Empirical priors and posterior concentration in a piecewise polynomial sequence model

Abstract: Inference on high-dimensional parameters in structured linear models is an important statistical problem. This paper focuses on the case of a piecewise polynomial Gaussian sequence model, and we develop a new empirical Bayes solution that enjoys adaptive minimax posterior concentration rates and improved structure learning properties compared to existing methods. Moreover, thanks to the conjugate form of the empirical prior, posterior computations are fast and easy. Numerical examples also highlight the method's strong finite-sample performance compared to existing methods across a range of different scenarios.