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Laplacian-P-splines (LPS) associate the P-splines smoother and the Laplaceapproximation in a unifying framework for fast and flexible inference under theBayesian paradigm. Gaussian Markov field priors imposed on penalized latentvariables and the Bernstein-von Mises theorem typically ensure a razor-sharpaccuracy of the Laplace approximation to the posterior distribution of thesevariables. This accuracy can be seriously compromised for some unpenalizedparameters, especially when the information synthesized by the prior and thelikelihood is sparse. We propose a refined version of the LPS methodology bysplitting the latent space in two subsets. The first set involves latentvariables for which the joint posterior distribution is approached from anon-Gaussian perspective with an approximation scheme that is particularly welltailored to capture asymmetric patterns, while the posterior distribution forparameters in the complementary latent set undergoes a traditional treatmentwith Laplace approximations. As such, the dichotomization of the latent spaceprovides the necessary structure for a separate treatment of model parameters,yielding improved estimation accuracy as compared to a setting where posteriorquantities are uniformly handled with Laplace. In addition, the proposedenriched version of LPS remains entirely sampling-free, so that it operates ata computing speed that is far from reach to any existing Markov chain MonteCarlo approach. The methodology is illustrated on the additive proportionalodds model with an application on ordinal survey data.

Penalty parameter selection and asymmetry corrections to Laplace approximations in Bayesian P-splines models