论文标题
转移的pólya树合奏的平滑和适应
Smoothing and adaptation of shifted Pólya Tree ensembles
论文作者
论文摘要
最近，S。Arlot和R. Guealer表明，随机森林模型在估计$α-$Hölder函数时的表现优于其单树模型，$α\ leq2 $。这支持了这样一个想法，即树估计器的集合比单个树更光滑。另一方面，基于贝叶斯树的方法上的大多数积极最佳性结果假定$α\ leq1 $。自然，人们想知道贝叶斯估计量的贝叶斯对应物在更顺畅的类别上是否最佳，就像$α\ leq 2 $的常见估计量被观察到的那样。我们介绍了密度估计的问题，并从贝叶斯非参数中的经典（截断）pólya树的结构引入了一个集成估计器。结果显示，由此产生的贝叶斯森林估计量可导致最佳的后部收缩率，最高为对数项，对于$ [0; 1）$ [0; 1）$的$ l^1 $距离，用于$ [0; 1）$ $ $α> 0 $。这改善了与Pólya树先验相关的结构的先前结果，其最佳性仅在$α\ leq 1 $的情况下得到证明。另外，我们介绍了这个新事先的自适应版本，即它不需要$α$的知识才能定义和达到最佳性。
Recently, S. Arlot and R. Genuer have shown that a model of random forests outperforms its single-tree counterpart in the estimation of $α-$Hölder functions, $α\leq2$. This backs up the idea that ensembles of tree estimators are smoother estimators than single trees. On the other hand, most positive optimality results on Bayesian tree-based methods assume that $α\leq1$. Naturally, one wonders whether Bayesian counterparts of forest estimators are optimal on smoother classes, just like it has been observed for frequentist estimators for $α\leq 2$. We dwell on the problem of density estimation and introduce an ensemble estimator from the classical (truncated) Pólya tree construction in Bayesian nonparametrics. The resulting Bayesian forest estimator is shown to lead to optimal posterior contraction rates, up to logarithmic terms, for the Hellinger and $L^1$ distances on probability density functions on $[0;1)$ for arbitrary Hölder regularity $α>0$. This improves upon previous results for constructions related to the Pólya tree prior whose optimality was only proven in the case $α\leq 1$. Also, we introduce an adaptive version of this new prior in the sense that it does not require the knowledge of $α$ to be defined and attain optimality.
