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تسجيل مستخدم جديدFluctuation geometry: A counterpart approach of inference geometry
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L Velazquez
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Starting from an axiomatic perspective, emph{fluctuation geometry} is developed as a counterpart approach of inference geometry. This approach is inspired on the existence of a notable analogy between the general theorems of emph{inference theory} and the the emph{general fluctuation theorems} associated with a parametric family of distribution functions $dp(I|theta)=rho(I|theta)dI$, which describes the behavior of a set of emph{continuous stochastic variables} driven by a set of control parameters $theta$. In this approach, statistical properties are rephrased as purely geometric notions derived from the emph{Riemannian structure} on the manifold $mathcal{M}_{theta}$ of stochastic variables $I$. Consequently, this theory arises as an alternative framework for applying the powerful methods of differential geometry for the statistical analysis. Fluctuation geometry has direct implications on statistics and physics. This geometric approach inspires a Riemannian reformulation of Einstein fluctuation theory as well as a geometric redefinition of the information entropy for a continuous distribution.
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Fluctuation geometry was recently proposed as a counterpart approach of Riemannian geometry of inference theory. This theory describes the geometric features of the statistical manifold $mathcal{M}$ of random events that are described by a family of
continuous distributions $dp(x|theta)$. A main goal of this work is to clarify the statistical relevance of Levi-Civita curvature tensor $R_{ijkl}(x|theta)$ of the statistical manifold $mathcal{M}$. For this purpose, the notion of emph{irreducible statistical correlations} is introduced. Specifically, a distribution $dp(x|theta)$ exhibits irreducible statistical correlations if every distribution $dp(check{x}|theta)$ obtained from $dp(x|theta)$ by considering a coordinate change $check{x}=phi(x)$ cannot be factorized into independent distributions as $dp(check{x}|theta)=prod_{i}dp^{(i)}(check{x}^{i}|theta)$. It is shown that the curvature tensor $R_{ijkl}(x|theta)$ arises as a direct indicator about the existence of irreducible statistical correlations. Moreover, the curvature scalar $R(x|theta)$ allows to introduce a criterium for the applicability of the emph{gaussian approximation} of a given distribution function. This type of asymptotic result is obtained in the framework of the second-order geometric expansion of the distributions family $dp(x|theta)$, which appears as a counterpart development of the high-order asymptotic theory of statistical estimation. In physics, fluctuation geometry represents the mathematical apparatus of a Riemannian extension for Einsteins fluctuation theory of statistical mechanics. Some exact results of fluctuation geometry are now employed to derive the emph{invariant fluctuation theorems}.
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