哈密顿蒙特卡罗的不可约性和几何遍历性

IF 3.2 1区数学 Q1 STATISTICS & PROBABILITY

Annals of Statistics Pub Date : 2020-12-01 DOI:10.1214/19-aos1941

Alain Durmus, É. Moulines, E. Saksman

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引用次数: 11

摘要

哈密顿蒙特卡罗算法(HMC)是目前最流行的用于连续状态空间上光滑分布采样的马尔可夫链蒙特卡罗算法之一。本文讨论了HMC算法的不可约性和几何遍历性。我们考虑StörmerVerlet积分器的步数是固定的或随机的情况。在与目标分布π相关的潜在U的温和条件下，我们首先证明了与HMC算法相关的马尔可夫核是不可约的和正循环的。在更严格的条件下，我们建立了Markov核是Harris递归的。我们在U上给出了HMC采样器几何遍历的可验证条件。最后，我们用几个例子来说明我们的结果。

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Irreducibility and geometric ergodicity of Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) is currently one of the most popular Markov Chain Monte Carlo algorithms to sample smooth distributions over continuous state space. This paper discusses the irreducibility and geometric ergodicity of the HMC algorithm. We consider cases where the number of steps of the StörmerVerlet integrator is either fixed or random. Under mild conditions on the potential U associated with target distribution π, we first show that the Markov kernel associated to the HMC algorithm is irreducible and positive recurrent. Under more stringent conditions, we then establish that the Markov kernel is Harris recurrent. We provide verifiable conditions on U under which the HMC sampler is geometrically ergodic. Finally, we illustrate our results on several examples.

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来源期刊

Annals of Statistics 数学-统计学与概率论

CiteScore

9.30

自引率

8.90%

发文量

119

审稿时长

6-12 weeks

期刊介绍： The Annals of Statistics aim to publish research papers of highest quality reflecting the many facets of contemporary statistics. Primary emphasis is placed on importance and originality, not on formalism. The journal aims to cover all areas of statistics, especially mathematical statistics and applied & interdisciplinary statistics. Of course many of the best papers will touch on more than one of these general areas, because the discipline of statistics has deep roots in mathematics, and in substantive scientific fields.