Variable selection consistency of Gaussian process regression

Bayesian nonparametric regression under a rescaled Gaussian process prior offers smoothness-adaptive function estimation with near minimax-optimal error rates. Hierarchical extensions of this approach, equipped stochastic variable selection, are known to also adapt the unknown intrinsic dimension sparse true function. But it remains unclear if such offer selection consistency, that is, subset important variables could be consistently learned from data. It is shown here consistency may indeed achieved models at least when has finite smoothness induce polynomially larger penalty on inclusion false positive predictors. Our result covers high-dimensional asymptotic setting where predictor allowed grow sample size. The proof utilizes Schwartz theory establish posterior probability wrong vanishes asymptotically. A necessary and challenging technical development involves providing sharp upper lower bounds small ball probabilities all rescaling levels prior, independent interest.