Monte Carlo approximation through Gibbs output in generalized linear mixed models

Geyer (J. Roy. Statist. Soc. 56 (1994) 291) proposed Monte Carlo method to approximate the whole likelihood function. His method is limited to choosing a proper reference point. We attempt to improve the method by assigning some prior information to the parameters and using the Gibbs output to evaluate the marginal likelihood and its derivatives through a Monte Carlo approximation. Vague priors are assigned to the parameters as well as the random effects within the Bayesian framework to represent a non-informative setting. Then the maximum likelihood estimates are obtained through the Newton Raphson method. Thus, out method serves as a bridge between Bayesian and classical approaches. The method is illustrated by analyzing the famous salamander mating data by generalized linear mixed models.Generalized linear mixed model Monte Carlo Newton Raphson Monte Carlo relative likelihood Gibbs sampler Metropolis-Hastings algorithm

Parallel variational Bayes for large datasets with an application to generalized linear mixed models

<div><p>The article develops a hybrid Variational Bayes algorithm that combines the mean-field and stochastic linear regression fixed-form Variational Bayes methods. The new estimation algorithm can be used to approximate any posterior without relying on conjugate priors. We propose a divide and recombine strategy for the analysis of large datasets, which partitions a large dataset into smaller subsets and then combines the variational distributions that have been learnt in parallel on each separate subset using the hybrid Variational Bayes algorithm. We also describe an efficient model selection strategy using cross validation, which is straightforward to implement as a by-product of the parallel run. The proposed method is applied to fitting generalized linear mixed models. The computational efficiency of the parallel and hybrid Variational Bayes algorithm is demonstrated on several simulated and real datasets. Supplementary material for this article is available online.</p></div