Invariant geometric structures on statistical models

0449264 - MÚ 2016 RIV CH eng C - Konferenční příspěvek (zahraniční konf.)
Schwachhöfer, L. - Ay, N. - Jost, J. - Le, Hong-Van
Invariant geometric structures on statistical models.
Geometric Science of Information. Cham: Springer, 2015 - (Nielsen, F.; Barbaresco, F.), s. 150-158. Lecture Notes in Computer Science, 9389. ISBN 978-3-319-25039-7.
[International Conference on Geometric Science of Information (GSI) 2015 /2./. Palaiseau (FR), 28.10.2015-30.10.2015]
Institucionální podpora: RVO:67985840
Klíčová slova: geometric structures * statistical models
Kód oboru RIV: BA - Obecná matematika
http://link.springer.com/chapter/10.1007/978-3-319-25040-3_17

We review the notion of parametrized measure models and tensor fields on them, which encompasses all statistical models considered by Chentsov [6], Amari [3] and Pistone-Sempi [10]. We give a complete description of n-tensor fields that are invariant under sufficient statistics. In the cases n = 2 and n = 3, the only such tensors are the Fisher metric and the Amari-Chentsov tensor. While this has been shown by Chentsov [7] and Campbell [5] in the case of finite measure spaces, our approach allows to generalize these results to the cases of infinite sample spaces and arbitrary n. Furthermore, we give a generalisation of the monotonicity theorem and discuss its consequences.
Trvalý link: http://hdl.handle.net/11104/0250851