Laplace approximation is a very useful tool in Bayesian inference and it claims a nearly Gaussian behavior of the posterior. \cite{SpLaplace2022} established some rather accurate finite sample results about the quality of Laplace approximation in terms of the so called effective dimension $p$ under the critical dimension constraint $p^{3} \ll n$. However, this condition can be too restrictive for many applications like error-in-operator problem or Deep Neuronal Networks. This paper addresses the question whether the dimensionality condition can be relaxed and the accuracy of approximation can be improved if the target of estimation is low dimensional while the nuisance parameter is high or infinite dimensional. Under mild conditions, the marginal posterior can be approximated by a Gaussian mixture and the accuracy of the approximation only depends on the target dimension. Under the condition $p^{2} \ll n$ or in some special situation like semi-orthogonality, the Gaussian mixture can be replaced by one Gaussian distribution leading to a classical Laplace result. The second result greatly benefits from the recent advances in Gaussian comparison from \cite{GNSUl2017}. The results are illustrated and specified for the case of error-in-operator model.

翻译：拉普拉斯近似是贝叶斯推断中一项非常有用的工具，它声称后验分布具有近似高斯行为。\cite{SpLaplace2022}在临界维数条件$p^{3} \ll n$下，以所谓的有效维度$p$为基础建立了关于拉普拉斯近似质量的一些相当精确的有限样本结果。然而，这一条件对于许多应用（如算子误差问题或深度神经网络）可能过于严格。本文探讨了在目标估计为低维而 nuisance 参数为高维或无限维的情况下，是否可放宽维数条件并改进近似精度的问题。在温和条件下，边际后验可通过高斯混合近似，且近似精度仅取决于目标维度。在条件$p^{2} \ll n$或某些特殊情形（如半正交性）下，高斯混合可被单一高斯分布替代，从而得到经典拉普拉斯结果。第二个结果极大受益于\cite{GNSUl2017}中高斯比较的最新进展。这些结果在算子误差模型中得以说明并具体化。