  The traditional activity of model selection aims at discovering a single
model superior to other candidate models. In the presence of pronounced noise,
however, multiple models are often found to explain the same data equally well.
To resolve this model selection ambiguity, we introduce the general approach of
model selection confidence sets (MSCSs) based on likelihood ratio testing. A
MSCS is defined as a list of models statistically indistinguishable from the
true model at a user-specified level of confidence, which extends the familiar
notion of confidence intervals to the model-selection framework. Our approach
guarantees asymptotically correct coverage probability of the true model when
both sample size and model dimension increase. We derive conditions under which
the MSCS contains all the relevant information about the true model structure.
In addition, we propose natural statistics based on the MSCS to measure
importance of variables in a principled way that accounts for the overall model
uncertainty. When the space of feasible models is large, MSCS is implemented by
an adaptive stochastic search algorithm which samples MSCS models with high
probability. The MSCS methodology is illustrated through numerical experiments
on synthetic data and real data examples.
