Graphical methods

Graph associahedra are natural generalizations of the classical associahedra. They provide polytopal realizations of the nested complex of a graph G, defined as the simplicial complex whose vertices are the tubes (i.e. connected induced subgraphs) of G and whose faces are the tubings (i.e. collections of pairwise nested or non-adjacent tubes) of G. They appeared in the work of M. Carr and S. Devadoss, and were further studied by A. Postnikov, by E.-M. Feichtner and B. Sturmfels, and by A. Zelevinsky. Recently, they also appeared in the work of T. Lam and P.

This lecture will describe two applications of Bayesian graphic models.

Graphical Markov models use graphs with nodes
corresponding to random variables, and edges that
encode conditional independence relationships between
those variables. Directed graphical models (aka Bayesian
networks) in particular have received considerable attention.
This lecture will review basic concepts in graphical
model theory such as Markov properties, equivalence,
and connections with causal inference.

Capturing complex interactions among a large set of variables is a

Learning high-dimensional probability distributions with a very
reduced number of samples is no more difficult than with a great
many. However, arranging for such models to generalize well in the
small-sample domain is hard. Our approach is motivated by
compositional models and Bayesian networks, and designed to adapt to
sample size. We start with a large, overlapping set of elementary
statistical building blocks, or primitives, which are
low-dimensional marginal distributions learned from data. Subsets of

The problem of recovering the sparsity pattern of an unknown signal arises in various domains, including graphical model selection, signal denoising, constructive approximation, compressive sensing, and subset selection in regression. The standard optimization-theoretic formulation of sparsity recovery involves l_0-constraints, and typically leads to computationally intractable optimization problems.