  Statistical inference of the high-dimensional regression coefficients is
challenging because the uncertainty introduced by the model selection procedure
is hard to account for. A critical question remains unsettled; that is, is it
possible and how to embed the inference of the model into the simultaneous
inference of the coefficients? To this end, we propose a notion of simultaneous
confidence intervals called the sparsified simultaneous confidence intervals.
Our intervals are sparse in the sense that some of the intervals' upper and
lower bounds are shrunken to zero (i.e., $[0,0]$), indicating the unimportance
of the corresponding covariates. These covariates should be excluded from the
final model. The rest of the intervals, either containing zero (e.g., $[-1,1]$
or $[0,1]$) or not containing zero (e.g., $[2,3]$), indicate the plausible and
significant covariates, respectively. The proposed method can be coupled with
various selection procedures, making it ideal for comparing their uncertainty.
For the proposed method, we establish desirable asymptotic properties, develop
intuitive graphical tools for visualization, and justify its superior
performance through simulation and real data analysis.
