Adaptive Bayesian semiparametric density estimation in sup-norm
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We investigate the problem of deriving adaptive posterior rates of contraction on L^∞ balls in density estimation. Although it is known that log-density priors can achieve optimal rates when the true density is sufficiently smooth, adaptive rates were still to be proven. Recent works have shown that the so called spike-and-slab priors can achieve optimal rates of contraction under L^∞ loss in white-noise regression and multivariate regression with normal errors. Here we show that a spike-and-slab prior on the log-density also allows for (nearly) optimal rates of contraction in density estimation under L^∞ loss. Interestingly, our results hold without lower bound on the smoothness of the true density.
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