# Tropical diagrams of probability spaces

## TL;DR

This paper introduces the tropical cone of diagrams of probability spaces, analyzing its large-scale geometry and potential applications in information theory and artificial intelligence.

## Contribution

It defines the tropical cone structure for diagrams of probability spaces and studies its asymptotic geometry, linking it to entropy and information optimization.

## Key findings

- Identification of the asymptotic cone as a convex cone in a Banach space
- Introduction of tropical diagrams as flexible objects within the tropical cone
- Potential applications to entropic cone analysis and information optimization

## Abstract

After endowing the space of diagrams of probability spaces with an entropy distance, we study its large-scale geometry by identifying the asymptotic cone as a closed convex cone in a Banach space. We call this cone the tropical cone, and its elements tropical diagrams of probability spaces. Given that the tropical cone has a rich structure, while tropical diagrams are rather flexible objects, we expect the theory of tropical diagrams to be useful for information optimization problems in information theory and artificial intelligence. In a companion article, we give a first application to derive a statement about the entropic cone.