Title: Congruent families and invariant tensors,
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.