[Paper Review] INLA or MCMC? A Tutorial and Comparative Evaluation for Spatial Prediction in log-Gaussian Cox Processes

Benjamín M. Taylor, Peter J. Diggle|arXiv (Cornell University)|Feb 8, 2012

Statistical Methods and Bayesian Inference20 references4 citations

TL;DR

This paper compares Markov chain Monte Carlo (MCMC) and integrated nested Laplace approximation (INLA) for Bayesian spatial prediction in log-Gaussian Cox processes, using a simulation study to evaluate predictive accuracy and computational speed. Despite INLA's faster runtime (4 minutes vs. 20 minutes for 100,000 MALA iterations), MCMC achieved significantly higher predictive accuracy, with median MSE 3× lower than INLA2 and 1.1× lower than INLA3, challenging the assumption that INLA is both faster and more robust in this context.

ABSTRACT

We investigate two options for performing Bayesian inference on spatial log-Gaussian Cox processes assuming a spatially continuous latent field: Markov chain Monte Carlo (MCMC) and the integrated nested Laplace approximation (INLA). We first describe the device of approximating a spatially continuous Gaussian field by a Gaussian Markov random field on a discrete lattice, and present a simulation study showing that, with careful choice of parameter values, small neighbourhood sizes can give excellent approximations. We then introduce the spatial log-Gaussian Cox process and describe MCMC and INLA methods for spatial prediction within this model class. We report the results of a simulation study in which we compare MALA and the technique of approximating the continuous latent field by a discrete one, followed by approximate Bayesian inference via INLA over a selection of 18 simulated scenarios. The results question the notion that the latter technique is both significantly faster and more robust than MCMC in this setting; 100,000 iterations of the MALA algorithm running in 20 minutes on a desktop PC delivered greater predictive accuracy than the default \verb=INLA= strategy, which ran in 4 minutes and gave comparative performance to the full Laplace approximation which ran in 39 minutes.

Motivation & Objective

• To objectively compare MCMC and INLA for Bayesian inference in spatial log-Gaussian Cox processes.
• To evaluate the trade-off between computational speed and predictive accuracy in spatial prediction.
• To assess the reliability of INLA’s approximations against MCMC’s asymptotically exact inference.
• To provide a tutorial on technical aspects of inference for latent Gaussian processes in spatial models.
• To challenge the prevailing assumption that INLA is both faster and more robust than MCMC in this setting.

Proposed method

• Approximate a spatially continuous Gaussian process using a Gaussian Markov random field (GMRF) on a regular grid, validated via simulation.
• Use the Metropolis-adjusted Langevin algorithm (MALA) for MCMC inference on the latent field, with 100,000 iterations per simulation.
• Apply INLA to the GMRF approximation using three variants: default INLA, full Laplace approximation, and a modified INLA2 strategy.
• Evaluate predictive performance using mean squared error (MSE) of estimated predictive probabilities across 18 simulated scenarios.
• Compare results across fixed spatial intensity functions λ₁(s) and λ₂(s) to assess robustness and bias.
• Implement all methods in the R package lgcp, with INLA integration via wrapper functions.

Experimental results

Research questions

• RQ1 Does INLA provide more accurate predictive probabilities than MCMC in spatial log-Gaussian Cox processes?
• RQ2 How does the computational speed of INLA compare to MCMC when both are used for spatial prediction?
• RQ3 To what extent do INLA’s approximations introduce bias compared to MCMC’s asymptotically exact inference?
• RQ4 How sensitive are the predictive results to the choice of spatial intensity function (λ₁(s) vs. λ₂(s))?
• RQ5 Can the GMRF approximation of a continuous Gaussian field yield reliable inference for spatial prediction?

Key findings

• MALA with 100,000 iterations achieved greater predictive accuracy than the default INLA strategy, which ran in 4 minutes.
• The median relative increase in MSE comparing MALA to INLA2 was 3 across all 18 scenarios, indicating MCMC’s superior accuracy.
• For fixed spatial λ₂(s), MALA outperformed INLA3 with a median MSE increase of 1.37, while for λ₁(s), INLA3 performed better with a median increase of 0.87.
• Despite INLA’s speed advantage, MALA showed less bias in predictive probability estimation, particularly in the central regions of the spatial domain.
• The INLA2 method exhibited noticeable upward bias in the center of the spatial domain, suggesting systematic error in approximation.
• The full Laplace approximation, which took 39 minutes, performed comparably to INLA, indicating that INLA’s speed does not necessarily come at the cost of accuracy in this setting.