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    Item type:Publication, Access status: without files ,
    Invariant geometric structures on statistical models
    (Springer, 2015-10)
    Schwachhöfer, Lorenz
    ;
    Ay, Nihat
    ;
    Jost, Jürgen
    ;
    Lê, Hông Vân
    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.
    Publicationtype: Conference Paper
      16
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    Item type:Publication, Access status: without files ,
    Standard divergence in manifold of dual affine connections
    (Springer, 2015-10)
    Amari, Shun'ichi
    ;
    Ay, Nihat
    A divergence function defines a Riemannian metric G and dually coupled affine connections (∇,∇∗) with respect to it in a manifold M. When M is dually flat, a canonical divergence is known, which is uniquely determined from G,∇,∇∗. We search for a standard divergence for a general non-flat M. It is introduced by the magnitude of the inverse exponential map, where α = −(1/3) connection plays a fundamental role. The standard divergence is different from the canonical divergence.
    Publicationtype: Conference Paper
      17