It is shown that the information geometrical notions such as Fisher metric, alpha-connections, divergence and exponential families are naturally extended to manifolds of Markov transition matrices, where the theory of dual connections works very well as in the usual information geometry for manifolds of probability distributions. Some statistical and probabilistic implications are also given in asymptotic settings.
We prove certain necessary and sufficient conditions for the existence of embedding of statistical manifolds. In particular we prove that any smooth statistical manifold can be embedded into the space of all probability measures on a finite set. As a result we get positive answers to the question by Amari on the existence of embedding of exponential families and to the Lauritzen question on realization of statistical manifolds as statistical models.
The variation distance closures of exponential families and their log-convex subfamilies will be presented. They provide nontrivial natural boundaries to general exponential families viewed as manifolds. Closures in a reversed information divergence, rI-closure, and its statistical relevance will be discussed. An exponential family is constructed such that its rI-closure, is neither rI-closed nor log-convex.
The Shannon-McMillan theorem is of fundamental importance in classical information theory, ergodic theory and probability theory. It is natural to ask, wether an analogous statement holds in the noncommutative setting of quantum information theory. For ergodic quantum sources we could prove a corresponding theorem with the von Neumann entropy replacing the classical metric entropy. Some generalizations and applications will be discussed.
We investigate the structure of the global maximizers of stochastic interdependence, which is measured by the Kullback-Leibler divergence of the underlying joint probability distribution from the exponential family of factorizable random fields (multi-information). As a consequence of our structure results, it comes out that random fields with globally maximal multi-information are contained in the topological closure of the exponential family of pair interactions.
Organizers
Nihat Ay
Max Planck Institute for Mathematics in the Sciences
Jürgen Jost
Max-Planck-Institut für Mathematik in den Naturwissenschaften
Administrative Contact
Antje Vandenberg
Max-Planck-Institut für Mathematik in den Naturwissenschaften
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