Oracle posterior contraction rates under hierarchical priors

We offer a general Bayes theoretic framework to derive posterior contraction rates under hierarchical prior design: the first-step serves assess model selection uncertainty, and second-step quantifies belief on strength of signals within chosen from first step. In particular, we establish non-asymptotic oracle (i) local Gaussianity condition log likelihood ratio statistical experiment, (ii) entropy dimensionality models, (iii) sufficient mass near best approximating signal for each model. The can be designed generically. distribution enjoys Gaussian tail behavior therefore resulting mean also satisfies an inequality, automatically serving as adaptive point estimator in frequentist sense. Model mis-specification is allowed these rates. unified attempt quantification experiments, easily verified various experiments considered [GvdV07a] beyond. results are applied problems including: trace regression, shape-restricted isotonic/convex high-dimensional partially linear (iv) covariance matrix estimation sparse factor model, (v) detection non-smooth polytopal image boundary, (vi) intensity Poisson process These new serve either theoretical justification practical proposals literature, or illustration generic construction scheme (nearly) minimax complicated experiment.