[Paper Review] Spatial Models with the Integrated Nested Laplace Approximation within Markov Chain Monte Carlo

Virgilio Gómez‐Rubio, Francisco Palmí‐Perales|arXiv (Cornell University)|Feb 13, 2017

Soil Geostatistics and Mapping23 references3 citations

TL;DR

This paper proposes integrating the Integrated Nested Laplace Approximation (INLA) within the Metropolis-Hastings algorithm to enable multivariate posterior inference for complex spatial models not natively supported by R-INLA. By using INLA to efficiently compute conditional posteriors while sampling key parameters via MCMC, the method achieves fast, accurate joint posterior estimation for spatial econometrics and disease mapping models with multiple spatial structures, demonstrating strong agreement with full MCMC results.

ABSTRACT

The Integrated Nested Laplace Approximation (INLA) is a convenient way to obtain approximations to the posterior marginals for parameters in Bayesian hierarchical models when the latent effects can be expressed as a Gaussian Markov Random Field (GMRF). In addition, its implementation in the R-INLA package for the R statistical software provides an easy way to fit models using INLA in practice. R-INLA implements a number of widely used latent models, including several spatial models. In addition, R-INLA can fit models in a fraction of the time than other computer intensive methods (e.g. Markov Chain Monte Carlo) take to fit the same model. Although INLA provides a fast approximation to the marginals of the model parameters, it is difficult to use it with models not implemented in R-INLA. It is also difficult to make multivariate posterior inference on the parameters of the model as INLA focuses on the posterior marginals and not the joint posterior distribution. In this paper we describe how to use INLA within the Metropolis-Hastings algorithm to fit spatial models and estimate the joint posterior distribution of a reduced number of parameters. We will illustrate the benefits of this new method with two examples on spatial econometrics and disease mapping where complex spatial models with several spatial structures need to be fitted.

Motivation & Objective

• To address the limitation of R-INLA in handling complex spatial models with multiple spatial components not natively implemented in the package.
• To enable multivariate posterior inference on key parameters—such as spatial weights and disease-specific effects—by combining INLA with MCMC.
• To reduce computational burden compared to full MCMC by leveraging INLA’s speed for conditional posteriors while sampling only a small subset of parameters.
• To demonstrate the method’s effectiveness on real-world spatial models in spatial econometrics and disease mapping with shared and specific spatial effects.
• To provide a practical, scalable alternative to full MCMC for large spatial datasets with complex dependence structures.

Proposed method

• Use the Metropolis-Hastings algorithm to sample from the joint posterior distribution of a reduced set of key parameters (e.g., spatial weights, precision parameters) in spatial models.
• At each MCMC iteration, condition on the current values of the sampled parameters and use INLA to compute the full conditional posterior distributions of all other model parameters.
• Leverage R-INLA’s efficient computation of posterior marginals for Gaussian Markov Random Fields (GMRFs), which underlie most spatial latent effects.
• Combine conditional posterior marginals across MCMC iterations via Bayesian model averaging to estimate the marginal distributions of non-sampled parameters.
• Apply the method to models with shared and specific spatial effects, such as intrinsic CAR models for multiple diseases or spatial econometrics with multiple spatial dependence structures.
• Ensure convergence and mixing by reducing the number of parameters simulated in MCMC, relying on INLA for fast, accurate conditional posterior approximations.

Experimental results

Research questions

• RQ1 Can INLA be effectively combined with MCMC to enable joint posterior inference for spatial models not directly supported by R-INLA?
• RQ2 How does the performance of INLA within MCMC compare to full MCMC in terms of accuracy and computational efficiency for complex spatial models?
• RQ3 To what extent can this hybrid approach capture multivariate dependencies, such as correlations between spatial weights or disease-specific effects?
• RQ4 Can this method be applied to real-world spatial data with multiple spatial components, such as joint disease mapping with shared and specific spatial patterns?
• RQ5 Does the method maintain accuracy in estimating posterior marginals and joint distributions when the number of parameters to be sampled is small but the model structure is complex?

Key findings

• The INLA within MCMC approach produced posterior marginals for key parameters that showed strong agreement with results from full MCMC, as evidenced by overlapping density estimates in Figure 5.
• The method successfully estimated the joint posterior distribution of the disease-specific weights (δ^(1), δ^(2), δ^(3)), enabling multivariate inference not possible with standard INLA.
• A strong positive correlation was observed between δ^(1) and δ^(2), reflecting the similar spatial patterns of lip/oral/pharynx and esophagus cancers, which the model successfully captured.
• The marginal distributions of disease-specific intercepts and precision parameters were accurately estimated by averaging conditional posteriors across MCMC iterations, as shown in Figure 6.
• The approach significantly reduced MCMC computational cost by minimizing the number of parameters to be updated in each iteration, while maintaining high accuracy through fast INLA-based conditional updates.
• The method proved effective for complex spatial models with shared and specific spatial effects, such as joint disease mapping with multiple cancers and spatial econometrics models with multiple spatial dependence structures.