Computationally tractable fitting of graphical models : the cost and benefits of decomposable Bayesian and penalized likelihood approaches : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Albany, New Zealand

Abstract

Gaussian graphical models are a useful tool for eliciting information about relationships in data with a multivariate normal distribution. In the rst part of this thesis we demonstrate that partial correlation graphs facilitate di erent and better insight into high-dimensional data than sample correlations. This raises the question of which method one should use to model and estimate the parameters. In the second, and major part, we take a more theoretical focus examining the costs and bene ts of two popular approaches to model selection and parameter estimation (penalized likelihood and decomposable Bayesian) when the true graph is non-decomposable. We rst consider the e ect a restriction to decomposable models has on the estimation of both the inverse covariance matrix and the covariance matrix. Using the variance as a measure of variability we compare non-decomposable and decomposable models. Here we nd that, if the true model is non-decomposable, the variance of estimates is demonstrably larger when a decomposable model is used. Although the cost in terms of accuracy is fairly small when estimating the inverse covariance matrix, this is not the case when estimation of the covariance matrix is the goal. In this case using a decomposable model caused up to 200-fold increases in the variance of estimates. Finally we compare the latest decomposable Bayesian method (the feature-inclusion stochastic search) with penalized likelihood methods (graphical lasso and adaptive graphical lasso) on measures of model selection and prediction performance. Here we nd that graphical lasso is clearly outclassed on all measures by both adaptive graphical lasso and feature-inclusion stochastic search. The sample size and the ultimate goal of the estimation will determine whether adaptive graphical lasso or feature-inclusion stochastic search is better.