Information geometry of graphs

Lead Research Organisation: University of Bristol
Department Name: Mathematics

Abstract

Entropy quantifies the way in which, for example, the outcome of tossing a fair coin is harder to predict than with a biased one. It plays a fundamental role in understanding how information is transmitted over noisy communication networks, and how large amounts of information can be stored in as small devices as possible (data compression). More recently, through the emerging field of Information Geometry, it has become clear that entropy can provide understanding of more fundamental questions of statistical inference. Specifically, Information Geometry offers a way to define a `distance' between distributions of random events, giving an unambiguous way to decide how different two models of randomness really are. However, these results are generally only understood in the context of real-valued random events, whereas many random events (those to do with counting, for example) take values in just the set 0,1,2,.... We propose to combine the expertise of the PI in the field of Information Theory with the RA's background in functional analysis, in order to define distance measures in a similar way for these counting processes, and to understand the properties of the resulting measures.

Planned Impact

This work will have an impact through resolving fundamental questions in mathematics, and combining ideas from two fields (functional analysis and information theory), to the benefit of both communities. We will present the results obtained at academic conferences and publish in leading academic journals. While the work is regarded as theoretical at this stage, we believe that a successful resolution of these problems would have major implications for statisticians and designers of communication systems, and we are well positioned to exploit these applications. This is largely due to the research environment in Bristol, which includes the highly rated Statistics group and an existing dynamic collaboration between the PI and Electrical Engineers. This project will raise the profile of the UK in the international mathematical landscape, by fostering a mutual understanding of successful approaches used by groups in Bristol and in Toulouse, bringing these two highly rated research groups closer together. This could lead to future collaborations including joint workshops, research visits and publications. Further, the project will have an element of training for the post-doc, who will be gaining new collaborators and extending the scope of his research.

Publications

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Hillion E (2015) A proof of the Shepp-Olkin entropy concavity conjecture in arXiv preprint arXiv:1503.01570

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Hillion E (2014) A natural derivative on [0,  n ] and a binomial Poincaré inequality in ESAIM: Probability and Statistics

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Hillion E (2014) Contraction of Measures on Graphs in Potential Analysis

 
Description We have discovered a new way of understanding relationships between different distributions of randomness. By doing this, we were able to prove a conjecture (the Shepp-Olkin conjecture) which had been an open problem for 30+ years.
Exploitation Route Our constructions may be used in other mathematical contexts, and we have made further conjectures regarding randomness and uncertainty which can spark future research.
Sectors Digital/Communication/Information Technologies (including Software)