Abstract
We demonstrate that the q-exponential family particularly admits natural geometrical structures among deformed exponential families. The property is the invariance of structures with respect to a general linear group, which transitively acts on the space of positive definite matrices. We prove this property via the correspondence between information geometry induced by a deformed potential on the space and the one induced by what we call β-divergence defined on the q-exponential family with q = β + 1. The results are fundamental in robust multivariate analysis using the q-Gaussian family.
Highlights
Generalizations of the exponential family have recently had much attention paid to them in mathematical statistics or statistical physics [1,2,3,4,5,6]. One of their goals is to attack a wider class of statistical problems that ranges outside the ones solved via the well-established theory of the exponential family [7]
We prove that information geometry on the q-Gaussian family induced by the β-divergence is equivalently characterized by the V -geometry on the space of positive definite matrices induced by the power potential
Studying the corresponding V -geometry, we show that some of the dually flat structures of the q-Gaussian family admit the GL(n, R)-invariances
Summary
Generalizations of the exponential family have recently had much attention paid to them in mathematical statistics or statistical physics [1,2,3,4,5,6] One of their goals is to attack a wider class of statistical problems that ranges outside the ones solved via the well-established theory of the exponential family [7]. We focus on investigating the geometric structures of the q-Gaussian family induced from the β-divergence [1] via the V -geometry following the above idea. For this purpose, we establish the correspondence between two geometries and derive explicit formulas of important geometric quantities, such as the Riemannian metric and mutually dual affine connections.
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