Statistical theory on finite alphabet structures: inference, algorithms, and applications

现代统计学的大量研究关注的是高度低估的问题，即未知参数的数量(远远)大于可观测数据的数量。这使得对这些参数本身的估计和推断是不可能的，因为这些参数是不可识别的。因此，包括额外的结构信息是相关的。在更广泛的意义上，这是通过某种稀疏性实现的：尽管感兴趣的参数是复杂的(例如，高维的)，但它具有简单的(例如，低维的)底层结构。该建议的重点是迄今为止相对较少受到关注的一种类型的稀疏性，即通过给定的有限字母表(FA)的信号的函数值的稀疏性。FA结构出现在许多不同的领域，例如，在癌症遗传学中，DNA拷贝数只能取几个已知整数值中的一个，以及在使用二进制信号的数字通信中。在这个项目的理论部分，我们想要与加州大学伯克利分校的Martin Wainwright教授和加州大学伯克利分校的Bin Yu教授一起分析FA结构如何在欠确定的统计模型中实现有意义的推理，取代并结合经典稀疏性。因此，我们想要重点研究盲源分离和高维线性模型。虽然FA结构解决了不可辨识性问题，但它们的组合性质导致了计算负担。因此，这一建议的一个基本研究目标是精确量化统计极小极大最优性和计算可行性之间的差距。特别是，我们希望开发快速算法，同时产生足够的静态效率。在此基础上，在本项目的分析部分，我们希望考虑对FA结构的修改：临床试验的子组检测经常会导致分割问题，其中特定的FA是由系统发生树诱导的。我们希望通过多规模的程序来解决这些问题。这些不仅提供了极小极大最优估计，而且还提供了置信度声明，这在医疗应用中可能特别关键。我们希望与加州大学伯克利分校的余斌教授和惠康人类遗传学信托中心(牛津大学)合作，用真实的数据例子展示FA-Procedure如何在个性化医疗方面提供显着的改进。

A vast amount of research of modern statistics is concerned with problems that are highly underdetermined, in the sense that the amount of unknown parameters is (much) larger than the amount of observable data. This renders estimation of and inference about such parameters impossible per se, as the parameters are not identifiable. Therefore, it is pertinent to include additional structural information. In a broader sense, this is achieved by some kind of sparsity: although the parameter of interest is complex (e.g., high-dimensional), it has a simple (e.g., low-dimensional) underlying structure. The focus of this proposal is on a type of sparsity that has received relatively few attention so far, namely, sparsity in the function values of a signal via a given finite alphabet (FA). FA structures appear in many different fields, for example, in cancer genetics, where DNA copy-numbers can only take one of a few known integer values, and in digital communications with binary signals.In the theoretical part of this project, we want to analyze, jointly with Prof. Martin Wainwright (UC Berkeley) and Prof. Bin Yu (UC Berkeley), how FA structures can enable meaningful inference in underdetermined statistical models, in place of and in combination with classical sparsity. Thereby, we want to focus on blind source separation and high-dimensional linear models. Although, FA structures solve the problem of non-identifiability, their combinatorial nature leads to a computational burden. Therefore, a fundamental research objective of this proposal is to precisely quantify this gap between statistical minimax optimality and computational feasibility. In particular, we want to develop fast algorithms, which, at the same time, yield adequate statical efficiency.On this basis, in the analytical part of this project, we want to consider a modification of FA structures: Subgroup detection for clinical trials often leads to segmentation problems, where a specific FA is induced by phylogenetic trees. We want to tackle those problems with multiscale procedures. Those do not just provide minimax optimal estimates, but also confidence statements, something which can be particularly crucial in medical applications. In cooperation with Prof. Bin Yu (UC Berkeley) and the Wellcome Trust Center for Human Genetics (Oxford) we want to demonstrate with real data examples how FA-procedures provide significant improvement in personalized medicine.