Functional Analysis of Markov Chain Monte Carlo algorithms

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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)