Functional Analysis of Markov Chain Monte Carlo algorithms

马尔可夫链蒙特卡罗(MCMC)技术的成功实施在很大程度上取决于它们收敛到均衡的速度。为了刻画马尔可夫链的收敛速度，人们已经做了许多尝试：当收敛速度是几何的时，人们可以研究马尔可夫算子的谱，并从它的第二大特征值开始得到收敛速度。见例如Douc，R.，E.Mouline，P.Priouret和P.Soulier(2018)。在这个框架中特别令人感兴趣的是一大类所谓的“模拟回火”算子，它们的光谱可以由在形成划分的状态空间集合内移动的单链的光谱来限定，沿着分子动力学和热力学中的马尔可夫过程研究的方法。例如，见Madras，N.和D.Randall(2002)。并不是所有的马尔可夫链都是几何收敛的。根据目标和建议分布的形状和结构，算法的收敛速度可能会变慢。在粒子MCMC区域中可以找到一个典型的示例，在该区域中，粒子吉布斯采样器可能是次几何收敛的，具体取决于某些重要权重的无界性。见Andrieu，C.，A.Lee和M.Vihola(2015)。粒子MCMC方法在这个项目中引起了极大兴趣。它们起源于物理学，是对薛定谔方程某些解的离散时间的蒙特卡罗近似，至今仍未被探索。见Del Moral，P.(2004)。这些算法的主要优点之一是，它们提供了对感兴趣的数量的无偏估计。在实践中，它们的实现依赖于重要性重采样技术，因此依赖于取决于目标和建议分布的某些权重。如上所述，这些权重经常在算法的性能中起到关键作用，即，根据它们的规格，可能会导致亚几何收敛。在亚几何情形下恢复收敛速度仍是一个相当未被探索的领域。直到最近，Andrieu，C.，A.Lee，S.Power和A.Q.Wang(2021)为我们提供了一种基于泛函分析结果的新技术，例如Poincaré不等式，它可以确定次几何收敛的马尔可夫链的收敛速度。这为将许多结果从几何情形推广到亚几何情形开辟了新的机会。参考文献Douc，R.，E.Mouline，P.Priouret和P.Soulier(2018)Madras，N.和D.Randall(2002)Del Moral，P.(2004)Andrieu，C.，A.Lee和M.Vihola(2015)Andrieu，C.，A.Lee，S.Power和A.Q.Wang(2021)

Successful implementation of Markov Chain Monte Carlo (MCMC) techniques relies largely on their speed of convergence to equilibrium. Many attempts have been done in order to characterise the convergence rate of a Markov Chain: when the rate of convergence is geometric, one can investigate the spectrum of the Markov operator and obtain the rate of convergence starting from its second largest eigenvalue. See e.g. Douc, R., E. Moulines, P. Priouret, and P. Soulier (2018). Of particular interest in this framework is the broad class of the so-called "simulated tempering" operators, whose spectrum can be bounded by the spectra of single chains moving within sets of the state space forming a partition, along the way of techniques for the study of Markov processes in Molecular Kinetics and Thermodynamics. See for example Madras, N. and D. Randall (2002). Not all Markov chains converge geometrically. An algorithm's speed of convergence may slow down depending on the shape and structure of the target and proposal distributions. A typical example can be found in the Particle MCMC area, where, depending on the unboundedness of some importance weights, the Particle Gibbs Sampler may be sub-geometrically convergent. See Andrieu, C., A. Lee, and M. Vihola (2015). Particle MCMC methods are of great interest in this project. Still quite unexplored, they arise from Physics as a Monte Carlo approximation of a discrete-time counterpart of some solution of the Schrödinger equation. See Del Moral, P. (2004). Among the major advantages of these algorithms is the fact that they provide unbiased estimates of quantities of interest. Their implementation, in practice, relies on importance resampling techniques, and hence on certain weights depending on the target and proposal distributions. These weights, as mentioned above, have very often a critical role in the performance of the algorithm, that is, may lead to sub-geometric convergence depending on their specification. To recover the convergence rate in the sub-geometric case is still a quite unexplored area. Only recently, Andrieu, C., A. Lee, S. Power, and A. Q. Wang (2021) provided us with a novel technique based on Functional Analysis results, such as Poincaré Inequalities, which allow to determine the convergence rate of a sub-geometrically convergent Markov Chain. This opens up new opportunities of generalising many of the results from the geometric case to the sub-geometric case. References Douc, R., E. Moulines, P. Priouret, and P. Soulier (2018) Madras, N. and D. Randall (2002) Del Moral, P. (2004) Andrieu, C., A. Lee, and M. Vihola (2015) Andrieu, C., A. Lee, S. Power, and A. Q. Wang (2021)