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平衡点的速度。为了证明马尔可夫链的收敛速度，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，并从其第二大特征值开始获得收敛速度。参见例如Douc，R.，E. Moulines，P. Priouret和P. Soulier（2018）。在这个框架中特别感兴趣的是广泛的一类所谓的“模拟回火”运营商，其频谱可以有界的单链移动的频谱内形成分区的状态空间的集合，沿着的方法的技术研究马尔可夫过程的分子动力学和热力学。参见例如Madras，N.和D. Randall（2002年）。并不是所有的马尔可夫链都几何收敛。算法的收敛速度可能会减慢，这取决于目标和建议分布的形状和结构。在粒子MCMC区域中可以找到一个典型的示例，其中，根据某些重要性权重的无界性，粒子吉布斯采样器可能是亚几何收敛的。参见Andrieu，C.，A. Lee和M. Vihola（2015年）。粒子MCMC方法在这个项目中有很大的兴趣。它们仍然是未被探索的，它们来自物理学，作为薛定谔方程某个解的离散时间对应物的蒙特卡罗近似。参见Del Moral，P.（2004）。这些算法的主要优点之一是它们提供感兴趣的量的无偏估计。在实践中，它们的实现依赖于重要性恢复技术，因此依赖于取决于目标和建议分布的某些权重。如上所述，这些权重通常在算法的性能中具有关键作用，也就是说，可以根据其规格导致子几何收敛。在亚几何情形下如何恢复收敛速度仍然是一个相当未探索的领域。最近，Andrieu，C.，A.李，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，例如庞加莱不等式，它允许确定子几何收敛马尔可夫链的收敛速度。这开辟了新的机会，推广许多结果从几何的情况下，子几何的情况。参考资料Douc，R.，E. Moulines，P. Priouret和P. Soulier（2018）Madras，N。和D. 04.第50集6.1 The Lord of the Lord（2004）A. Lee和M. 03 The Dog（2015）A.李，S。Power和A. Q.王（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)