Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions
Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions
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DOI：
10.1007/s40072-021-00211-z
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发表时间：
2020-03
期刊：
影响因子：
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通讯作者：
N. Glatt-Holtz;Cecilia F. Mondaini
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文献类型：
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作者：
N. Glatt-Holtz;Cecilia F. Mondaini
我们建立了在无限维希尔伯特空间上定义的预处理哈密顿蒙特卡罗 (HMC) 算法的几何遍历性，如 Beskos 等人开发的那样。 （Stoch Process Appl 121(10):2201–2230, 2011）。该算法可以用作从相对于高斯测量绝对连续的某些目标测量类别进行采样的基础。我们的工作解决了 Beskos 等人提出的一个悬而未决的问题。 (2011)，并提供了基于 Bou-Rabee 和 Eberle 中给出的精确耦合技术的最新证明的替代方案（无限维度中预条件哈密顿蒙特卡罗的两尺度耦合，2019）。这里的方法通过使用弱 Harris 定理和广义耦合论证，在合适的 Wasserstein 距离上建立收敛。我们还表明，根据我们的主要收敛结果，可以导出大数定律和中心极限定理。此外，我们的方法为经典有限维 HMC 算法提供了混合率的新颖证明。因此，我们开发的方法提供了一个灵活的框架来解决其他马尔可夫链蒙特卡罗算法的严格收敛问题。此外，我们表明我们的结果范围包括贝叶斯方法中出现的逆偏微分方程问题的某些度量，参见。斯图尔特（Acta Number 19：451–559，2010）。特别是，我们验证了一类涉及从无源标量恢复散度自由矢量场的逆问题所需的所有假设，Borggaard 等人。 （SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020）。
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. (Stoch Process Appl 121(10):2201–2230, 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. (2011), and provides an alternative to a recent proof based on exact coupling techniques given in Bou-Rabee and Eberle (Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019). 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 19:451–559, 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, Borggaard et al. (SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020).