[Paper Review] Approximate Bayesian Computation and Model Validation for Repulsive Spatial Point Processes

Shinichiro Shirota, Alan E. Gelfand|arXiv (Cornell University)|Apr 24, 2016

Point processes and geometric inequalities3 citations

TL;DR

This paper proposes an approximate Bayesian computation (ABC) framework for fitting and validating repulsive spatial point processes, specifically Gibbs and determinantal point processes (DPPs), using simulation-based inference. By leveraging easy simulation from these models and summary statistics like $K$-functions, ABC enables robust Bayesian inference and model assessment, outperforming traditional MCMC in computational efficiency and convergence, as demonstrated on a Duke Forest tree dataset with improved model fit for Strauss and DPP models.

ABSTRACT

In many applications involving spatial point patterns, we find evidence of inhibition or repulsion. The most commonly used class of models for such settings are the Gibbs point processes. A recent alternative, at least to the statistical community, is the determinantal point process. Here, we examine model fitting and inference for both of these classes of processes in a Bayesian framework. While usual MCMC model fitting can be available, the algorithms are complex and are not always well behaved. We propose using approximate Bayesian computation (ABC) for such fitting. This approach becomes attractive because, though likelihoods are very challenging to work with for these processes, generation of realizations given parameter values is relatively straightforward. As a result, the ABC fitting approach is well-suited for these models. In addition, such simulation makes them well-suited for posterior predictive inference as well as for model assessment. We provide details for all of the above along with some simulation investigation and an illustrative analysis of a point pattern of tree data exhibiting repulsion. R-code and datasets are included in the supplementary material.

Motivation & Objective

• To address the challenge of intractable likelihoods in repulsive spatial point processes like Gibbs and determinantal point processes (DPPs).
• To develop a unified, simulation-based Bayesian inference approach using Approximate Bayesian Computation (ABC) that bypasses complex MCMC algorithms with intractable normalizing constants.
• To enable posterior inference, model checking, and model comparison through posterior predictive simulation and Monte Carlo tests.
• To demonstrate the method’s effectiveness on real spatial data, particularly the Duke Forest tree dataset, showing improved model fit over Poisson processes.

Proposed method

• Adopt ABC-MCMC for Bayesian inference, using parameter proposals and rejection sampling based on distance between observed and simulated summary statistics.
• Use simulation to generate point patterns from both Gibbs and DPP models given parameter values, avoiding direct likelihood evaluation.
• Employ summary statistics such as the $K$-function and $L(r)-r$ to capture second-order properties and compare observed vs. simulated patterns.
• Apply Monte Carlo tests with $p$-values to assess model adequacy, using $s_r(\mathbf{y})$ to evaluate second-order characteristics.
• Use in-sample relative predictive score (RPS) to compare model performance, with HPP as baseline.
• Tune ABC using percentiles of the distance distribution to select $\epsilon$, and use pilot runs to initialize ABC-MCMC efficiently.

Experimental results

Research questions

• RQ1 Can ABC provide a reliable and computationally efficient alternative to MCMC for Bayesian inference in Gibbs and DPP models with intractable likelihoods?
• RQ2 How well can ABC distinguish between competing repulsive point process models (e.g., Strauss vs. DPP) using summary statistics?
• RQ3 What is the performance of ABC in detecting model inadequacy through posterior predictive checks and $p$-values for summary statistics?
• RQ4 How does model fit vary with different parameterizations and radii in the Strauss process?
• RQ5 Can ABC-based inference be extended to nonhomogeneous repulsive point processes using nonhomogeneous $K$-functions?

Key findings

• The ABC approach successfully recovered the true model structure in simulation studies, with posterior distributions centered near true parameter values.
• For the Duke Forest dataset, the Strauss process with $R=0.053$ showed the best model fit, with $p$-values of 0.3574 ($r=0.03$) and 0.3923 ($r=0.05$), indicating no significant lack of fit.
• The DPP-G and DPP-PE models showed moderate fit with $p$-values around 0.18–0.28, suggesting acceptable but less optimal performance than the Strauss model.
• The HPP was rejected for small radii ($r=0.03, 0.05$) with $p$-values of 0.0428 and 0.0130, indicating strong evidence of repulsion.
• In-sample RPS ratios were near 1.00, but the Strauss model with $R=0.053$ showed the most favorable relative performance, indicating better predictive accuracy.
• The ABC-MCMC algorithm achieved stable convergence without burn-in when initialized with pilot-run estimates, enhancing computational efficiency.