Information geometric duality of $\varphi$-deformed exponential families

Abstract

In the world of generalized entropies---which, for example, play a role in physical systems with sub- and super-exponential phasespace growth per degree of freedom---there are two ways for implementing constraints in the maximum entropy principle: linear- and escort constraints. Both appear naturally in different contexts. Linear constraints appear e.g. in physical systems, when additional information about the system is available through higher moments. Escort distributions appear naturally in the context of multifractals and information geometry. It was shown recently that there exists a fundamental duality that relates both approaches on the basis of the corresponding deformed logarithms (deformed-log duality). Here we show that there exists another duality that arises in the context of information geometry, relating the Fisher information of $\varphi$-deformed exponential families that correspond to linear constraints (as studied by J. Naudts), with those that are based on escort constraints (as studied by S.-I. Amari). We explicitly demonstrate this information geometric duality for the case of $(c,d)$-entropy that covers all situations that are compatible with the first three Shannon-Khinchin axioms, and that include Shannon, Tsallis, Anteneodo-Plastino entropy, and many more as special cases. Finally, we discuss the relation between the deformed-log duality and the information geometric duality, and mention that the escort distributions arising in the two dualities are generally different and only coincide for the case of the Tsallis deformation.

arXiv comment9 pages, 2 figures

References in corpus (20)

• Superstatistics
• Black hole thermodynamical entropy
• Statistical mechanics in the context of special relativity
• Statistical mechanics in the context of special relativity II
• Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive
• Constructing a statistical mechanics for Beck-Cohen superstatistics
• Generalized entropies and the transformation group of superstatistics
• Deformed exponentials and logarithms in generalized thermostatistics
• When do generalized entropies apply? How phase space volume determines entropy
• Geometry of escort distributions
• Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example
• The three faces of entropy for complex systems -- information, thermodynamics and the maxent principle
• Generalized entropies and logarithms and their duality relations
• Generalized Shannon-Khinchin Axioms and Uniqueness Theorem for Pseudo-additive Entropies
• A simple probabilistic construction yielding generalized entropies and divergences, escort distributions and q-Gaussians
• On the robustness of q-expectation values and Renyi entropy
• Classification of complex systems by their sample-space scaling exponents
• Generalization of the possible algebraic basis of $q$-triplets
• On the uniqueness theorem for pseudo-additive entropies
• Towards an Information Geometric characterization/classification of Complex Systems. I. Use of Generalized Entropies

Cited by in corpus (4)

• Thermodynamics of exponential Kolmogorov-Nagumo averages
• Information geometry of scaling expansions of non-exponentially growing configuration spaces
• On the equivalence between four versions of thermostatistics based on strongly pseudo-additive entropies
• Conditions for the existence of a generalization of Rényi divergence