Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonparametric Bayesian intensity estimation for covariate-driven inhomogeneous point processes

Matteo Giordano, Alisa Kirichenko, Judith Rousseau

TL;DR

This paper develops a comprehensive nonparametric Bayesian framework for estimating the intensity of covariate-driven inhomogeneous Poisson processes from a single realisation observed over an expanding domain. It introduces a general theory (covariate-dependent L1 contraction) and derives concrete rates under several priors: bounded covariates with Gaussian priors, unbounded covariates with mixtures of Gaussians, and ergodic covariates with Gaussian wavelet priors, achieving minimax-optimal rates in many settings. It also proposes Polya tree priors tailored to multivariate covariate spaces, enabling adaptive pointwise and global (L1) contraction rates and providing tractable posterior computations. The results rely on a blend of concentration inequalities for spatial averages, entropy bounds, and hypothesis testing strategies adapted to covariate-driven Poisson models, and they extend to several covariate regimes including Gaussian, Poisson tessellations, and finite covariate spaces. Overall, the work advances theoretical guarantees for Bayesian intensity estimation in Cox processes with covariates, offering practical priors and strategies for uncertainty quantification and computation in spatial statistics.

Abstract

This work studies nonparametric Bayesian estimation of the intensity function of an inhomogeneous Poisson point process in the important case where the intensity depends on covariates, based on the observation of a single realisation of the point pattern over a large area. It is shown how the presence of covariates allows to borrow information from far away locations in the observation window, enabling consistent inference in the growing domain asymptotics. In particular, optimal posterior contraction rates under both global and point-wise loss functions are derived. The rates in global loss are obtained under conditions on the prior distribution resembling those in the well established theory of Bayesian nonparametrics, combined with concentration inequalities for functionals of stationary processes to control certain random covariate-dependent loss functions appearing in the analysis. The local rates are derived with an ad-hoc study that builds on recent advances in the theory of Pólya tree priors, extended to the present multivariate setting with a novel construction that makes use of the random geometry induced by the covariates.

Nonparametric Bayesian intensity estimation for covariate-driven inhomogeneous point processes

TL;DR

This paper develops a comprehensive nonparametric Bayesian framework for estimating the intensity of covariate-driven inhomogeneous Poisson processes from a single realisation observed over an expanding domain. It introduces a general theory (covariate-dependent L1 contraction) and derives concrete rates under several priors: bounded covariates with Gaussian priors, unbounded covariates with mixtures of Gaussians, and ergodic covariates with Gaussian wavelet priors, achieving minimax-optimal rates in many settings. It also proposes Polya tree priors tailored to multivariate covariate spaces, enabling adaptive pointwise and global (L1) contraction rates and providing tractable posterior computations. The results rely on a blend of concentration inequalities for spatial averages, entropy bounds, and hypothesis testing strategies adapted to covariate-driven Poisson models, and they extend to several covariate regimes including Gaussian, Poisson tessellations, and finite covariate spaces. Overall, the work advances theoretical guarantees for Bayesian intensity estimation in Cox processes with covariates, offering practical priors and strategies for uncertainty quantification and computation in spatial statistics.

Abstract

This work studies nonparametric Bayesian estimation of the intensity function of an inhomogeneous Poisson point process in the important case where the intensity depends on covariates, based on the observation of a single realisation of the point pattern over a large area. It is shown how the presence of covariates allows to borrow information from far away locations in the observation window, enabling consistent inference in the growing domain asymptotics. In particular, optimal posterior contraction rates under both global and point-wise loss functions are derived. The rates in global loss are obtained under conditions on the prior distribution resembling those in the well established theory of Bayesian nonparametrics, combined with concentration inequalities for functionals of stationary processes to control certain random covariate-dependent loss functions appearing in the analysis. The local rates are derived with an ad-hoc study that builds on recent advances in the theory of Pólya tree priors, extended to the present multivariate setting with a novel construction that makes use of the random geometry induced by the covariates.
Paper Structure (48 sections, 29 theorems, 342 equations)

This paper contains 48 sections, 29 theorems, 342 equations.

Table of Contents

  1. Introduction
  2. Related literature
  3. Our contributions
  4. Covariate-driven Poisson processes and Bayesian inference
  5. Preliminaries and notation
  6. The observation model
  7. Nonparametric Bayesian inference on $\rho$
  8. Posterior contraction rates in global loss
  9. Posterior contraction rates in a covariate-dependent $L^1$-distance
  10. Bounded covariate spaces; Gaussian process priors
  11. Unbounded covariate spaces; nonparametric mixtures of Gaussians priors
  12. $L^1$-posterior contraction rates under ergodicity
  13. Ergodic Gaussian covariates; Gaussian wavelet priors
  14. Poisson random tessellations; Gaussian process priors
  15. Finite covariate spaces
  16. ...and 33 more sections

Key Result

Theorem 3.1

Let $\rho_0\in L^\infty(\mathcal{Z})$ be non-negative valued. Consider data $D^{(n)}=(N^{(n)},Z^{(n)})\sim P^{(n)}_{\rho_0}$ from the observation model Eq:PointProc with $\rho=\rho_0$ and $Z$ a stationary, almost surely locally bounded, random field with invariant measure $\nu$. Assume that the prio for some $C_1>0$. Further assume that there exist measurable sets $\mathcal{R}_n\subseteq\mathcal{R

Theorems & Definitions (69)

  • Remark 2.1: Continuous observations of the covariates
  • Remark 2.2: Stationarity of the covariates
  • Remark 2.3: Deterministic covariates
  • Theorem 3.1
  • Remark 3.2: Small ball probability lower bound
  • Remark 3.3: Empirical loss function
  • Proposition 3.4
  • Remark 3.5: Gaussian priors in the intensity estimation literature
  • Example 3.6: Truncated Gaussian wavelet series priors
  • Proposition 3.7
  • ...and 59 more