Model and variable selection are important topics in Bayesian inference. In particular, the selection of different fixed, random effects and hyperparameters in hierarchical models can be difficult because of their complex structure. In this paper, we introduce the use of approximate Bayesian inference for model and variable selection for hierarchical Bayesian models. The approach is based on the application of Markov chain Monte Carlo methods on the model space. In particular, the Metropolis-Hastings algorithm is used to sample from the set of model indices. In this way, the marginal likelihood is used to compute the acceptance probability so that it is not required to estimate the parameter models directly. For this, the integrated nested Laplace approximation (INLA) is used because it provides accurate estimates of the marginal likelihood and it can also provide estimates of the posterior marginal of the model parameters. This method can be applied not only to variable selection but to a wide range of problems subject to model uncertainty. To illustrate the potential of this approach, examples on variable selection, changepoint models and log-Gaussian Cox processes are developed.

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