$U$-entropy and maximum entropy model

  • Shinto Eguchi (Institute of Statistical Mathematics, Japan)
University n.n. Universität Leipzig (Leipzig)


Boltzmann-Shannon entropy leads to an exponential model as the maximum entropy model with the constraint to the space of pdfs under which expectations of a given statistic $t(x)$ become a common vector. The maximum likelihood estimator for the expectation parameter of $t(x)$ under the exponential model is characterized by specific properties such as the attainment the Cramer-Rao bound. Any generator function $U$ defines $U$-entropy and $U$-divergence from the assumption of convexity of $U$. In this framework, $U$-entropy leads to $U$-model as the maximum entropy model under which the minimum $U$-divergence estimator for the expectation parameter is characterized by a structure of orthogonal foliation. If $U(s) = \exp(s)$, then this reduces to the case of Boltzmann-Shannon entropy. Surprisingly, we observe that the minimum $U$-divergence estimator under the $U$-model has a unique form, that is, the sample mean of $t(x)$. Alternatively if the minimum $U$-divergence estimator is employed under another $U$-model, then the estimator has a different form with the weighted mean of $t(x)$ over the sample. This talk discusses information geometric understandings for this aspect with Pythagoras identity, minimax game and robustness.

02.08.10 06.08.10

Information Geometry and its Applications III

Universität Leipzig (Leipzig) University n.n. University n.n.

Antje Vandenberg

Max-Planck-Institut für Mathematik in den Naturwissenschaften Contact via Mail

Nihat Ay

Max Planck Institute for Mathematics in the Sciences, Germany

Paolo Gibilisco

Università degli Studi di Roma "Tor Vergata", Italy

František Matúš

Academy of Sciences of the Czech Republic, Czech Republic