  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 populations
calls for comparing these estimated GGMs. Our goal is to identify differences
in GGMs known to have similar structure. We characterize the uncertainty of
differences with confidence intervals obtained using a parametric distribution
on parameters of a sparse estimator. Sparse penalties enable statistical
guarantees and interpretable models even in high-dimensional and low-sample
settings. Characterizing the distributions of sparse models is inherently
challenging as the penalties produce a biased estimator. Recent work invokes
the sparsity assumptions to effectively remove the bias from a sparse estimator
such as the lasso. These distributions can be used to give 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 derive
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, whose distribution can be characterized in an
efficient manner. We then show how the debiased lasso and multi-task fused
lasso can be used to obtain confidence intervals on edge differences in GGMs.
We validate the techniques proposed on a set of synthetic examples as well as
neuro-imaging dataset created for the study of autism.
