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We investigate the frequentist coverage properties of Bayesian credible sets in a general, adaptive, nonparametric framework. It is well known that the construction of adaptive and honest confidence sets is not possible in general. To overcome this problem we introduce an extra assumption on the functional parameters, the so called general polished tail condition. We then show that under standard assumptions both the hierarchical and empirical Bayes methods results in honest confidence sets for sieve type of priors in general settings and we characterize their size. We apply the derived abstract results to various examples, including the nonparametric regression model, density estimation using exponential families of priors, density estimation using histogram priors and nonparametric classification model, for which we show that their size is near minimax adaptive with respect to the considered specific semi-metrics.
For estimating a lower bounded location or mean parameter for a symmetric and logconcave density, we investigate the frequentist performance of the $100(1-alpha)%$ Bayesian HPD credible set associated with priors which are truncations of flat priors
Neural networks are one of the most popularly used methods in machine learning and artificial intelligence nowadays. Due to the universal approximation theorem (Hornik et al. (1989)), a neural network with one hidden layer can approximate any continu
In the sparse normal means model, coverage of adaptive Bayesian posterior credible sets associated to spike and slab prior distributions is considered. The key sparsity hyperparameter is calibrated via marginal maximum likelihood empirical Bayes. Fir
We develop singular value shrinkage priors for the mean matrix parameters in the matrix-variate normal model with known covariance matrices. Our priors are superharmonic and put more weight on matrices with smaller singular values. They are a natural
Variable selection in the linear regression model takes many apparent faces from both frequentist and Bayesian standpoints. In this paper we introduce a variable selection method referred to as a rescaled spike and slab model. We study the importance