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HEAVY-TAILED BAYESIAN NONPARAMETRIC ADAPTATION
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DOI：10.1214/24-AOS2397.png)
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We propose a new Bayesian strategy for adaptation to smoothness in nonparametric models based on heavy-tailed series priors. We illustrate it in a variety of settings, showing in particular that the corresponding Bayesian posterior distributions achieve adaptive rates of contraction in the minimax sense (up to logarithmic factors) without the need to sample hyperparameters. Unlike many existing procedures, where a form of direct model (or estimator) selection is performed, the method can be seen as performing a soft selection through the prior tail. In Gaussian regression, such heavy-tailed priors are shown to lead to (near-)optimal simultaneous adaptation both in the L-2- and L-infinity-sense. Results are also derived for linear inverse problems, for anisotropic Besov classes, and for certain losses in more general models through the use of tempered posterior distributions. We present numerical simulations corroborating the theory.
Keyword:
Bayesian nonparametrics
frequentist analysis of posterior distributions
adaptation to smoothness
heavy tails
fractional posteriors
期刊
IF:
3.7
论文数:
2.8K
被引数:
2.9W
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分享收藏ADAPTIVE BAYESIAN ESTIMATION USING A GAUSSIAN RANDOM FIELD WITH INVERSE GAMMA BANDWIDTH.png)
ANNALS OF STATISTICS
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ANNALS OF STATISTICSIF3.7
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