  This paper proposes a unified framework to quantify local and global
inferential uncertainty for high dimensional nonparanormal graphical models. In
particular, we consider the problems of testing the presence of a single edge
and constructing a uniform confidence subgraph. Due to the presence of unknown
marginal transformations, we propose a pseudo likelihood based inferential
approach. In sharp contrast to the existing high dimensional score test method,
our method is free of tuning parameters given an initial estimator, and extends
the scope of the existing likelihood based inferential framework. Furthermore,
we propose a U-statistic multiplier bootstrap method to construct the
confidence subgraph. We show that the constructed subgraph is contained in the
true graph with probability greater than a given nominal level. Compared with
existing methods for constructing confidence subgraphs, our method does not
rely on Gaussian or sub-Gaussian assumptions. The theoretical properties of the
proposed inferential methods are verified by thorough numerical experiments and
real data analysis.
