We focus on Bayesian inverse problems with Gaussian likelihood, linear forward model, and priors that can be formulated as a Gaussian mixture. Such a mixture is expressed as an integral of Gaussian density functions weighted by a mixing density over the mixing variables. Within this framework, the corresponding posterior distribution also takes the form of a Gaussian mixture, and we derive the closed-form expression for its posterior mixing density. To sample from the posterior Gaussian mixture, we propose a two-step sampling method. First, we sample the mixture variables from the posterior mixing density, and then we sample the variables of interest from Gaussian densities conditioned on the sampled mixing variables. However, the posterior mixing density is relatively difficult to sample from, especially in high dimensions. Therefore, we propose to replace the posterior mixing density by a dimension-reduced approximation, and we provide a bound in the Hellinger distance for the resulting approximate posterior. We apply the proposed approach to a posterior with Laplace prior, where we introduce two dimension-reduced approximations for the posterior mixing density. Our numerical experiments indicate that samples generated via the proposed approximations have very low correlation and are close to the exact posterior.

翻译：本文研究具有高斯似然、线性前向模型且先验可表述为高斯混合的贝叶斯反问题。此类混合通过混合变量上的混合密度加权高斯密度函数的积分表示。在此框架下，对应的后验分布同样呈现高斯混合形式，我们推导了其后验混合密度的闭式表达式。为从后验高斯混合中采样，我们提出一种两步采样方法：首先从后验混合密度中抽取混合变量，随后根据已采样的混合变量从条件高斯密度中抽取目标变量。然而，后验混合密度在高维情形下采样尤为困难。为此，我们提出用降维近似替代后验混合密度，并给出近似后验在Hellinger距离上的误差界。我们将所提方法应用于具有拉普拉斯先验的后验分布，针对后验混合密度构建了两种降维近似。数值实验表明，通过所提近似生成的样本相关性极低，且与精确后验高度吻合。