Approximate Bayesian Inference for Latent Gaussian Models

Abstract

This thesis consists of five papers, presented in chronological order. Their content is summarised in this section.

Paper I introduces the approximation tool for latent GMRF models and discusses, in particular, the approximation for the posterior of the hyperparameters θ in equation (1). It is shown that this approximation is indeed very accurate, as even long MCMC runs cannot detect any error in it. A Gaussian approximation to the density of χi|θ, y is also discussed. This appears to give reasonable results and it is very fast to compute. However, slight errors are detected when comparing the approximation with long MCMC runs. These are mostly due to the fact that a possible - skewed density is approximated via a symmetric one. Paper I presents also some details about sparse matrices algorithms.

The core of the thesis is presented in Paper II. Here most of the remaining issues present in Paper I are solved. Three different approximation for χi|θ, y with different degrees of accuracy and computational costs are described. Moreover, ways to assess the approximation error and considerations about the asymptotical behaviour of the approximations are also discussed. Through a series of examples covering a wide range of commonly used latent GMRF models, the approximations are shown to give extremely accurate results in a fraction of the computing time used by MCMC algorithms.

Paper III applies the same ideas as Paper II to generalised linear mixed models where χ represents a latent variable at n spatial sites on a two dimensional domain. Out of these n sites k, with n >> k , are observed through data. The n sites are assumed to be on a regular grid and wrapped on a torus. For the class of models described in Paper III the computations are based on discrete Fourier transform instead of sparse matrices. Paper III illustrates also how marginal likelihood π (y) can be approximated, provides approximate strategies for Bayesian outlier detection and perform approximate evaluation of spatial experimental design.

Paper IV presents yet another application of the ideas in Paper II. Here approximate techniques are used to do inference on multivariate stochastic volatility models, a class of models widely used in financial applications. Paper IV discusses also problems deriving from the increased dimension of the parameter vector θ, a condition which makes all numerical integration more computationally intensive. Different approximations for the posterior marginals of the parameters θ, π(θi)|y), are also introduced. Approximations to the marginal likelihood π(y) are used in order to perform model comparison.

Finally, Paper V is a manual for a program, named inla which implements all approximations described in Paper II. A large series of worked out examples, covering many well known models, illustrate the use and the performance of the inla program. This program is a valuable instrument since it makes most of the Bayesian inference techniques described in this thesis easily available for everyone.