We study sampling from posterior distributions in Bayesian linear inverse problems where $A$, the parameters to observables operator, is computationally expensive. In many applications, $A$ can be factored in a manner that facilitates the construction of a cost-effective approximation $\tilde{A}$. In this framework, we introduce Latent-IMH, a sampling method based on the Metropolis-Hastings independence (IMH) sampler. Latent-IMH first generates intermediate latent variables using the approximate $\tilde{A}$, and then refines them using the exact $A$. Its primary benefit is that it shifts the computational cost to an offline phase. We theoretically analyze the performance of Latent-IMH using KL divergence and mixing time bounds. Using numerical experiments on several model problems, we show that, under reasonable assumptions, it outperforms state-of-the-art methods such as the No-U-Turn sampler (NUTS) in computational efficiency. In some cases, Latent-IMH can be orders of magnitude faster than existing schemes.

翻译：我们研究贝叶斯线性逆问题中后验分布的采样，其中参数到观测量的算子$A$计算开销很大。在许多应用中，$A$可以分解，从而有助于构建成本效益高的近似算子$\tilde{A}$。在此框架下，我们提出潜在-IMH，一种基于Metropolis-Hastings独立（IMH）采样的采样方法。潜在-IMH首先使用近似算子$\tilde{A}$生成中间潜在变量，然后使用精确算子$A$对其进行精炼。其主要优势在于将计算成本转移到离线阶段。我们利用KL散度和混合时间界从理论上分析了潜在-IMH的性能。通过在几个模型问题上的数值实验，我们证明在合理假设下，它在计算效率上优于最先进的方法，如无U型转弯采样器（NUTS）。在某些情况下，潜在-IMH可比现有方案快数个数量级。