# Congruent families and invariant tensors

## TL;DR

This paper generalizes classical invariance results in information geometry, showing that invariant tensor families under congruent Markov morphisms are generated by canonical tensors for any degree n.

## Contribution

It extends the characterization of invariant tensors from 2- and 3-tensors to arbitrary degree n, linking them to canonical tensor fields.

## Key findings

- Invariant tensor families are algebraically generated by canonical tensors.
- Classical invariance results are extended to higher-degree tensors.
- The work unifies the understanding of invariant tensors in statistical models.

## Abstract

Classical results of Chentsov and Campbell state that -- up to constant multiples -- the only $2$-tensor field of a statistical model which is invariant under congruent Markov morphisms is the Fisher metric and the only invariant $3$-tensor field is the Amari-Chentsov tensor. We generalize this result for arbitrary degree $n$, showing that any family of $n$-tensors which is invariant under congruent Markov morphisms is algebraically generated by the canonical tensor fields defined in an earlier paper.