Laplace expansions in MCMC algorithms
Chantal Guihenneuc‐Jouyaux, Judith Rousseau · 2004
Complex hierarchical models lead to complicated likelihood and then, in a Bayesian analysis, to complicated posterior distributions. To obtain Bayes estimates such as the posterior mean or bayesian confidence regions, it is therefore necessary to simulate the posterior distribution using a MCMC algorithm. These algorithms get often slower as the number of observations increases, specially in the case of latent variables. To improve the convergence of the algorithm, we propose to decrease the number of parameters to simulate at each iteration by using a Laplace approximation on the nuisance parameters. We therefore study, theoretically the impact that such an approximation has on the target posterior distribution. We prove that the distance between the true target distribution and the approximated one becomes of order O(N −a) with a ∈ (0, 1), a close to 1, as the number of observations N increases. A simulation study illustrates the theoretical results. It turns out, that the approximated algorithm behaves extremely well, at least in the example considered in the paper, which is driven by a study on HIV patients.