Variable selection confidence sets for high dimensional data

© 2017 Dr Chao ZhengThe traditional activity of model selection aims at discovering a single model superior to other candidate models. In the presence of noise in data, however, it is typically difficult to declare a single model significantly superior to all possible competitors, due to the prevailing effect of the model selection uncertainty. In this situation, multiple or even a large number of models may be equally supported by data. To resolve this model selection ambiguity, we extend the general approach of variable selection confidence sets (VSCSs) to general parametric family and ultra-high dimensional scenario. A VSCS 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 VSCS contains all the relevant information about the true model structure. In addition, we propose an important subset of models (LBMs) and natural statistics based on the VSCS to measure importance of variables in a principled way that accounts for the overall model uncertainty. When the space of feasible models is large, VSCS is implemented by an adaptive stochastic search algorithm which samples VSCS models with high probability. The VSCS methodology is illustrated through numerical experiments on synthetic data and real data examples