Scalable approximation of complex probability distributions

不确定性在我们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-这一直是统计领域的一个核心探索。一个经常用来描述不确定性的关键概念是概率分布的概念，它可以被认为是为系统的每一种可能状态测量置信度。因此，能够询问概率分布在统计学中至关重要，因此在许多依赖统计学和不确定性量化的科学和工程领域中也是如此。随着科学模型变得越来越复杂，查询概率分布所需的计算变得令人望而却步，以至于这些计算成为许多学科的瓶颈。我的研究领域关注的是通过利用随机性的算法来打破这些瓶颈的计算方法。这项拟议的研究将为复杂概率分布的自动和可扩展近似开发新的方法。具体地说，这项研究是由组合组件(系统发育树、网络、序列比对和重建等)、高维参数或更广泛的非凸问题(在这里，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)的贝叶斯推理问题推动的，要么是因为凸性的概念不适用，如非欧几里得、组合空间，和/或因为多个模式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)，在这些问题中，开发可伸缩的非凸近似是推进科学项目的主要瓶颈。通过这项应用研究，我很好地了解了高维随机效应模型中贝叶斯统计和推断中一系列相互关联的、具有很高影响潜力的研究问题。我最近在分布连续体的不可逆抽样和优化方面所做的工作为探讨这些研究问题提供了有前途的新视角。我的研究小组将研究如何使用这些新的方法来提高黑盒方法中可以实现的并行度，即不对问题的结构做出额外假设的方法。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上进行测试，例如来自CRISPR-Cas9干预分析的问题。绝大多数资金将投资于培训不同研究小组的研究生和本科生。

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.