In order to solve tasks like uncertainty quantification or hypothesis tests in Bayesian imaging inverse problems, we often have to draw samples from the arising posterior distribution. For the usually log-concave but high-dimensional posteriors, Markov chain Monte Carlo methods based on time discretizations of Langevin diffusion are a popular tool. If the potential defining the distribution is non-smooth, these discretizations are usually of an implicit form leading to Langevin sampling algorithms that require the evaluation of proximal operators. For some of the potentials relevant in imaging problems this is only possible approximately using an iterative scheme. We investigate the behaviour of a proximal Langevin algorithm under the presence of errors in the evaluation of proximal mappings. We generalize existing non-asymptotic and asymptotic convergence results of the exact algorithm to our inexact setting and quantify the bias between the target and the algorithm's stationary distribution due to the errors. We show that the additional bias stays bounded for bounded errors and converges to zero for decaying errors in a strongly convex setting. We apply the inexact algorithm to sample numerically from the posterior of typical imaging inverse problems in which we can only approximate the proximal operator by an iterative scheme and validate our theoretical convergence results.

翻译：为解决贝叶斯成像逆问题中的不确定性量化或假设检验等任务，通常需要从后验分布中采样。对于通常为对数凹但高维的后验分布，基于朗之万扩散时间离散化的马尔可夫链蒙特卡洛方法是常用工具。若定义分布的势能函数非光滑，这些离散化通常采用隐式形式，从而得到需评估近端算子的朗之万采样算法。对于成像问题中某些相关势能函数，这一评估只能通过迭代方案近似实现。本文研究了在近端映射评估存在误差的情况下近端朗之万算法的行为。我们将精确算法的现有非渐近与渐近收敛性结果推广至近似设定，量化了由误差导致的算法平稳分布与目标分布之间的偏差。在强凸设定下，我们证明当误差有界时额外偏差保持有界，当误差衰减时额外偏差收敛至零。通过将近似算法应用于典型成像逆问题的后验数值采样（其中仅能通过迭代方案近似近端算子），验证了我们的理论收敛性结论。