Invariant geometric structures on statistical models

Publikationstyp
Conference Paper
Date Issued
2015-10
Sprache
English
Author(s)
Schwachhöfer, Lorenz  
Ay, Nihat 
Jost, Jürgen  
Lê, Hông Vân  
TORE-URI
http://hdl.handle.net/11420/14098
First published in
Lecture notes in computer science  
Number in series
9389 LNIP
Start Page
150
End Page
158
Citation
Lecture Notes in Computer Science 9389 LNIP: 150-158 (2015)
Contribution to Conference
2nd International Conference on Geometric Science of Information, GSI 2015  
Publisher DOI
10.1007/978-3-319-25040-3_17
Scopus ID
2-s2.0-84950311310
Publisher
Springer
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.
DDC Class
004: Informatik
510: Mathematik