Graphical models, exponential families

Exponential Families and Graphical Models 2 Define d | A(θ) exponential family is where Ω is an open set. A minimal exponential family is where the φ’s are linearly independent, namely there does not exist a nonzero α ∈ Rd such that αφ(x) = constant.  · Graphical Models, Exponential Families, and Variational Inference. The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including . The ‘bible’ of graphical models, and much of the first half of this course is based on this. One complication is that the book makes a distinction between two different types of vertex, which can make some ideas look more complicated. 2. M.J. Wainwright and M.I. Jordan, Graphical Models, Exponential Families, and Vari-.

Abstract. We provide a classification of graphical models according to their representation as subfamilies of exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical models and chain graphs with no hidden variables, including Bayesian networks with several families of local distributions, are curved. Graphical Models, Exponential Families, and Variational Inference Abstract: The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical. [Show full abstract] graphical models with hidden variables are stratified exponential families (SEFs). A SEF is a finite union of CEFs of various dimensions satisfying some regularity conditions.

We show that some graphical models with no hidden variables including Bayesian networks with several families of local distributions are Curved Exponential Families (CEFs). We also show that Baysian networks with hidden variables, and several other types of graphical models including non-chordal undirected graphical models are Stratified Exponential Families (SEFs). Graphical Models, Exponential Families, and Variational Inference. The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, Graphical Models, Exponential Families and Variational Inference develops general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations.