Computational Information Geometry/Neuroinformatics

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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.