Conformal Flattening for Deformed Information Geometries on the Probability Simplex ^†

Area of Science:

• Statistical Science
• Information Geometry

Background:

• Generalized exponential functions are advancing statistical models.
• Representing functions are key to deforming information geometry structures.

Purpose of the Study:

• Investigate invariance and dual flatness in information geometry.
• Characterize representing functions for invariant geometries.
• Construct dually flat geometries from non-flat ones.

Main Methods:

• Solving ordinary differential equations to characterize invariant geometry.
• Proposing conformal flattening for constructing dually flat geometries.

Main Results:

• A method to identify representing functions for invariant geometries.
• A technique to create dually flat geometries using conformal flattening.
• Demonstration of gradient flow properties on probability simplexes.

Conclusions:

• The study provides a theoretical framework for constructing specific information geometries.
• The findings have implications for understanding complex statistical models and their geometric properties.