Minimax Bayes, Asymptotic Minimax and Sparse Wavelet Priors

Abstract

Pinsker (1980) gave a precise asymptotic evaluation of the minimax mean squared error of estimation of a signal in Gaussian noise when the signal is known a priori to lie in a compact ellipsoid in Hilbert space. This ‘Minimax Bayes’ method can be applied to a variety of global non-parametric estimation settings with parameter spaces far from ellipsoidal. For example it leads to a theory of exact asymptotic minimax estimation over norm balls in Besov and Triebel spaces using simple co-ordinatewise estimators and wavelet bases.This paper outlines some features of the method common to several applications. In particular, we derive new results on the exact asymptotic minimax risk over weak lp- balls in Rn as n → ∞, and also for a class of ‘local’ estimators on the Triebel scale.By its very nature, the method reveals the structure of asymptotically least favorable distributions. Thus we may simulate ‘least favorable’ sample paths. We illustrate this for estimation of a signal in Gaussian white noise over norm balls in certain Besov spaces. In wavelet bases, when p < 2, the least favorable priors are sparse, and the resulting sample paths strikingly different from those observed in Pinsker’s ellipsoidal setting (p = 2).