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

这项研究的愿景是正式确定计算统计的几何基础，并提供实现开发先进统计方法的雄心所需的工具和分析结果，这对于解决跨科学和行业的重要新兴推理问题至关重要。随着对现有统计推断方法的要求越来越高和越来越雄心勃勃的应用的考虑，计算统计工具的能力不断被拉伸，往往超出了实际可行的范围。例如，深入了解细胞功能机制的潜力，阐明生态动力学;改善神经学诊断，揭开宇宙的深层奥秘，这些只是正在进行的科学研究中的一部分，这些研究严重依赖统计推断方法，并对现有统计方法的能力提出了前所未有的要求。这种情况促使人们在发展量化不确定性的统计方法方面不断创新。这项研究的目的是更加雄心勃勃，并进一步建立一个新的范式，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本研究中以超越统计和数学科学之间传统界面的方式正式化。

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.