Adaptive Bayesian density estimation in sup-norm

Jeudi 11 avril 15:45-16:45 - Zacharie Naulet (University of Toronto)

Résumé : We investigate the problem of deriving adaptive posterior rates of contraction on $L^\infty$ 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. Here we establish a generic $L^\infty$-contraction result for log-density priors
with independent wavelet coefficients. The result is then applied to the so called spike-and-slab prior to obtain adaptive and minimax rates. Interestingly, our approach is different from previous works on $L^\infty$-contraction and is reminiscent to the classical test-based approach used in Bayesian nonparametrics. Moreover, we require no lower bound on the smoothness of the true density, albeit the rates are deteriorated by an extra log(n) factor in the case of low smoothness.

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