Gaussian approximations are routinely employed in Bayesian statistics to ease inference when the target posterior is intractable. Although these approximations are asymptotically justified by Bernstein-von Mises type results, in practice the expected Gaussian behavior may poorly represent the shape of the posterior, thus affecting approximation accuracy. Motivated by these considerations, we derive an improved class of closed-form approximations of posterior distributions which arise from a new treatment of a third-order version of the Laplace method yielding approximations in a tractable family of skew-symmetric distributions. Under general assumptions which account for misspecified models and non-i.i.d. settings, this family of approximations is shown to have a total variation distance from the target posterior whose rate of convergence improves by at least one order of magnitude the one established by the classical Bernstein-von Mises theorem. Specializing this result to the case of regular parametric models shows that the same improvement in approximation accuracy can be also derived for polynomially bounded posterior functionals. Unlike other higher-order approximations, our results prove that it is possible to derive closed-form and valid densities which are expected to provide, in practice, a more accurate, yet similarly-tractable, alternative to Gaussian approximations of the target posterior, while inheriting its limiting frequentist properties. We strengthen such arguments by developing a practical skew-modal approximation for both joint and marginal posteriors that achieves the same theoretical guarantees of its theoretical counterpart by replacing the unknown model parameters with the corresponding MAP estimate. Empirical studies confirm that our theoretical results closely match the remarkable performance observed in practice, even in finite, possibly small, sample regimes.

翻译：高斯近似在贝叶斯统计中被常规用于简化后验分布难以处理时的推断问题。尽管这些近似通过伯恩斯坦-冯·米塞斯类型的结果在渐近意义上得到验证，但在实际中，预期的高斯行为可能无法良好地反映后验分布的形状，从而影响近似精度。基于这些考虑，我们推导出一类改进的后验分布闭合形式近似，其源于对拉普拉斯方法三阶版本的新处理，从而在可处理的偏斜对称分布族中生成近似。在考虑模型误设和非独立同分布设定的一般假设下，我们证明了这类近似与目标后验之间的总变差距离的收敛速率，相比经典伯恩斯坦-冯·米塞斯定理所建立的速率至少提升一个数量级。将该结果特化到正则参数模型情形表明，对于多项式有界后验泛函，同样可推导出近似精度的类似提升。与其他高阶近似不同，我们的结果证明，可以推导出闭合形式且有效的密度函数，这些函数在实践中预计能提供比目标后验的高斯近似更精确且同样易于处理的替代方案，同时继承其极限频率学派性质。我们通过开发适用于联合后验和边缘后验的实用偏斜模态近似来强化这些论证，该近似通过将未知模型参数替换为相应的最大后验估计，达到了与其理论对应物相同的理论保证。实证研究表明，即使在有限（可能较小）样本场景下，我们的理论结果也能紧密匹配实践中观察到的显著表现。