Geometry of deformed exponential families: Invariant, dually-flat and conformal geometries

An information-geometrical foundation is established for the deformed exponential families of probability distributions. Two different types of geometrical structures, an invariant geometry and a flat geometry, are given to a manifold of a deformed exponential family. The two different geometries provide respective quantities such as deformed free energies, entropies and divergences. The class belonging to both the invariant and flat geometries at the same time consists of exponential and mixture families. Theq-families are characterized from the viewpoint of the invariant and flat geometries. The q-exponential family is a unique class that has the invariant and flat geometries in the extended class of positive measures. Furthermore, it is the only class of which the Riemannian metric is conformally connected with the invariant Fisher metric.

[1]  Filtrations for which All ℋ2 Martingales Are of Integrable Variation; Distances between σ-Algebras , 2008 .

[2]  H. Matsuzoe Geometry of contrast functions and conformal geometry , 1999 .

[3]  Shun-ichi Amari,et al.  Geometry of q-Exponential Family of Probability Distributions , 2011, Entropy.

[4]  Takashi Kurose Conformal-Projective Geometry of Statistical Manifolds , 2002 .

[5]  Takashi Kurose ON THE DIVERGENCES OF 1-CONFORMALLY FLAT STATISTICAL MANIFOLDS , 1994 .

[6]  J. Naudts Deformed exponentials and logarithms in generalized thermostatistics , 2002, cond-mat/0203489.

[7]  A. Ohara Geometric study for the Legendre duality of generalized entropies and its application to the porous medium equation , 2009, 0904.1530.

[8]  I. Csiszár Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems , 1991 .

[9]  Noboru Murata,et al.  A Generalization of Independence in Naive Bayes Model , 2010, IDEAL.

[10]  Shun-ichi Amari,et al.  A dually flat structure on the space of escort distributions , 2010 .

[11]  S. Amari,et al.  Asymptotic theory of sequential estimation : Differential geometrical approach , 1991 .

[12]  Christian Beck,et al.  Generalised information and entropy measures in physics , 2009, 0902.1235.

[13]  Jan Naudts Generalised Exponential Families and Associated Entropy Functions , 2008, Entropy.

[14]  S. Eguchi Information Geometry and Statistical Pattern Recognition , 2004 .

[15]  C. Tsallis Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World , 2009 .

[16]  Shun-ichi Amari,et al.  $\alpha$ -Divergence Is Unique, Belonging to Both $f$-Divergence and Bregman Divergence Classes , 2009, IEEE Transactions on Information Theory.

[17]  Estimators, escort probabilities, and phi-exponential families in statistical physics , 2004, math-ph/0402005.

[18]  F. Topsøe Entropy and equilibrium via games of complexity , 2004 .

[19]  Giovanni Pistone,et al.  An Infinite-Dimensional Geometric Structure on the Space of all the Probability Measures Equivalent to a Given One , 1995 .

[20]  H. Matsuzoe On realization of conformally-projectively flat statistical manifolds and the divergences , 1998 .

[21]  C. Tsallis Possible generalization of Boltzmann-Gibbs statistics , 1988 .

[22]  A. Ohara Geometry of distributions associated with Tsallis statistics and properties of relative entropy minimization , 2007 .

[23]  A. Dawid,et al.  Game theory, maximum entropy, minimum discrepancy and robust Bayesian decision theory , 2004, math/0410076.

[24]  Giovanni Pistone,et al.  Exponential statistical manifold , 2007 .

[25]  A. M. Scarfone,et al.  Deformed logarithms and entropies , 2004, cond-mat/0402418.

[26]  Hirohiko Shima,et al.  Geometry of Hessian Structures , 2013, GSI.

[27]  Giovanni Pistone κ-exponential models from the geometrical viewpoint , 2009, 0903.2012.

[28]  Dual connections in nonparametric classical information geometry , 2001, math-ph/0104031.

[29]  N. N. Chent︠s︡ov Statistical decision rules and optimal inference , 1982 .