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引用次数: 6
摘要
摘要:在密度估计模型中,考虑了利用Pólya树型先验分布进行自适应推理的问题。介绍了一类具有树形结构的先验密度,称为spike-and-slab Pólya树。对于这一类,得到了两类结果:第一,对于0和1之间的任何Hölder真密度的正则性,贝叶斯后验分布以自适应的方式收敛于最大范数的极小极大率,从而提供了[5]中经典Pólya树的结果的自适应对应。其次,考虑了不确定度的量化问题。导出了一个自适应非参数Bernstein - von Mises定理。其次,在真密度的自相似条件下,来自后验分布的某些可信集是自适应置信带,具有规定的覆盖水平,直径在极小极大意义上以最优速率收缩。
Spike and slab Pólya tree posterior densities: Adaptive inference
Abstract: In the density estimation model, the question of adaptive inference using Pólya tree–type prior distributions is considered. A class of prior densities having a tree structure, called spike–and–slab Pólya trees, is introduced. For this class, two types of results are obtained: first, the Bayesian posterior distribution is shown to converge at the minimax rate for the supremum norm in an adaptive way, for any Hölder regularity of the true density between 0 and 1, thereby providing adaptive counterparts to the results for classical Pólya trees in [5]. Second, the question of uncertainty quantification is considered. An adaptive nonparametric Bernstein– von Mises theorem is derived. Next, it is shown that, under a self-similarity condition on the true density, certain credible sets from the posterior distribution are adaptive confidence bands, having prescribed coverage level and with a diameter shrinking at optimal rate in the minimax sense.