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.
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.
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.
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.
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.
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.
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.
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.
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."
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.
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.
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.
Organizers
Nihat Ay
Max Planck Institute for Mathematics in the Sciences, Leipzig
František Matúš
Academy of Sciences of the Czech Republic, Prague
Administrative Contact
Antje Vandenberg
Max-Planck-Institut für Mathematik in den Naturwissenschaften
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