Model Selection for Multi-Attribute Gaussian Graphical Models

Abstract

In this thesis, we examine a common challenge in the estimation of conditional independence graphs, namely the selection of the regularization parameter for sparse precision matrix estimation in multi-attribute Gaussian graphical models (MA-GGMs). Even in the classical single-attribute setting, choosing an appropriate model selection criterion remains difficult; the multi-attribute high-dimensional setting further complicates this task. We compare three widely used approaches: stability selection, the Bayesian information criterion, and cross-validation. These techniques are applied to penalized Gaussian likelihood estimation with both convex (lasso) and non-convex (log-sum) penalties. Using extensive simulations on Erdős–Rényi, Barabási–Albert, and chain graphs, we assess how topology, sparsity, sample size, and penalty choice affect the performance of each selector. We further evaluate the methods on a real-world financial time series dataset consisting of multi-attribute stock returns from the S&P 100 index. The empirical results show that stability selection performs well in sparse or small sample regimes; BIC combined with the log-sum penalty provides strong and stable structural recovery; and cross-validation becomes reliable only when the sample size is sufficiently large. These findings provide practical guidance for model selection in high-dimensional multi-attribute graphical models, particularly in applications where accurate recovery of dependence structure is essential.