A Deformed Exponential Statistical Manifold

Area of Science:

• Probability theory
• Differential geometry
• Information geometry

Background:

• The set of strictly positive probability densities (Pμ) lacks a standard manifold structure.
• Deformed exponential functions offer a flexible way to model probability distributions.

Purpose of the Study:

• To equip the set of probability densities (Pμ) with a C∞-Banach manifold structure.
• To explore generalized divergence measures using deformed exponential functions.

Main Methods:

• Utilizing a φ-connection with a deformed exponential function (φ) to define the manifold structure.
• Calculating the tangent space and tangent bundle of the probability density manifold.
• Defining a new divergence based on the q-exponential function.

Main Results:

• Established Pμ as a C∞-Banach manifold.
• Characterized the tangent space and tangent bundle of Pμ.
• Demonstrated that the newly defined q-exponential divergence is related to the existing q-divergence.
• Showcased the generalization of Rényi divergence using q-exponential and κ-exponential functions.

Conclusions:

• The φ-connection provides a robust method for manifold construction in probability theory.
• The generalized divergence measures have implications for statistical inference and machine learning.