Inexact Markov Chain Monte Carlo methods rely on Markov chains that do not exactly preserve the target distribution. Examples include the unadjusted Langevin algorithm (ULA) and unadjusted Hamiltonian Monte Carlo (uHMC). This paper establishes bounds on Wasserstein distances between the invariant probability measures of inexact MCMC methods and their target distributions with a focus on understanding the precise dependence of this asymptotic bias on both dimension and discretization step size. Assuming Wasserstein bounds on the convergence to equilibrium of either the exact or the approximate dynamics, we show that for both ULA and uHMC, the asymptotic bias depends on key quantities related to the target distribution or the stationary probability measure of the scheme. As a corollary, we conclude that for models with a limited amount of interactions such as mean-field models, finite range graphical models, and perturbations thereof, the asymptotic bias has a similar dependence on the step size and the dimension as for product measures.

翻译：非精确马尔可夫链蒙特卡洛方法依赖于不完全保持目标分布的马尔可夫链，例如未调整的朗之万算法(ULA)和未调整的哈密顿蒙特卡洛(uHMC)。本文建立了非精确MCMC方法的不变概率测度与其目标分布之间的Wasserstein距离界，重点阐明了该渐近偏差对维度和离散化步长的精确依赖关系。基于精确或近似动力学收敛到平衡态的Wasserstein界假设，我们证明对于ULA和uHMC，渐近偏差取决于与目标分布或方案平稳概率测度相关的关键量。作为推论，我们得出结论：对于具有有限相互作用量的模型（如平均场模型、有限范围图模型及其扰动模型），渐近偏差对步长和维度的依赖关系与乘积测度情形类似。