Scalable skewed Bayesian inference for latent Gaussian models using INLA and variational Bayes

Approximate Bayesian inference for the class of latent Gaussian models can be achieved effciently with integrated nested Laplace approximations (INLA). Based on recent reformulations in the INLA methodology, we propose a further extension that is necessary in some cases like heavy-tailed likelihoods or binary regression with imbalanced data, among others. This extension formulates a skewed version of the Laplace method, such that some marginals are skewed and some are kept Gaussian, while the dependence is maintained with the Gaussian copula from the Laplace method. Our approach is formulated to be scalable in model complexity and data size, using a variational inferential framework enveloped in INLA. We illustrate the necessity and performance using simulated cases, as well as applications to a rare disease where class imbalance is naturally present, and a large diabetes dataset.