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We consider the problem of improving the Gaussian approximate posterior marginals computed by expectation propagation and the Laplace method in latent Gaussian models and propose methods that are similar in spirit to the Laplace approximation of Tierney and Kadane (1986). We show that in the case of sparse Gaussian models, the computational complexity of expectation propagation can be made comparable to that of the Laplace method by using a parallel updating scheme. In some cases, expectation propagation gives excellent estimates where the Laplace approximation fails. Inspired by bounds on the correct marginals, we arrive at factorized approximations, which can be applied on top of both expectation propagation and the Laplace method. The factorized approximations can give nearly indistinguishable results from the non-factorized approximations and their computational complexity scales linearly with the number of variables. We experienced that the expectation propagation based marginal approximations we introduce are typically more accurate than the methods of similar complexity proposed by Rue et al. (2009). [ABSTRACT FROM AUTHOR]

Abstract: Bayesian logistic regression with a multivariate Laplace prior is introduced as a multivariate approach to the analysis of neuroimaging data. It is shown that, by rewriting the multivariate Laplace distribution as a scale mixture, we can incorporate spatio-temporal constraints which lead to smooth importance maps that facilitate subsequent interpretation. The posterior of interest is computed using an approximate inference method called expectation propagation and becomes feasible due to fast inversion of a sparse precision matrix. We illustrate the performance of the method on an fMRI dataset acquired while subjects were shown handwritten digits. The obtained models perform competitively in terms of predictive performance and give rise to interpretable importance maps. Estimation of the posterior of interest is shown to be feasible even for very large models with thousands of variables. [Copyright &y& Elsevier]