Advancing the Geometric Framework for Computational Statistics: Theory, Methodology and Modern Day Applications

这项研究的愿景是使计算统计学的几何基础正式化，并提供必要的工具和分析结果，以实现开发先进统计方法的雄心，这对于解决整个科学和工业中重大的新推理问题至关重要。随着对现有统计推断方法日益苛刻和雄心勃勃的应用的考虑，计算统计工具的能力不断扩大，往往超出了实际可行的范围。例如，深入了解细胞功能的机制、阐明生态动力学、改进神经学诊断和揭开宇宙的深层奥秘只是正在进行的科学研究的一部分，这些研究严重依赖统计推断方法，并对现有统计方法的当前能力提出了无与伦比的要求。这种情况促使人们不断创新，制定量化不确定性的统计方法。这项拟议研究的目的是更雄心勃勃，更进一步地建立一种新的范式，通过形式化和开发先进的蒙特卡罗方法来支持下一代计算统计方法的进步。计算统计学的几何基础将在这项拟议的研究中以一种超越统计和数学科学之间传统接口的方式得到形式化。

The vision of this research is to formalise the geometric foundations of computational statistics and provide the tools and analytic results required to realise the ambition of developing the advanced statistical methodology that is essential to address emerging inference problems of major importance across the sciences and industry. As ever more demanding and ambitious applications of existing statistical inference methods are being considered, the capabilities of computational statistics tools are constantly being stretched, often beyond what is practically feasible. For example the potential to gain insights into the mechanisms of cellular function, elucidating ecological dynamics; improving neurological diagnostics, and uncovering the deep mysteries of the cosmos are only some of the ongoing scientific studies that are heavily reliant on statistical inference methods and are placing unparalleled demand on the current capabilities of available statistical methodology. This situation motivates continual innovation in the development of statistical methods for the quantification of uncertainty. The aim of this proposed research is to be more ambitious and go much further in establishing a novel paradigm that underpins the advancement of next generation computational statistical methods by formalising and developing advanced Monte Carlo methods. The geometric foundations of computational statistics will be formalised within this proposed research in a way that reaches beyond traditional interfaces between statistical and mathematical sciences.