Maximum likelihood estimation in Gaussian AMP chain graph models and Gaussian ancestral graph models

Abstract

Graphical Markov models use graphs to represent dependencies between stochastic variables. Via Markov properties, missing edges in the graph are translated into conditional independence statements, which, in conjunction with a distributional assumption, define a statistical model. This thesis considers maximum likelihood (ML) estimation of the parameters of two recently introduced classes of graphical Markov models in the case of continuous variables with a joint multivariate Gaussian distribution. The two new model classes are the AMP chain graph models, based on chain graphs equipped with a new Markov property, and the ancestral graph models, based on a new class of graphs. Both classes generalize the widely used models based on acyclic directed graphs (Bayesian networks) and undirected graphs (Markov random fields).In this thesis, we first show that the likelihood of AMP chain graph and ancestral graph models may be multimodal. Next, we combine existing techniques (iterative proportional fitting, generalized least squares) into an algorithm for ML estimation in AMP chain graph models. For the ancestral graphs, we develop an ML estimation algorithm based on a new iterative conditional fitting (ICF) idea, which in the considered Gaussian case can be implemented using least squares regression on synthetic variables. We derive the ICF algorithm in the special case of bidirected graphs, also termed covariance graphs, and subsequently generalize it to cover arbitrary ancestral graphs.