Congruent families and invariant tensors

Abstract

A classical result of Chentsov states 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 Ay, Jost, Lê, Schwachhöfer (Bernoulli, 24:1692–1725, 2018, [4]).