Studying, Improving, and Applying Markov chain Monte Carlo methods

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Markov chain Monte Carlo (MCMC) methods allow for data generation from highly complex probability distributions (the target distribution). Metropolis-Hastings (MH) samplers form an important class of MCMC algorithms, which are renowed for their versatility (they can be applied to virtually any probability distribution of interest) and ease of implementation. Researchers and practitioners in various fields of applications such as biostatistics, computer science, physics, finance, and applied statistics extensively use such methods. In applying MH samplers, it is necessary to choose a preferred proposal distribution; the normal distribution is a popular choice due to its accessibility. The idea is to generate, at every iteration, a candidate from this proposal distribution; this candidate is then accepted as a suitable value for the sample (according to a certain acceptance probability), or simply discarded. Existing samplers do not always perform efficienctly in applications. In order to deal with increasingly complex distributions and massive datasets arising in practice, it is necessary to develop new samplers and/or improve existing methods. It is also important to collaborate actively with other disciplines so as to develop tools that are useful for their problematics. Reversible-jump MCMC (RJ-MCMC) algorithms constitute an extension of the MCMC strategy, as they allow sampling from target distributions of varying dimensions. These algorithms obviously constitute a great tool in Bayesian model selection, where the dimensionality of the parameter vector is typically not fixed. They can be used in multiple change-point analysis, where the models considered allow different parts of a dataset to obey different probability laws. Although extensively used in practice, the RJ-MCMC has not been studied theoretically. As a result, the mechanism from moving from one dimension to another is usually chosen by trial and error. One goal of this proposal is to provide users with a theoretical guideline for tuning the RJ-MCMC sampler. A second aspect of my research, besides theoretically improving existing methods, is to improve the computational efficiency of samplers. The multiple-try Metropolis (MTM) strategy allows generating multiple candidates in a given iteration (by opposition to only one in the usual MH sampler). A drawback of this method is the necessity of generating an auxiliary sample at every iteration, which significantly increases the computational intensity of the method. Some researchers have eliminated the need for this auxiliary sample by reexpressing the problem, but the resulting method does not outperform the original MTM algorithm. I believe that their method could be improved by encouraging movements within the Markov chain, so as to obtain an algorithm more efficient than the usual MTM. An interesting new avenue in MCMC theory aims at sampling in spaces of functions (by opposition to sampling points from a density of interest). These situations may be found in weather forecasting, oceanography (goundwater flow), medicine and security (image registration), physics, finance (sampling from some stochastic volatility models), etc. Standard MCMC algorithms become arbitrarily slow under the mesh refinement used to obtain increasingly accurate samples from such infinite-dimensional problems. There is thus a crying need for designing new MCMC techniques that can be applied in such contexts. A last goal is to effectively apply the methodology developed in areas such as Biology; specifically I intend to use and refine MCMC methods for investigating large-scale colored networks labeled with certain organisms, bacteria, and viruses. Of particular interest is the distribution of various types of paths in such networks.