Information geometry of graphs

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.

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.