Computational Information Geometry/Neuroinformatics

这项研究的动机是相信几何，独特地，给我们提供了适当的工具来理解：统计模型的选择，统计和科学方法之间的相互作用，特别是经验信息的限制-所有这些都在给定的科学问题的背景下。我们的目标是实现博克斯的观点，他指出“既然所有的模型都是错误的，科学家必须警惕哪些是重要的错误”。对于给定的科学背景，我们用适当的几何结构定义了一个“所有模型的空间”。对于任何有效的统计模型，我们寻找所有对感兴趣的推理问题有重大影响的小模型变化。在这里，大小是通过数据、几何和科学背景的结合来校准的。要全面理解这一点，是一项极具挑战性的任务，因此我们将重点放在神经科学的一个特定领域，我们已经拥有了理论和实践专业知识。神经元是大脑中的细胞，连接在一个巨大的网络中，基本上以二元方式活动，活跃或安静，非常短的时间尺度活动周期被称为峰值。实验工具的最新发展使我们能够记录大量神经元和长时间的这些峰值。这是现代大数据问题的一个很好的例子。对于任何给定的感兴趣的科学问题，统计学家必须在分析这些数据时做出建模选择，而我们的研究重点是充分理解这些选择的后果。我们最近的工作已经在简单的例子中证明了我们的几何方法的概念，这项研究将把我们的突破扩展到真实的、复杂的科学背景中。我们在统计上下文中使用几何似乎很奇怪，但事实上，这种使用非常普遍。欧几里得几何是线性回归、大部分时间序列分析和应用统计学中使用的许多近似方法的基础。经典信息几何将这种欧几里得几何推广到更丰富的模型类别，在我们最近的工作中，我们进一步扩展了该理论，给出了包含重要边界和奇点的“所有模型空间”的几何。这为我们提供了研究模型构建、灵敏度和不确定性所需的所有几何工具。参与这项研究的学生将接触到统计思想的基础及其在真实和复杂科学问题背景下的实际应用。他们还将获得计算和数据分析技能，并进一步获得与从业人员合作的宝贵经验。这项培训将为他们的学术生涯做好充分的准备，或者帮助填补加拿大目前对数据科学家的巨大需求。此外，由于这是基础性工作，从长远来看，我们的结果将为应用统计学的整个学科提供信息，这些学科与神经科学相去甚远。

This research is motivated by the conviction that geometry, uniquely, gives us appropriate tools to understand: statistical model choice, the interplay between the statistical and scientific methods, and, especially, the limits to empirical information - all in the context of a given scientific problem. We aim to implement Box's view, where he states 'since all models are wrong the scientist must be alert to what is importantly wrong'. For a given scientific context, we define a 'space of all models' with an appropriate geometric structure. For any working statistical model we search for all small modelling changes which have big effects on the inferential question of interest. Here big and small are calibrated by a combination of data, geometry and scientific context. To understand this, in its full generality, is a highly challenging task, and so we focus on a particular area of Neuroscience where we already have both theoretical and practical expertise. Neurons are cells in the brain, connected in a huge network, which act in a basically binary way, being active or quiet, with the very short timescale activity periods being called spikes. Recent developments in experimental tools enable us to record these spikes across large numbers of neurons and long periods of time. This is a good example of a modern Big Data problem.  For any given scientific problem of interest, the statistician has to make modelling choices during the analysis of such data, and our research is focused on fully understanding the consequences of these choices. Our recent work has shown proof-of-concept of our geometric approach in simple examples, and this research will extend our breakthroughs to real, complex scientific contexts.  It may seem surprising that we use geometry in this statistical context but this use is, in fact, very common.  Euclidean geometry underlies linear regression, much of time series analysis, and many approximation methods used in applied statistics. Classical Information Geometry has generalised this Euclidean geometry to much richer classes of models and in our recent work we have further extended the theory to give a geometry of 'the space of all models' including important boundary and singular points. This gives us all the geometric tools that we need to study model building, sensitivity and uncertainty. Students involved in this research will be exposed to the foundations of statistical thought and its practical application in the context of real and complex scientific problems. They will also gain computational and data-analytic skills and, further, get invaluable experience working with practitioners. This training will fully prepare them for either an academic career or to help fill the, currently huge, demand in Canada for Data Scientists. Furthermore, since this is foundational work, our results will, in the long term, inform the whole discipline of applied statistics in areas far removed from neuroscience.