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The geometry of uncertainty

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Uncertainty exists in all fields of applied science, as our knowledge is inherently limited, data are often missing, witnesses are characterized by uncertain reliability. Still, decisions often need to be made even in these adverse scenarios – think of climatic change, infrastructure planning, or modelling of extremely rare events such as volcanic eruptions. Standard probability theory does provide a way of reasoning with uncertainty in a “quantitative” way.

However, more general approaches to the mathematics of uncertainty have been developed as an alternative to Bayesian probability – collectively known as “imprecise probabilities”. Examples are Dempster-Shafer belief functions, fuzzy measures, random sets, monotone capacities possibility theory. Imprecise probability measures can be seen as points living in a suitable geometrical space, where they can be handled by geometric means.

The “geometrical language” we describe here may, on one side, serve to “pedagogical” purposes by pictorially illustrating the nature of these objects. More importantly, these geometrical methods can be useful as a tool for specialists in the field to operate with imprecise probabilities in a convenient way. While partial geometrical analyses have been conducted in related fields in the past, the comprehensive geometrical framework for imprecise probabilities presented here is an original contribution of the author.


Dr Fabio Cuzzolin received a laurea magna cum laude degree in Computer Engineering from the University of Padua, Italy in 1997. He was awarded a Ph.D. degree for the thesis “Visions of a generalized probability theory” by the same institution in 2001. After conducting research with Politecnico di Milano, Italy; Washington University in St. Louis, USA ; the University of California at Los Angeles, USA ; and INRIA Rhone-Alpes, France, he joined the Torr Computer Vision group in Oxford in 2008. He is currently a Reader and the Head of the Artificial Intelligence and Vision research group at Oxford Brookes University, and Associate member of the TVG .

His research interests include artificial intelligence, machine learning, and computer vision. He is a world expert in the theory of belief functions, to which he has contributed with a general geometric approach to uncertainty measures recently published in a Lambert and a Springer monograph. He is currently the author of about 90 peer-reviewed publications. Dr. Cuzzolin has won a number of awards for his work. He was the Program Chair of the 3rd International Conference on Belief Functions, he is currently Associate Editor of the IEEE Transactions on Fuzzy Systems and Guest Editor for the International Journal of Approximate Reasoning.

This talk is part of the Machine Learning @ CUED series.

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