  Various computational challenges arise when applying Bayesian inference
approaches to complex hierarchical models. Sampling-based inference methods,
such as Markov Chain Monte Carlo strategies, are renowned for providing
accurate results but with high computational costs and slow or questionable
convergence. On the contrary, approximate methods like the Integrated Nested
Laplace Approximation (INLA) construct a deterministic approximation to the
univariate posteriors through nested Laplace Approximations. This method
enables fast inference performance in Latent Gaussian Models, which encode a
large class of hierarchical models. R-INLA software mainly consists of three
strategies to compute all the required posterior approximations depending on
the accuracy requirements. The Simplified Laplace approximation (SLA) is the
most attractive because of its speed performance since it is based on a Taylor
expansion up to order three of a full Laplace Approximation. Here we enhance
the methodology by simplifying the computations necessary for the skewness and
modal configuration. Then we propose an expansion up to order four and use the
Extended Skew Normal distribution as a new parametric fit. The resulting
approximations to the marginal posterior densities are more accurate than those
calculated with the SLA, with essentially no additional cost.
