# Information Geometric Complexity of Entropic Motion on Curved Statistical Manifolds under Different Metrizations of Probability Spaces

## TL;DR

This paper compares how different metrics on probability spaces affect the complexity and convergence of entropic motion on curved statistical manifolds, revealing tradeoffs between complexity growth and convergence speed.

## Contribution

It provides a comparative analysis of Fisher-Rao and alpha-order entropy metrics on Gaussian manifolds, highlighting their impact on information geometric entropy and convergence behavior.

## Key findings

- Fisher-Rao metric leads to linear growth of IGE and fast convergence.
- Alpha-order entropy metric results in logarithmic IGE growth and slow convergence.
- Insights into the tradeoff between complexity and convergence speed in entropic inference.

## Abstract

We investigate the effect of different metrizations of probability spaces on the information geometric complexity of entropic motion on curved statistical manifolds. Specifically, we provide a comparative analysis based upon Riemannian geometric properties and entropic dynamical features of a Gaussian probability space where the two distinct dissimilarity measures between probability distributions are the Fisher-Rao information metric and the alpha-order entropy metric. In the former case, we observe an asymptotic linear temporal growth of the information geometric entropy (IGE) together with a fast convergence to the final state of the system. In the latter case, instead, we note an asymptotic logarithmic temporal growth of the IGE together with a slow convergence to the final state of the system. Finally, motivated by our findings, we provide some insights on a tradeoff between complexity and speed of convergence to the final state in our information geometric approach to problems of entropic inference.