out log-likelihood.