Popular deterministic approximations of posterior distributions from, e.g. the Laplace method, variational Bayes and expectation-propagation, generally rely on symmetric approximating families, 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 a reduced accuracy when considering symmetric approximations. Recent research has moved towards more flexible approximating families which incorporate skewness. However, current solutions are often model specific, lack a general supporting theory, increase the computational complexity of the optimization problem, and do not provide a broadly applicable solution to incorporate skewness in any symmetric approximation. This article addresses such a gap by introducing a general and provably optimal strategy to perturb any off-the-shelf symmetric approximation of a generic posterior distribution. This novel perturbation scheme is derived without additional optimization steps, and yields a similarly tractable approximation within the class of skew-symmetric densities that provably 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 $\sqrt{n}$ factor, in asymptotic regimes. These advancements are illustrated in numerical studies focusing on skewed perturbations of state-of-the-art Gaussian approximations.

翻译：流行的后验分布确定性逼近方法，例如拉普拉斯方法、变分贝叶斯和期望传播，通常依赖于对称逼近族，且常采用高斯分布。这一选择虽便于优化与推断，但通常会影响整体逼近的质量。事实上，即使在基础参数模型中，后验分布也常表现出不对称性，这在使用对称逼近时会导致偏差并降低精度。近期研究已转向纳入偏斜度的更灵活逼近族。然而，现有方案往往针对特定模型、缺乏通用理论支撑、增加了优化问题的计算复杂度，且未能提供广泛适用的方法将偏斜度纳入任意对称逼近。本文通过引入一种通用且可证明最优的策略来填补这一空白，该策略可对任意通用后验分布的现成对称逼近进行扰动。这一新颖的扰动方案无需额外优化步骤即可导出，并在偏斜对称密度类中产生同样易于处理的逼近，可证明其能提升原始对称逼近的有限样本精度。此外，在适当假设下，该方法在渐近情形下至少以$\sqrt{n}$因子提高了逼近精确后验的收敛速率。这些进展在聚焦于最先进高斯逼近的偏斜扰动数值研究中得到了验证。