3. Maximum Likelihood Inference

This chapter delves into the optimization procedures for maximum likelihood inference in discrete regression graph models, focusing on gradient-ascent algorithms to maximize the log-likelihood function with independence constraints specified through Lagrange multipliers. It discusses the recursive application of the algorithm to multivariate regression models induced by the basic factorization of a regression graph and explores model selection based on structural learning of the DAG of chain components. The text also covers the likelihood function for discrete regression graph models, highlighting the implementation of algorithms for maximum likelihood estimation (MLE) in the space of the logarithm of expected cell counts. It provides a detailed explanation of the Lagrangian log-likelihood maximization algorithm for a broad class of multivariate regression models for categorical data, emphasizing the use of the multivariate logistic transformation as a link function. Additionally, the chapter addresses the asymptotic properties and model comparison, including the use of penalized likelihood criteria such as Akaike’s Information Criterion (AIC) and the Bayesian Information Criterion (BIC). Practical examples, such as fitting a regression chain model to data from the General Social Survey, illustrate the application of these methods. The chapter also discusses strategies for structure learning of regression chain graphs, including constraint-based and score-based algorithms, and provides insights into the interpretation of dependencies and the selection of reduced regression models. Overall, the chapter offers a comprehensive guide to understanding and applying maximum likelihood inference in discrete regression graph models, making it an invaluable resource for professionals in data analysis and statistical modeling.