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]).
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Notes
- 1.
Since we do not consider non-covariant n-tensor fields in this paper, we shall suppress the attribute covariant.
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Acknowledgements
This work was mainly carried out at the Max Planck Institute for Mathematics in the Sciences in Leipzig, and we are grateful for the excellent working conditions provided at that institution. The research of H.V. Lê was supported by the GAČR project 18-01953J and RVO: 67985840. L. Schwachhöfer acknowledges partial support by grant SCHW893/5-1 of the Deutsche Forschungsgemeinschaft.
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Schwachhöfer, L., Ay, N., Jost, J., Vân Lê, H. (2018). Congruent Families and Invariant Tensors. In: Ay, N., Gibilisco, P., Matúš, F. (eds) Information Geometry and Its Applications . IGAIA IV 2016. Springer Proceedings in Mathematics & Statistics, vol 252. Springer, Cham. https://doi.org/10.1007/978-3-319-97798-0_6
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