  Laplacian-P-splines (LPS) associate the P-splines smoother and the Laplace
approximation in a unifying framework for fast and flexible inference under the
Bayesian paradigm. Gaussian Markov field priors imposed on penalized latent
variables and the Bernstein-von Mises theorem typically ensure a razor-sharp
accuracy of the Laplace approximation to the posterior distribution of these
variables. This accuracy can be seriously compromised for some unpenalized
parameters, especially when the information synthesized by the prior and the
likelihood is sparse. We propose a refined version of the LPS methodology by
splitting the latent space in two subsets. The first set involves latent
variables for which the joint posterior distribution is approached from a
non-Gaussian perspective with an approximation scheme that is particularly well
tailored to capture asymmetric patterns, while the posterior distribution for
parameters in the complementary latent set undergoes a traditional treatment
with Laplace approximations. As such, the dichotomization of the latent space
provides the necessary structure for a separate treatment of model parameters,
yielding improved estimation accuracy as compared to a setting where posterior
quantities are uniformly handled with Laplace. In addition, the proposed
enriched version of LPS remains entirely sampling-free, so that it operates at
a computing speed that is far from reach to any existing Markov chain Monte
Carlo approach. The methodology is illustrated on the additive proportional
odds model with an application on ordinal survey data.
