Slice sampling: theoretical and methodological advances

Markov chain Monte Carlo (MCMC) is a standard approach for approximate sampling from intractable target distributions. Slice sampling is an umbrella term for a large variety of MCMC samplers, between which the only universal commonality is the use of a particular auxiliary variable construction. These samplers are commonly divided into two classes, ideal slice sampling (ISS) and hybrid slice sampling (HSS). ISS methods are conceptually simple and their theoretical properties can be analyzed correspondingly well, but it is generally deemed infeasible to implement them as computationally efficient samplers. In contrast, HSS methods are typically much more complicated and, accordingly, harder to analyze, but their more involved methodology enables executing them in a computationally efficient manner. The present thesis is divided into two parts, one focused on improving the theoretical understanding of ISS, and one primarily concerned with the development of new methodology for HSS. In the first part, we establish new and improved bounds on the spectral gap of two ISS variants, uniform and polar slice sampling. Motivated by our results on the latter, we begin the second part of the thesis by developing a computationally efficient HSS method designed to mimic it, which we call Gibbsian polar slice sampling (GPSS). Finally, in an effort to increase the effectiveness of GPSS in practical applications, we develop parallel affine transformation tuning (PATT), a sampling scheme which utilizes non-trivial parallelization to adaptively learn an affine transformation designed to transform a given target density into one that is more amenable to an underlying MCMC method, and internally applies this MCMC method to the transformed target. Through a series of numerical experiments with both synthetic and real-world data, we demonstrate that the performance of PATT's combination with GPSS is often at least on par with those of several methods currently considered state of the art.