Monte Carlo approximation of likelihood function in spatial GLMMs through an empirical Bayes method

ABSTRACT

In spatial generalized linear mixed models (SGLMMs), statistical inference encounters problems, since random effects in the model imply high-dimensional integrals to calculate the marginal likelihood function. In this article, we temporarily treat parameters as random variables and express the marginal likelihood function as a posterior expectation. Hence, the marginal likelihood function is approximated using the obtained samples from the posterior density of the latent variables and parameters given the data. However, in this setting, misspecification of prior distribution of correlation function parameter and problems associated with convergence of Markov chain Monte Carlo (MCMC) methods could have an unpleasant influence on the likelihood approximation. To avoid these challenges, we utilize an empirical Bayes approach to estimate prior hyperparameters. We also use a computationally efficient hybrid algorithm by combining inverse Bayes formula (IBF) and Gibbs sampler procedures. A simulation study is conducted to assess the performance of our method. Finally, we illustrate the method applying a dataset of standard penetration test of soil in an area in south of Iran.

KEYWORDS:

• Generalized linear mixed models
• Gibbs sampler
• Inverse Bayes formula
• Likelihood approximation
• Spatial data

MATHEMATICS SUBJECT CLASSIFICATION:

• 62H11
• 62H12
• 62J12
• 62F15

Acknowledgment

The authors are grateful to the reviewer for his valuable and precise comments and suggestions that greatly improved this article.

Funding

This research was in part supported by a grant from IPM (no. 92620036).