Nonlinear Optimization of Exponential Family Graphical Models

This project explores methods for carrying out projections arising in the information geometry of the exponential family of probability models. Kullback-Leibler divergence serves as the distance measure between probability models in this context. The applications include maximum likelihood parameter estimation given sample paths of an unknown density as well as model reduction where one wishes to fit a lower-order exponential model to a given higher-order model. These are fundamental problems arising in the context of the graphical modeling literature. Here, one considers exponential family models defined on graphs such that the random process is constrained to be Markov with respect to some interaction graph. In this context, we will show that the fundamental problem of minimizing the Kullback-Leibler divergence may be reduced to a certain moment-matching problem which may be posed as a convex programming problem. This minimization problem is then well-suited to solution by a variety of generic nonlinear programming methods. We will explore these methods in the context of Gaussian processes which are Markov with respect to a given graph and employ both gradient and Hessian based techniques (gradient descent, conjugate gradients, preconditioned conjugate gradients and Newton’s method). We compare the performance of these methods to the standard iterative proportional fitting method for solving this moment-matching problem. Also, we extend these methods to explore the problem of structure estimation employing either the Akaike or Bayesian Information Criterion to estimate the graphical structure of a Gaussian process from data without any prior knowledge of the Markov structure of the process.