Hybrid random fields. A scalable approach to structure and parameter learning in probabilistic graphical models
This book presents novel probabilistic graphical models, i.e., hybrid random fields. These graphical models realize the tight binding between probability and graph theory by representing random variables as the nodes of a graph, and statistical dependencies among the variables by the graph's edges. However, the expression ``graphical model'' includes a number of learning machines, such as inductive logic programming, statistical relational learning, machine learning over structured domains, recursive neural networks, graph neural networks, etc. Before given a more precise picture of this book, it may be good to point out a number of problems presented in the text. Chapter 2 introduces the idea of Bayesian networks in a qualitative manner. The given formal definition involves hybrid random fields, namely the Markov condition, \(d\)-separation, the Markov blanket, and their relationship with the factorization of the overall, joint distribution of the variables represented by the whole network. Additionally, the chapter gives a survey on generalizations of the basic model (such as dynamic Bayesian networks and hidden Markov models). Chapter 3 provides a step-by-step introduction to the major ideas behind the concept of Markov random field, starting from stochastic processes. The role of the potential energy of local neighborhoods over an undirected graph is pointed out, along with the motivations behind the choice of modelling the overall joint density via the Gibbs distribution. Moreover, the theoretical justification for the definition of Markov random fields in the light of the equivalence between the assumption of the Gibbs distribution and the emergence of the Markov property is provided. In Chapter 4, the hybrid random field (HRF) model is introduced. The HRF model represents the joint probability distribution of its variables over a set of random variables that satisfies certain conditions involving the concepts of directed and undirected union. The chapter discusses also the difference between this definition of HRF and a looser definition presented earlier in the literature. Chapter 5 generalizes the HRF model to domains involving continuous variables. In the given approach, some estimation techniques for modeling the conditional probability density functions associated with the nodes of the graphs are provided. An emphasis is put on normal distributions and Gaussian mixture models. The maximum-likelihood estimation of the parameters of normal densities is studied in detail. It is interesting to note that the ``change of variables'' theorem allows for the development of a non-paranormal technique for the estimation of joint and conditional probability density functions without making any prior assumption on the form of the modeled density functions. Chapter 6 treats learning machines that learn effectively and prove to be useful in real-world scenarios. Some experiments based on the identification of Markov blankets are described. Among others, the discrimination between positive and negative individuals in a in an application regarding lung cancer diagnosis is presented. These results bring evidence that hybrid random fields scale up well to large datasets, giving a level of accuracy which is at least as good as that achieved by traditional graphical models. Chapter 7 attempts to make the philosophical reflection on some issues concerning the relationships between statistical machine learning and the cognitive sciences. Chapter 8 summarizes the main contributions of the book. In summary, the authors have written a very valuable book -- rigorous in the treatment on the mathematical background, but also enriched with a very open view of the field, full of stimulating connections.
scientific article; zbMATH DE number 5887505