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：我们研究从贝叶斯线性反演问题的后验分布中采样，其中参数到观测量的算子 $A$ 计算成本高昂。在许多应用中，$A$ 可以按某种方式分解，从而便于构建一个经济高效的近似算子 $\tilde{A}$。在此框架下，我们提出了 Latent-IMH，一种基于 Metropolis-Hastings 独立（IMH）采样器的采样方法。Latent-IMH 首先使用近似算子 $\tilde{A}$ 生成中间潜变量，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至离线阶段。我们使用 KL 散度和混合时间界对 Latent-IMH 的性能进行了理论分析。通过在若干模型问题上进行数值实验，我们表明，在合理的假设下，其在计算效率上优于诸如 No-U-Turn 采样器（NUTS）等先进方法。在某些情况下，Latent-IMH 可比现有方案快数个数量级。