INTRODUCTION The article GRAPHICAL MODELS: PARAMETER LEARNING discussed the learning of parameters for a xed graphical model. In this article, we discuss the simultaneous learning of parameters and structure. Real-world applications of such learning abound and can be found in (e.g.) the Proceedings of the Conference on Uncertainty in Arti cial Intelligence (1991 and after). An index to software...

for some functions f and g. Probabilistic graphical models are a way of representing conditional independence assumptions using graphs. Nodes represent random variables and lack of edges represent conditional independence assumptions, in a way which we will define below. There are many kinds of graphical model, but the two most popular are Bayesian (belief) networks1, which are based on directe...

Two real-valued or vector-valued random variables X, Y are independent for probability measure P (written: X ⊥ Y [P ]) if for all sets A and B, P[X ∈ A, Y ∈ B] = P[X ∈ A] · P[Y ∈ B]. For jointly discrete or jointly continuous random variables this is equivalent to factoring of the joint probability mass function or probability density function, respectively. The variables X and Y are conditiona...

We combine the ideas behind trees and Gaussian graphical models to form a new nonparametric family of graphical models. Our approach is to attach nonparanormal “blossoms”, with arbitrary graphs, to a collection of nonparametric trees. The tree edges are chosen to connect variables that most violate joint Gaussianity. The non-tree edges are partitioned into disjoint groups, and assigned to tree ...

Graphical modeling is an important method to efficiently represent and analyze uncertain information in knowledge-based systems. Its most prominent representatives are Bayesian networks and Markov networks for probabilistic reasoning, which have been well-known for over ten years now. However, they suffer from certain deficiencies, if imprecise information has to be taken into account. Therefor...

This chapter discusses the use of directed acyclic graphs (DAGs) for causal inference in the observational social sciences. It focuses on DAGs’ main uses, discusses central principles, and gives applied examples. DAGs are visual representations of qualitative causal assumptions: They encode researchers’ beliefs about how the world works. Straightforward rules map these causal assumptions onto t...

Probabilistic graphical models are being used widely in artiicial intelligence, for instance, in diagnosis and expert systems, as a uniied qualitative and quantitative framework for representing and reasoning with probabilities and independencies. Their development and use spans several elds including artiicial intelligence, decision theory and statistics, and provides an important bridge betwe...

We study a rich family of distributions that capture variable interactions significantly more expressive than those representable with low-treewidth or pairwise graphical models, or log-supermodular models. We call these cooperative graphical models. Yet, this family retains structure, which we carefully exploit for efficient inference techniques. Our algorithms combine the polyhedral structure...

I am grateful for the opportunity to respond to these two excellent papers. Although graphical models are intuitively compelling for conceptualizing statistical associations, the scientiic community generally views such models with hesitancy and suspicion. The two papers before us demonstrate the use of graphs { speciically, directed acyclic graphs (DAGs) { as a mathematical tool of great versa...