Importance Sampling Cross-Validation for LGCP

Cross-validation for Log-Gaussian Cox process (LGCP) is often related to the inherent spatial dependence of point patterns and the prohibitive computational cost of repeated model refitting (Roberts et al. 2017; Liu et al. 2025). While regional thinning provides an intuitive validation mechanism, its reliance on independent thinning assumptions is unattainable in the presence of complex spatial clustering or sparse data (Cronie et al. 2024). Until now, extending this to a dependent thinning mechanism has been theoretically appealing, but the resulting intensity lacks a closed-form analytical solution. Addressing these challenges and leveraging recent proof that the logarithmic score is strictly proper for point processes (Brehmer et al. 2024), we propose a new estimator for the expected log predictive density (ELPD) termed Leave-One-Block-Out-Cross-Validation (LOBO-CV). This framework leverages Pareto-smoothed leave-one-out importance sampling (PS-LOOIS) extended to spatially structured blocks. By factorizing the log-likelihood on the B-th spatial region, we can efficiently compute an estimator of the ELPD, with corresponding standard errors for uncertainty quantification (Vehtari et al. 2017). By reweighting MCMC samples from the full-data posterior, our approach approximates the predictive distribution for specific blocks without requiring full model refitting, provided the Pareto shape parameter (k) remains stable (k ≤ 0.7). In cases where k > 0.7, indicating that the importance weights have reached infinite variance and the approximation is unreliable due to significant divergence between the full-data posterior and the leave-B-out posterior, the framework triggers a targeted model refit for that specific fold. The resulting framework is computationally efficient, independent on MCMC implementation, and provides a robust tool for evaluating LGCP performance in diverse applied spatial settings