利用精确条件分布改进条件化：通过从边缘分布采样实现可证的快速混合时间界
Exploiting Exact Conditionals Improves Conditioning: Provably Fast Mixing Time Bounds By Sampling from the Marginal
AI总结：
该研究提出MarCo算法，利用目标分布的边缘-条件结构，通过从边缘分布采样结合精确条件分布采样，实现了更优的MCMC混合时间界与收敛性能，数值结果验证了其有效性。
AI中文摘要：
从概率分布中采样的问题出现在诸多应用场景中，例如分层贝叶斯逆问题中的后验采样以及机器学习中的高斯过程。马尔可夫链蒙特卡洛（MCMC）算法常被用于从目标概率分布中采样，但其实现的计算开销可能很高，尤其是在大规模问题中。在某些应用中，目标分布可自然分解为低维边缘分布和允许精确采样的条件分布。我们描述了一种名为MarCo的MCMC算法，该算法利用此类结构，通过从边缘分布进行Metropolis-Hastings采样，再从精确条件分布中采样来生成马尔可夫链。MarCo的设计使其在联合空间上构建的马尔可夫链继承了边缘MCMC算法的收敛行为，这带来了多重理论和计算优势。我们证明，与直接从联合分布采样相比，MarCo可实现更优的混合时间上界。此外，与同样利用边缘-条件结构的单块方法相比，我们通过Peskun-Tierney序框架表明，MarCo具有更大的右谱间隙和更小的渐近方差，从而具备更优异的收敛特性。数值结果展示了MarCo的性能优势，相关结果针对各类问题给出，包括半盲图像去模糊示例。
英文摘要：
The problem of sampling from a probability distribution arises in many applications such as posterior sampling in hierarchical Bayesian inverse problems and Gaussian processes for machine learning. Markov chain Monte Carlo (MCMC) algorithms are often used for sampling from a target probability distribution, but implementations can be computationally expensive, especially for large-scale problems. In certain applications, the target distribution naturally factorizes into a lower dimensional marginal distribution and a conditional distribution that allows exact sampling. We describe an MCMC algorithm called MarCo that exploits such a structure and generates a Markov chain via Metropolis-Hastings sampling from the marginal distribution, followed by sampling from the exact conditional distribution. By design, MarCo constructs a Markov chain on the joint space that inherits the convergence behavior of the marginal MCMC algorithm. This provides multiple theoretical and computational advantages. We prove that MarCo can achieve improved mixing time upper bounds compared to direct sampling from the joint distribution. Moreover, compared to one-block methods that also exploit marginal-conditional structure, we use the framework of Peskun-Tierney ordering to show that MarCo has a larger right spectral gap and smaller asymptotic variance, thus leading to superior convergence properties. Numerical results illustrate the performance benefits of MarCo and are provided for various problems, including a semi-blind image deblurring example.
