Scalable approximation of complex probability distributions

不确定性在我们这个世界的普遍存在，推动了复杂数学方法的发展，以理解和驯服不确定性——这一直是统计学领域的核心追求。经常用来描述不确定性的一个关键概念是概率分布的概念，它可以被认为是对系统的每一个可能状态的可信度的度量。因此，能够询问概率分布在统计学中是至关重要的，因此在许多依赖于统计和不确定性量化的科学和工程领域也是如此。随着科学模型变得越来越复杂，查询概率分布所需的计算越来越难以计算，以至于这些计算成为许多学科的瓶颈。我的研究领域是通过使用利用随机性的算法来打破这些瓶颈的计算方法。提出的研究将为复杂概率分布的自动化和可扩展近似开发新的方法。具体来说，本研究的动机是贝叶斯推理问题与组合组件（系统发育树，网络，序列比对和重建等），高维参数的组合，或者更广泛地说，非凸问题(这里我们使用“非凸问题”指的是感兴趣的概率分布的负对数密度是非凸的情况，要么是因为凹凸的概念不适用，如非欧几里得；组合空间，和/或由于多模态或由于不可识别而集中在子流形上。我遇到了几个应用统计学问题（特别是在癌症基因组学中），其中开发可扩展的非凸近似是推进科学项目的主要瓶颈。从这个应用研究中，我很好地认识到贝叶斯统计和高维随机效应模型推理中一系列相互关联且具有高影响潜力的研究问题。我最近在分布连续体的非可逆采样和优化方面的工作为解决这些研究问题提供了有希望的新角度。我的研究小组将研究如何使用这些新颖的方法来提高黑盒方法（即不对问题结构进行额外假设的方法）中可以实现的并行度。这项研究将在癌症基因组学中出现的具有挑战性的计算问题中进行测试，例如来自CRISPR-Cas9干预分析的问题。绝大部分资金将投入到研究生和本科生在一个多样化的研究小组的培训。

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.