Statistical Theory for Astrophysical Problems

非参数推断已成为研究宇宙的重要工具，该项目由两个相互交织的部分组成：（a）开发新的理论工具和非参数方法，这些工具和方法受到天体物理学问题的启发，但适用范围更广，（B）将这些工具应用于两个重要的天体物理学问题，这将确定新统计理论的需要并指导其发展。 具体来说，研究人员将专注于暗能量状态方程的推断，并从点过程数据（如星系巡天产生的数据）中识别出非线性结构。 第一个问题引起了一个具有挑战性的非线性逆问题，并要求一个非参数的方法，鉴于什么是鲜为人知的暗能量状态方程。研究人员将开发新的非线性反问题理论，允许对未知函数进行准确的估计和尖锐的置信度声明。 然后，这些技术将应用于Ia型超新星数据，可能与其他数据源相结合，以推断暗能量。 第二个问题引起了具有挑战性的空间和推理问题。目前统计学文献中的理论只适用于单根暗条，天文学文献中的技术没有理论支持。 这个项目的研究人员将缩小这一差距，发展理论来定义，识别和推断大脑结构。 研究人员将测试这种技术，并将其应用于星系巡天数据。宇宙学中最重要的问题之一是理解暗能量。可观测量与暗能量之间的关系产生了一个具有挑战性的非线性逆问题。 由于关于暗能量性质的先验信息非常少，解决这个问题的参数方法是有限的，而且是次优的。 在不久的将来，随着更大的数据集的承诺，将有必要和机会提取暗能量状态方程的精细尺度特征。 研究人员将为这些问题开发新的推理理论，重点是形状约束下的估计，尖锐的假设检验和准确的置信集。 其目标是在准确性上比目前最好的技术有实质性的提高。 特别是，研究人员将专注于理解暗能量的问题，并确定物质分布中的结构。 前者是现代宇宙学的核心问题之一，需要最先进的统计技术才能从数据中获得最多。 研究人员将开发新的统计理论和方法，大大提高从超新星数据和其他数据源估计暗能量特征的精度。 后一个问题是理解宇宙中物质分布的核心。目前的统计理论只适用于有限版本的问题，目前的天文学方法没有强有力的理论支持。研究人员将缩小这一差距，并开发一种方法和相应的理论，可以处理问题的现实版本，并提供最佳或接近最佳的性能。

Nonparametric inference has become an essential tool for studying the cosmos.This project consists of two intertwined components: (a) development of new theoretical tools and nonparametric methodologies that are inspired by problems in astrophysics but apply more broadly, and (b) application of these tools to two important astrophysical problems, which will frame the need for and guide the development of new statistical theory. Specifically, the investigators will focus on inference for the dark energy equation of state and on identifying filamentary structures from point process data such as that produced by galaxy surveys. The first problem gives rise to a challenging nonlinear inverse problem and demands a nonparametric approach, given what little is known about the dark energy equation of state. The investigators will develop new theory for nonlinear inverse problems that allow for accurate estimates and sharp confidence statements about the unknown function. These techniques will then be applied to Type Ia supernova data, possibly combined with other data sources, to make inferences about dark energy. The second problem gives rise to challenging spatial and inference problems. Current theory in the statistical literature applies to a single filament only, and techniques in the astronomical literature are not supported by theory. The investigators on this project will close that gap, developing theory for defining, identifying, and making inferences about the filamentary structures. The investigators will test this technique and apply it to galaxy survey data.One of the most important problems in cosmology is understanding dark energy.The relationship between observable quantities and dark energy produces a challenging nonlinear inverse problem. With very little strong a priori information about the nature of dark energy, parametric approaches to the problem are limited and suboptimal. And with the promise of much larger data sets in the near future, there will be need and opportunity to extract fine-scale features of the dark energy equation of state. The investigators will develop new theory of inference for such problems, with a focus on estimation under shape constraints, sharp hypothesis testing, and accurate confidence sets. The goal is a substantial improvement in accuracy over the current best techniques. In particular, the investigators will focus on the problem of understanding dark energy and on identifying filamentary structures in distribution of matter. The former is one of the central problems in modern cosmology and demands state of the art statistical techniques to get the most from the data. The investigators will develop new statistical theory and methodologies that substantially improve the precision with which features of dark energy can be estimated from supernova data and other data sources. The latter problem is central to understanding the distribution of matter in the universe. Current statistical theory only applies to a limited version of the problem, and current astronomical methodologies do not have strong theoretical support. The investigators will close that gap and develop a method and corresponding theory that can handle realistic versions of the problem and give optimal or near-optimal performance.