Markov chain Monte Carlo (MCMC) sampling of densities restricted to linearly constrained domains is an important task arising in Bayesian treatment of inverse problems in the natural sciences. While efficient algorithms for uniform polytope sampling exist, much less work has dealt with more complex constrained densities. In particular, gradient information as used in unconstrained MCMC is not necessarily helpful in the constrained case, where the gradient may push the proposal's density out of the polytope. In this work, we propose a novel constrained sampling algorithm, which combines strengths of higher-order information, like the target's log-density's gradients and curvature, with the Hit-&-Run proposal, a simple mechanism which guarantees the generation of feasible proposals, fulfilling the linear constraints. Our extensive experiments demonstrate improved sampling efficiency on complex constrained densities over various constrained and unconstrained samplers.

翻译：马尔可夫链蒙特卡洛（MCMC）采样在自然科学中反问题的贝叶斯处理中，对线性约束域内的密度进行采样是一项重要任务。虽然存在高效的均匀多面体采样算法，但针对更复杂约束密度的研究却少得多。特别是在无约束MCMC中使用的梯度信息，在约束情形下未必有效，因为梯度可能将提议的密度推出多面体。本文提出一种新颖的约束采样算法，该算法将高阶信息（如目标对数密度的梯度和曲率）的优势与Hit-&-Run提议机制相结合，后者是一种能保证生成满足线性约束的可行提议的简单机制。我们的大量实验表明，该算法在复杂约束密度上的采样效率优于多种约束和无约束采样器。