The Geometry of Statistical Data and Information: A Large Deviation Perspective (2501.01556v2)

Abstract: Combinatorics, probabilities, and measurements are fundamental to understanding information. This work explores how the application of large deviation theory (LDT) in counting phenomena leads to the emergence of various entropy functions, including Shannon's entropy, mutual information, and relative and conditional entropies. In terms of these functions, we reveal an inherent geometrical structure through operations, including contractions, lift, change of basis, and projections. Legendre-Fenchel (LF) transform which is central to both LDT and Gibbs' method of thermodynamics, offers a novel energetic description of data. The manifold of empirical mean values of statistical data ad infinitum has a parametrization using LF conjugates w.r.t. an entropy function; this gives rise to a family of models as a dual space and the additivity known in statistical thermodynamic energetics. This work clearly introduces data into the current information geometry, and includes information projection defined through conditional expectations in Kolmogorov's probability theory.

• The paper integrates large deviation theory, information geometry, and statistical thermodynamics to characterize empirical counting frequencies and analyze their geometrical structure.
• The work extends information geometry by defining empirical frequency manifolds and establishes a duality between empirical frequencies and their conjugate internal energies via the Legendre-Fenchel transform.
• The authors propose the Fisher information matrix, derived from rate function Hessians, as the appropriate metric for the space of empirical counting frequencies.

Extended Information Geometry: Large Deviation Theory, Statistical Thermodynamics, and Empirical Counting Frequencies

The paper explores the theoretical framework that integrates large deviation theory (LDT), information geometry (IG), and statistical thermodynamics to understand the characterization of empirical counting frequencies. This analysis reveals an elaborate geometrical structure, expanding on the traditional paper of probability distributions within the field of IG by incorporating entropy functions derived through combinatorial principles.

Key Contributions and Findings

1. Entropy Functions from Combinatorial Principles:
   • The authors utilize large deviation theory to conceptualize entropy functions, including Shannon's entropy, mutual information, and Kullback-Leibler divergence, arising from the elementary act of counting phenomena.
   • These entropy measures are mathematically framed as rate functions, reflecting the exponential growth or decreasing probability of observing sequences of data.
2. Geometric Interpretation in Statistical Models:
   • The work extends conventional information geometry by defining empirical frequency manifolds, providing a unique perspective distinct from probability-driven models.
   • A significant aspect is the establishment of a duality between empirical frequencies (derived from data) and internal energies (conceptualized in thermodynamic terms).
3. Role of the Legendre-Fenchel Transform:
   • Central to the paper is the use of the Legendre-Fenchel transform, linking empirical frequencies and their conjugate internal energies.
   • This duality elucidates the energetic description of data, highlighting a form of additivity akin to that seen in statistical thermodynamics.
4. Implications for Information Theory and Thermodynamics:
   • Large deviation theory, through contractions and projections, relates to classical IG in capturing the geometry of probability distributions.
   • The authors connect these notions to statistical mechanics and thermodynamics, leveraging the intrinsic properties of entropy as a measure of uncertainty and dynamical behavior.
5. Metric and Divergence Functions:
   • The Fisher information matrix, emerging from the Hessian of the rate functions, is posited as the appropriate metric for the space of empirical frequencies.
   • This choice of metrics, informed by statistical uncertainties, refines the Riemannian structure traditionally applied in statistical manifold theories.

Implications for AI and Future Work

The methodologies and insights presented in this paper offer fertile ground for advancements in AI as it involves nuanced understanding and manipulation of probabilistic models. For instance, expanding IG to more data-centric approaches enhances machine learning frameworks that include adaptive metrics and efficiently model dynamical data inputs. Potential pathways for future exploration also emerge in the form of more systematic ties between statistical thermodynamics and deep learning networks, particularly in unsupervised learning scenarios and in the design of systems grounded in probabilistic logic.

Furthermore, the concepts outlined may challenge the community to revisit the intersections between probabilistic reasoning and information processing, ultimately fostering implementations where AI systems can seamlessly integrate these foundational principles for optimized data handling and inference making.

In a nutshell, the paper proposes an enhanced theoretical scaffold for empirical data analysis, marrying traditional concepts in information theory with the rigor of thermodynamic principles, potentially driving innovations in both the cognitive and practical dimensions of AI research.