Bayesian Strategies for Repulsive Spatial Point Processes

TL;DR

The paper tackles Bayesian inference for repulsive spatial point processes with intractable likelihoods, focusing on two canonical models: Strauss Point Process and Determinantal Point Process with Gaussian kernel (DPPG). It analyzes and compares several computational strategies, including the exchange algorithm, Noisy Metropolis-Hastings, and ABC-MCMC, and identifies a critical issue in Shirota and Gelfand's ABC-MCMC, proposing a corrected framework that incorporates intractable multiplicative terms in the acceptance ratio. Monte Carlo estimations and parallelization are proposed to implement the corrected ABC-MCMC, and approximate variants of exchange and NMH are developed for efficiency. Empirical results from simulation studies and a real Duke Forest dataset show that the corrected ABC-MCMC can achieve competitive accuracy and superior mixing for small tolerance levels, while Noisy M-H with a small number of auxiliary draws delivers favorable efficiency; approximate variants offer substantial speed advantages with acceptable accuracy trade-offs, suggesting these methods are viable tools for analyzing doubly-intractable RSPP models. The work advances practical Bayesian inference for repulsive spatial processes and highlights avenues for parallel computing and extension to other point-process families.

Abstract

There is increasing interest to develop Bayesian inferential algorithms for point process models with intractable likelihoods. A purpose of this paper is to illustrate the utility of using simulation based strategies, including Approximate Bayesian Computation (ABC) and Markov Chain Monte Carlo (MCMC) methods for this task. Shirota and Gelfand (2017) proposed an extended version of an ABC approach for Repulsive Spatial Point Processes (RSPP), but their algorithm was not correctly detailed. In this paper, we correct their method and, based on this, we propose a new ABC-MCMC algorithm to which Markov property is introduced compared to a typical ABC method. Though it is generally impractical to use, Monte Carlo approximations can be leveraged for intractable terms. Another aspect of this paper is to explore the use of the exchange algorithm and the noisy Metropolis-Hastings algorithm (Alquier et al., 2016) on RSPP. Comparisons to ABC-MCMC methods are also provided. We find that the inferential approaches outlined above yield good performance for RSPP in both simulated and real data applications and should be considered as viable approaches for the analysis of these models.

Figures (5)

• Figure 1: Left: Strauss point process simulated point locations $\bm{y_1}$. Right: Profile pseudo likelihood estimator $\hat{R}$ for $\bm{y_1}$.
• Figure 2: Determinantal point process: Left: Plot of the point positions contained in the dataset $\bm{y_2}$ which was randomly generated from a DPPG. Right: Trace plots of the Metropolis-Hastings algorithm (M-H), the Exchange algorithm (Ex), the Noisy M-H $K=2$ algorithm ($\text{NMH}_{\text{K2}}$), the approximate Exchange algorithm ($\text{Ex}^{app}$), the approximate Noisy M-H $K=2$ algorithm ($\text{NMH}^{app}_{\text{K2}}$), the F&P ABC-MCMC $p=0.5$ algorithm ($\text{F\&P}_{\text{p0.5}}$) and the corrected S&G ABC-MCMC $p=0.5$ algorithm ($\text{cS\&G}_{\text{p0.5}}$). The first and second rows correspond to the posterior samples of $\tau$ and $\sigma$, respectively.
• Figure 3: Determinantal point process: Posterior density plots of the Ground Truth (GT), the Metropolis-Hastings algorithm (M-H), the Exchange algorithm (Ex), the Noisy M-H $K=2$ algorithm ($\text{NMH}_{\text{K2}}$), the approximate Exchange algorithm ($\text{Ex}^{app}$), the approximate Noisy M-H $K=2$ algorithm ($\text{NMH}^{app}_{\text{K2}}$), the F&P ABC-MCMC algorithm ($\text{F\&P}_{\text{p}1.5}$, $\text{F\&P}_{\text{p}0.5}$) and the corrected S&G ABC-MCMC algorithm ($\text{cS\&G}_{\text{p}1.5}$, $\text{cS\&G}_{\text{p}0.5}$).
• Figure 4: Left: Plot of the tree positions of real Duke Forest dataset $\bm{y}_{obs}$. Right: Profile pseudo likelihood estimator $\hat{R}$ for $\bm{y}_{obs}$.
• Figure 5: Real dataset: Posterior density plots of the Ground Truth (GT), the Exchange algorithm (Ex), the Noisy M-H $K=2$ algorithm ($\text{NMH}_{\text{K}2}$), the F&P ABC-MCMC algorithm ($\text{F\&P}_{\text{p}2.5}$, $\text{F\&P}_{\text{p}1}$, $\text{F\&P}_{\text{p}0.5}$) and the corrected S&G ABC-MCMC algorithm ($\text{cS\&G}_{\text{p}2.5}$, $\text{cS\&G}_{\text{p}1}$, $\text{cS\&G}_{\text{p}0.5}$).