线性贝叶斯反演问题的连续高斯混合解及其在 Laplace 先验中的应用
Continuous Gaussian mixture solution for linear Bayesian inversion with application to Laplace priors
AI总结：
针对线性贝叶斯反演问题，提出基于高斯混合的两步采样方法及降维近似策略，推导了后验混合密度闭式表达式，并在 Laplace 先验下验证了近似样本的低相关性与高精度。
摘要：
我们关注具有高斯似然、线性前向模型以及可表述为高斯混合的先验的贝叶斯反演问题。这种混合表示为关于混合变量的、由混合密度加权的高斯密度函数的积分。在此框架下，相应的后验分布也表现为高斯混合形式，我们推导了其后验混合密度的闭式表达式。为了从后验高斯混合中采样，我们提出了一种两步采样方法。首先，我们从后验混合密度中采样混合变量，然后从以采样混合变量为条件的高斯密度中采样感兴趣的变量。然而，后验混合密度相对难以采样，特别是在高维情况下。因此，我们提出用降维近似替换后验混合密度，并为所得的近似后验提供了 Hellinger 距离上的界限。我们将所提方法应用于带有 Laplace 先验的后验，其中我们为后验混合密度引入了两种降维近似。我们的数值实验表明，通过所提近似生成的样本具有极低的相关性，且接近精确后验。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
