Introduction to Imprecise Probabilities

Chapter 9Probabilistic graphical models

Alessandro Antonucci, Cassio P. de Campos and Marco Zaffalon

Instituto Dalle Molle di Studi sull'Intelligenza Artificiale (IDSIA), Manno-Lugano, Switzerland

9.1 Introduction

In the previous chapters of this book the reader has been introduced to a number of powerful tools for modelling uncertain knowledge with imprecise probabilities. These have been formalized in terms of sets of desirable gambles (Chapter 1), lower previsions (Chapter 2), and sets of linear previsions (Section 1.6.2), while their relations with other uncertainty models have been also described (Chapter 4). In the discrete multivariate case, a direct specification of models of this kind might be expensive because of too a high number of joint states, this being exponential in the number of variables. Yet, a compact specification can be achieved if the model displays particular invariance (Chapter 3) or composition properties. The latter is exactly the focus of this chapter: defining a model over its whole set of variables by the composition of a number of sub-models each involving only fewer variables. More specifically, we focus on the kind of composition induced by independence relations among the variables. Graphs are particularly suitable for the modelling of such independencies, so we formalize our discussion within the framework of probabilistic graphical models. Following these ideas, we introduce a class of probabilistic graphical models with imprecision based ...