ABSTRACT
The objective of Chapter 10 is to introduce concepts for working with models of stochastic data generating processes. The beginning of the chapter introduces probability modeling concepts including: parametric probability models, misspecified models, local probability models, and models of missing data generating processes. Next, major types of probability models are introduced as special cases of the Gibbs probability model including: exponential family models, multinomial logistic models, Poisson models, multivariate Gaussian models, Dirichlet models, Gamma models, Chi-squared models, Laplace models, and hyperbolic secant models. Following an overview of widely used probability models, Bayesian network and Markov random field methods are introduced as tools for factoring a joint density. Using these methods, it is shown how to construct the joint distribution of a collection of random variables from local conditional probabilistic relationships among small groups of random variables. A unique feature of this chapter is a careful presentation of a new version of the Hammersley-Clifford theorem intended to support the analysis and design of Markov random fields consisting of both discrete and absolutely continuous random variables. At the end of the chapter, special topics such as conditional random fields and mixtures of Markov random fields are briefly discussed.
