Markov chain Monte Carlo algorithms and locally informed proposal distributions

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Markov chain Monte Carlo (MCMC) methods allow for data generation from highly complex probability distributions (the target). They are used in several fields of application such as weather forecasting, medicine, physics, security, etc. I aim at contributing to the theoretical, methodological, and applied MCMC literature through different projects. Theoretical results, besides leading to a better understanding of samplers, often point towards an improvement of the sampling scheme. In terms of methodology, the main challenge is to propose new sampling schemes that require minimum input and remain computationally affordable. A third objective is contributing to interdisciplinary research by proposing efficient samplers in specific modeling contexts. The Metropolis-Hastings (MH) algorithm is the most popular sampler in the MCMC toolbox and the underlying method in the proposed research. It has been enhanced in countless ways, producing cutting edge samplers. At each iteration, a candidate is drawn from a selected proposal distribution and then accepted as a suitable value for the sample according to a specific acceptance probability. To obtain a sample that is representative of the target, careful tuning of the proposal distribution is required. Lately, there has been interest in local tunings that evolve from one iteration to the next; these offer interesting efficiency gains at low computational costs. Locally-balanced proposal distributions in MH samplers produce smarter candidates in high-dimensional regimes, reducing the impact of the accept/reject step and producing quality samples in an efficient way. I wish to study the connection between locally-balanced proposals, local tunings, and gradient-informed samplers. I intend to propose a flexible proposal distribution that is a function of the dimension, converging towards a locally-balanced proposal as the dimensionality increases. Generally, I expect this promising local balance concept to lead to an informed way of selecting the various tuning parameters in variants of the MH sampler (e.g. weight function in samplers with pools of candidates, transitions between models in the reversible-jump MCMC sampler, etc.) The traditional approach to study the theoretical behavior of samplers is restrictive as it focuses on one target component at a time, making it difficult to study targets with correlation. Recently, we proposed a new approach that allows studying all components simultaneously; I intend to investigate the extent of the sampler and/or target complexity (with copulas) this proof can handle. This could lead to informed design schemes and open a whole new handling of correlation in MCMC samplers. Two applied projects are planned. The first involves the use of a statistical framework in a modeling financial context; the second one, in Ecology, will model the trajectory of migratory birds using real geolocation data by light, based on measurements of sunlight intensity over time.