论文标题
在有限和无限尺寸中的哈密顿蒙特卡洛算法的混合速率
Mixing Rates for Hamiltonian Monte Carlo Algorithms in Finite and Infinite Dimensions
论文作者
论文摘要
我们建立了预处理的汉密尔顿蒙特卡洛（HMC）算法的几何形状，定义在无限二维的希尔伯特空间上，如[Beskos等人，随机过程。 Appl。，2011]。该算法可用作从某些类别的目标度量中进行采样的基础，这些算法对于高斯度量绝对连续。我们的工作解决了[Beskos等人，随机过程中提出的一个空旷的问题。 Appl。，2011]，并根据Arxiv：1909.07962中给出的精确耦合技术提供了最新证明的替代方法。这里的方法通过使用弱harris定理以及广义耦合参数在合适的瓦斯坦距离中建立了收敛。我们还表明，由于我们的主要收敛结果，可以得出大量和中心限制定理的定律。此外，我们的方法可以为经典有限维HMC算法提供混合速率的新颖证明。因此，我们开发的方法提供了一个灵活的框架，以应对其他马尔可夫链蒙特卡洛算法的严格合并。此外，我们表明结果的范围包括贝叶斯逆PDE问题的某些措施，请参见。 [Stuart，Acta Numer。，2010]。特别是，我们验证了某些类别的逆问题所需的假设，涉及从被动标量恢复自由矢量场，ARXIV：1808.01084V3。
We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in [Beskos et al., Stochastic Process. Appl., 2011]. This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in [Beskos et al., Stochastic Process. Appl., 2011], and provides an alternative to a recent proof based on exact coupling techniques given in arXiv:1909.07962. The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. [Stuart, Acta Numer., 2010]. Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, arXiv:1808.01084v3.
