# Mini-Workshop: Information Geometry 2003

## Abstracts for the talks

### Maximizing multi-information

**N. Ay** (joint work with *A. Knauf, F. Matúš*)

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.

### The quantum Shannon-McMillan-theorem and related topics

**T. Krüger** (joint work with *I. Bjelakovic, R. Siegmund-Schultze, A. Szkola*)

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.

### Extensions and closures of exponential families

**F. Matúš** (joint work with *I. Csiszár *)

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.

### Information geometry of Markov chains

**H. Nagaoka**

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.

### Monotone invariants and embeddings of statistical manifolds

**H. Van Lè**

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.

## Date and Location

**August 27, 2003**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Nihat Ay**

Max Planck Institute for Mathematics in the Sciences

**Jürgen Jost**

Max Planck Institute for Mathematics in the Sciences

## Administrative Contact

**Antje Vandenberg**

Max Planck Institute for Mathematics in the Sciences