Skew-symmetric approximations of posterior distributions

Routinely-implemented deterministic approximations of posterior distributions from, e.g., Laplace method, variational Bayes and expectation-propagation, generally rely on symmetric approximating densities, often taken to be Gaussian. This choice facilitates optimization and inference, but typically affects the quality of the overall approximation. In fact, even in basic parametric models, the posterior distribution often displays asymmetries that yield bias and reduced accuracy when considering symmetric approximations. In this work we introduce a general strategy to perturb any off-the-shelf symmetric approximation of a generic posterior distribution. Crucially, this  perturbation 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 approximation. Furthermore, under suitable assumptions, improves its convergence rate to the exact posterior by at least a \sqrt{n} factor, in asymptotic regimes. Our results are illustrated in numerical studies focusing on skewed perturbations of state-of-the-art Gaussian approximations.  Joint work with Daniele Durante and Francesco Pozza