# Geometric Aspects of Conditional Independence and Information

## Abstracts

### A Refinement of the Common Cause Principle

**Nihat Ay** *(MPI MiS Leipzig, Germany)*

Thursday, March 13, 2008

I study the interplay between stochastic dependence and causal relations within the setting of Bayesian networks and in terms of information theory. The application of a recently defined causal information flow measure provides a quantitative refinement of Reichenbach's common cause principle, see the working paper at Santa Fe Institute.

### On minimization of entropy functionals under moment constraints

**Imre Csiszár** *(Hungarian Academy of Sciences, Hungary)* (joint work with *František Matúš *)

Friday, March 14, 2008

Minimization problems for entropy-like integrals and Bregman
distances subject to a finite number of moment constraints are addressed in
a general setting. Analogues of the authors' previous results on information
projections to families determined by linear constraints, and reverse
information projections to exponential families, are established. No
constraint qualification is assumed.

### Decision Geometry

**A. Philip Dawid** *(University College London, United Kingdom)*

Friday, March 14, 2008

A decision problem is defined in terms of an outcome space, an action
space and a loss function. Starting from these simple ingredients, we can
construct: Proper Scoring Rule; Entropy Function; Divergence Function;
Riemannian Metric; and Unbiased Estimating Equation. From an abstract
viewpoint, the loss function defines a duality between the outcome and
action spaces, while the correspondence between a distribution and its
Bayes act induces a self-duality. Together these determine a decision
geometry for the family of distributions on outcome space. This allows
generalisation of many standard statistical concepts and properties. In
particular we define and study generalised exponential families.

### Exponential families with semigroup-valued sufficient statistics

**Steffen Lauritzen** *(University of Oxford, United Kingdom)*

Thursday, March 13, 2008

We consider an algebraic extension of the notion of an exponential family where
the sufficient statistic takes values in an Abelian semigroup rather than a
vector space. The characters of the semigroup play a role of exponential
functions. As opposed to standard exponential families, characters are not
necessarily everywhere positive. Still we show how basic results concerning
existence and uniqueness of the MLE can be established, in some sense yielding a
simpler theory. We shall describe examples which are very different from
standard families and others which appear as natural extensions of standard
families. In particular we shall examine graphical and hierarchical log-linear
models in this light.

### Entropy functions, information inequalities and polymatroids

**František Matúš** *(Academy of Sciences of the Czech Republic, Czech Republic)*

Friday, March 14, 2008

Shannon entropies of all subvectors of a random vector are considered
for the coordinates of an entropic point in a Euclidean space. The
problem to find all entropic points will be reviewed and its relation
to conditional independence structures discussed. Inequalities for
the entropic points and their applications will be presented from the
viewpoint of cones of polymatroids.

### Relations among conditional probabilities

**Jason Morton** *(Stanford University, USA)*

Friday, March 14, 2008

We describe a Gröbner basis of relations among conditional
probabilities in a discrete probability space, with any set of
conditioned-upon events. They may be specialized to the random
variable case, the purely conditional case, and other
special cases. We also investigate the connection to generalized
permutohedra and describe a "conditional probability simplex."

### Optimizing secret sharing schemes for general access structure

**Carles Padró** *(Universitat Politècnica de Catalunya, Spain)*

Friday, March 14, 2008

A secret sharing scheme is a method to distribute a secret value into
shares in such a way that only some qualified subsets of participants
are able to recover the secret from their shares. The family of the
qualified subsets is the access structure of the scheme. Determining the
optimal complexity of secret sharing schemes for any given access
structure is a very difficult and long-standing open problem, which
involves varied and deep mathematical techniques. This talk is a survey
about the last results on this problem.
A special stress will be put on the connections between matroids and
ideal secret sharing schemes, that is, schemes with minimum-length
shares. In particular, some recent results about the length of the
shares in secret sharing schemes for the Vamos matroid, in a joint work
with Amos Beimel and Noam Livne, will be presented. Specifically,
non-Shannon inequalities for the entropy function are used for the first
time in secret sharing to find the first example of a matroid in which
the length of the shares is larger than the length of the secret by a
constant factor.

### From f-divergence to quantum quasi-entropy

**Dénes Petz** *(Hungarian Academy of Sciences, Hungary)*

Friday, March 14, 2008

The talk gives a review about the quantum setting compared
with the classical. As particular cases of quasi-entropy, generalized
covariance, Fisher information and skew information are discussed.

### Factorization criteria for Markov models induced by directed acyclic graphs via marginalization

**Thomas Richardson** *(University of Washington, USA)*

Thursday, March 13, 2008

We will give a characterization of the distributions with Markovian
structure corresponding to directed acyclic graphs under
marginalization. These Markov models correspond to graphs containing directed
and bi-directed edges, but no directed cycles, also called, acyclic
directed mixed graphs. This factorization criterion leads directly to a
parameterization in the binary case.

### Some remarks on algebraic approach to learning Bayesian nets

**Milan Studený** *(Academy of Sciences of the Czech Republic, Czech Republic)*

Thursday, March 13, 2008

I plan to present the idea of algebraic approach
to learning Bayesian networks by maximization of a quality criterion. The idea is
transform the problem to a classic linear programming problem. Some open question
should be presented.

### Marginal Likelihood Integrals for Mixtures of Independence Models

**Bernd Sturmfels** *(University of California at Berkeley, USA)*

Thursday, March 13, 2008

Evaluation of marginal likelihood integrals is central to
Bayesian statistics. It is generally assumed that these integrals cannot
be evaluated exactly, except in trivial case, and many techniques
(e.g. MCMC)
have been developed to obtain asymptotics and approximations. This
lecture
argues that exact integration is more feasible than is widely believed.
We present an exact algebraic method for the computation of marginal
likelihood integrals for a class of mixture models for discrete data.
This is joint work in progress with Shaowei Lin and Zhiqiang Xu.

### Counterexamples to additivity of minimum output p-Renyi entropies of channels, for parameter p>2, p>1 and p near 0

**Andreas Winter** *(University of Bristol, United Kingdom)*

Thursday, March 13, 2008

The 'standard' additivity conjectures are about entropic quantities associated to channels: Holevo capacity and minimum output entropy. While these are still open, there is a long-standing approach using Renyi entropies. For a long time they have been conjectured to obey similar additivity relations, but this has recently been disproved. The talk is based on the author's preprint arXiv:0707.0402, and Patrick Hayden's more recent arXiv:0707.3291, plus as yet unpublished work by Chris King, and further counterexamples at p=0 due to the author with Toby Cubitt, Aram Harrow, Debbie Leung and Ashley Montanaro.

## Date and Location

**March 13 - 14, 2008**

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

Information Theory of Cognitive Systems Group

Leipzig

Contact by Email

**František Matúš**

Academy of Sciences of the Czech Republic

Institute of Information Theory and Automation

Prague

Contact by Email

## Administrative Contact

**Antje Vandenberg**

Max Planck Institute for Mathematics in the Sciences

Contact by Email