Scalable approximation of complex probability distributions

点击翻译按钮获取中文摘要

The prevalence of uncertainty in our world has fuelled the development of sophisticated mathematical methods to understand and tame uncertainty---this has been a central quest in the field of statistics. A key concept often used to depict uncertainty is the notion of a probability distribution, which can be thought of as measuring, for each possible state of the system, a degree of belief. Being able to interrogate probability distributions is therefore of paramount importance in statistics, and hence in the many fields of science and engineering that depend on statistics and uncertainty quantification. As scientific models become increasingly complex, the calculations required to query probability distributions are getting computationally prohibitive, to the point that these computations are the bottleneck in many disciplines. My field of research is concerned with computational methods that break these bottlenecks, by making use of algorithms exploiting randomness. The proposed research will develop new methodologies for automated and scalable approximations of complex probability distributions. Specifically, this research is motivated by Bayesian inference problems with a combination of  combinatorial components (phylogenetic trees, networks, sequence alignments and reconstruction, etc), high-dimensional parameters, or, more broadly, non-convex problems (here we use 'non-convex problems' to refer to situations where the negative log density of a probability distribution of interest is non-convex, either because the notion of convexity does not apply, such as non-Euclidean, combinatorial spaces, and/or because of multiple modes or concentration on a sub-manifold due to non-identifiability. I have encountered several applied statistics problems (in particular in cancer genomics) where developing scalable non-convex approximations is a main bottleneck in advancing scientific projects. From this applied research I have a good awareness of a set of interrelated research problems with high impact potential in Bayesian statistics and inference in high-dimensional random effects models. My recent work on non-reversible sampling and optimization of distribution continua provides promising novel angles to approach these research problems. My research group will investigate how these novel methodologies can be used to increase the degree of parallelism that can be achieved in black-box methods, i.e. methods that do not make additional assumptions on the structure of the problem. The research will be tested in challenging computational problems that arise in cancer genomics, for example problems coming from analysis of CRISPR-Cas9 interventions. The vast majority of the funding will be invested in the training of graduate and undergraduate students in a diverse research group.