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不确定的问题，即未知参数的数量（远）大于可观察数据的数量。这使得对这些参数的估计和推断本身是不可能的，因为这些参数是不可识别的。因此，包含额外的结构信息是相关的。从更广泛的意义上来说，这是通过某种稀疏性来实现的：尽管感兴趣的参数很复杂（例如，高维），但它具有简单（例如，低维）的基础结构。该提案的重点是迄今为止受到相对较少关注的一种稀疏性，即通过给定的有限字母表（FA）的信号函数值的稀疏性。 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之一，以及在二进制信号的数字通信中。在本项目的理论部分，我们希望与 Martin Wainwright 教授（加州大学伯克利分校）和 Bin Yu 教授（加州大学伯克利分校）共同分析 FA 结构如何在不确定的统计模型中代替经典稀疏性并与经典稀疏性相结合，从而实现有意义的推理。 因此，我们希望重点关注盲源分离和高维线性模型。虽然 FA 结构解决了不可识别性问题，但它们的组合性质导致了计算负担。 因此，该提案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，在该项目的分析部分，我们希望考虑对FA结构进行修改：临床试验的亚组检测通常会导致分割问题，其中特定的FA是由系统发育树诱导的。我们希望通过多尺度程序来解决这些问题。这些不仅提供极小极大最优估计，还提供置信度陈述，这在医疗应用中尤其重要。我们与 Bin Yu 教授（加州大学伯克利分校）和威康人类遗传学信托中心（牛津）合作，希望用真实的数据示例来证明 FA 程序如何显着改善个性化医疗。

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.