Time-Embedded Information Geometry: A Compatible Geometric Extension of the Fisher–Rao Framework

Introduction
Classical information geometry provides a differential-geometric framework for statistical models based on structures such as statistical manifolds and the Fisher–Rao metric. However, time is typically treated as an external parameter, which limits the representation of evolving statistical systems and dynamic probabilistic structures.

Methods
We propose a conceptual extension called Time-Embedded Information Geometry (TEIG), in which temporal evolution is incorporated directly into the structure of the statistical model. Instead of treating time as an external variable, it is embedded as an additional coordinate within the model. This formulation enables a unified geometric representation that captures both parameter variation and temporal change, while preserving the underlying statistical structure.

Results
We show that this formulation remains compatible with the classical Fisher–Rao geometry and provides a consistent way to describe time-dependent statistical configurations. The proposed framework maintains structural coherence with existing theory while extending its expressive capability. In particular, when temporal effects are neglected, the framework reduces naturally to standard information geometry.

Conclusions
TEIG offers a geometrically consistent extension of Fisher–Rao-based models and provides a new perspective for analyzing time-dependent statistical systems. This approach may contribute to broader applications in mathematical modeling, dynamical systems, and computational science. Furthermore, it provides a flexible framework for integrating temporal dynamics into geometric modeling.