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Open AccessFeature PaperArticle

On the Limiting Behaviour of the Fundamental Geodesics of Information Geometry

1
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, Buckinghamshire MK7 6AA, UK
2
Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 2G1, Canada
*
Author to whom correspondence should be addressed.
Entropy 2017, 19(10), 524; https://doi.org/10.3390/e19100524
Received: 13 July 2017 / Revised: 13 September 2017 / Accepted: 28 September 2017 / Published: 30 September 2017
(This article belongs to the Special Issue Information Geometry II)
The Information Geometry of extended exponential families has received much recent attention in a variety of important applications, notably categorical data analysis, graphical modelling and, more specifically, log-linear modelling. The essential geometry here comes from the closure of an exponential family in a high-dimensional simplex. In parallel, there has been a great deal of interest in the purely Fisher Riemannian structure of (extended) exponential families, most especially in the Markov chain Monte Carlo literature. These parallel developments raise challenges, addressed here, at a variety of levels: both theoretical and practical—relatedly, conceptual and methodological. Centrally to this endeavour, this paper makes explicit the underlying geometry of these two areas via an analysis of the limiting behaviour of the fundamental geodesics of Information Geometry, these being Amari’s (+1) and (0)-geodesics, respectively. Overall, a substantially more complete account of the Information Geometry of extended exponential families is provided than has hitherto been the case. We illustrate the importance and benefits of this novel formulation through applications. View Full-Text
Keywords: extended exponential families; information geometry; Riemannian Markov Chain Monte Carlo extended exponential families; information geometry; Riemannian Markov Chain Monte Carlo
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Critchley, F.; Marriott, P. On the Limiting Behaviour of the Fundamental Geodesics of Information Geometry. Entropy 2017, 19, 524.

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