Skew-symmetric approximations of posterior distributions

Popular deterministic approximations of posterior distributions from, e.g. the Laplace method, variational Bayes and expectation-propagation, generally rely on symmetric families, often taken to be Gaussian. This choice facilitates optimization and inference, but typically affects the quality of the approximation. In fact, even in basic parametric models, posterior distributions often display asymmetries that yield bias and reduced accuracy when considering symmetric approximations. Recent research has moved towards more flexible approximations incorporating skewness. However, current solutions are often model specific, lack general supporting theory, increase the computational complexity of the optimization problem, and do not provide broadly applicable solutions to incorporate skewness in any symmetric approximation. This article addresses such a gap by introducing a general and provably optimal strategy to perturb any symmetric approximation of a generic posterior distribution. The proposed solution is derived without additional optimization steps, and yields a similarly tractable approximation within the class of skew-symmetric densities that enhances the finite sample accuracy of the original symmetric counterpart. Furthermore, under suitable assumptions, it improves the convergence rate to the exact posterior by at least a √n factor, in asymptotic regimes. These advancements are illustrated in numerical studies focusing on skewed perturbations of state-of-the-art Gaussian approximations.