Approssimazioni asimmetriche delle distribuzioni a posteriori

In Bayesian statistics, routinely implemented deterministic approximations of posterior distributions typically rely on symmetric densities, often taken to be Gaussian. Such a choice facilitates optimization and inference, but may compromise the quality of the overall approximation. In fact, even in simple parametric models, the posterior distribution can displays substantial asymmetries that yield major bias and reduced accuracy when considering symmetric approximations. Recent research has moved toward more flexible classes of approximating densities incorporating skewness. However, current solutions are model specific, lack general supporting theory and usually increase the computational challenges and complexity of the optimization problem. This thesis aims to fill such a gap by developing a general, and theoretically supported, family of skew-symmetric approximations. To accomplish this goal, Chapter 1 demonstrates that in the idealized framework where the true data generating mechanism is known, the posterior distribution converges, in an appropriate sense, to a specific sequence of skew-symmetric distributions at a rate that is faster than the classical Gaussian one derived under the Bernstein-Von Mises theorem. In Chapter 2, these findings further motivate the development of practical plug-in versions that, besides enjoying the same theoretical guarantees, can approximate the posterior distribution in real-world scenarios. The approximations developed in the first two chapters are derived by exploiting asymptotic arguments. Chapter 3 offers a different perspective by introducing a general and provably optimal strategy to perturb any off-the-shelf symmetric approximation of a generic posterior distribution. Such a novel perturbation is derived without additional optimization steps and yields a similarly-tractable approximation within the class of skew-symmetric densities that provably improves