Testing for Differences in Gaussian Graphical Models: Applications to Brain Connectivity

Functional brain networks are well described and estimated from data with Gaussian Graphical Models (GGMs), e.g. using sparse inverse covariance estimators. Comparing functional connectivity of subjects in two population calls for comparing these estimated GGMs. We study the problem of identifying differences in Gaussian Graphical Models (GGMs) known to have similar structure. We aim to characterize the uncertainty of differences with confidence intervals obtained using a para-metric distribution on parameters of a sparse estimator. Sparse penalties enable statistical guarantees and interpretable models even in high-dimensional and low-number-of-samples settings. Quantifying the uncertainty of the parameters selected by the sparse penalty is an important question in applications such as neuroimaging or bioinformatics. Indeed, selected variables can be interpreted to build theoretical understanding or to make therapeutic decisions. Characterizing the distributions of sparse regression models is inherently challenging since the penalties produce a biased estimator. Recent work has shown how one can invoke the sparsity assumptions to effectively remove the bias from a sparse estimator such as the lasso. These distributions can be used to give us confidence intervals on edges in GGMs, and by extension their differences. However, in the case of comparing GGMs, these estimators do not make use of any assumed joint structure among the GGMs. Inspired by priors from brain functional connectivity we focus on deriving the distribution of parameter differences under a joint penalty when parameters are known to be sparse in the difference. This leads us to introduce the debiased multi-task fused lasso. We show that we can debias and characterize the distribution in an efficient manner. We then go on to show how the debiased lasso and multi-task fused lasso can be used to obtain confidence intervals on edge differences in Gaussian graphical models. We validate the techniques proposed on a set of synthetic examples as well as neuro-imaging dataset created for the study of autism.