Smoothing and adaptation of shifted Pólya tree ensembles
Smoothing and adaptation of shifted Pólya tree ensembles
复制标题
平移 Pólya 树系综的平滑和适应
DOI：
10.3150/21-bej1426
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发表时间：
2020
期刊：
影响因子：
1.5
通讯作者：
Thibault Randrianarisoa
中科院分区：
文献类型：
--
作者：
Thibault Randrianarisoa
最近，S.Arlot和R.Genuer已经证明了随机森林模型在估计$\α-$H\“旧函数$\α\leq2$方面优于其单树模型。这支持了这样一种观点，即树木估计器的集合比单一的树木估计器更平滑。另一方面，基于贝叶斯树方法的大多数正最优性结果都假设$\α\leq1$。自然，人们想知道森林估计量的贝叶斯对应是否在更平滑的类别上是最优的，就像对于频率估计量所观察到的$α=2$一样。我们讨论了密度估计问题，并从贝叶斯非参数中经典的(截断)P‘Olya树结构中引入了一种集成估计。由此得到的贝叶斯森林估计，对于Hellinger和$[0；1)$上的概率密度函数的$L^1$距离，对于任意H“较老的正则性$α>；0$，得到了最优的后验压缩速率，直到对数项。这改进了以前与P-olya树相关的构造的结果，其最优性仅在$\α\leq1$的情况下被证明。此外，我们还引入了这一新先验知识的自适应版本，因为它不需要定义$\α$的知识并达到最优性。
Recently, S. Arlot and R. Genuer have shown that a model of random forests outperforms its single-tree counterpart in the estimation of $\alpha-$H\"older functions, $\alpha\leq2$. 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 $\alpha\leq1$. Naturally, one wonders whether Bayesian counterparts of forest estimators are optimal on smoother classes, just like it has been observed for frequentist estimators for $\alpha\leq 2$. We dwell on the problem of density estimation and introduce an ensemble estimator from the classical (truncated) P\'olya 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 H\"older regularity $\alpha>0$. This improves upon previous results for constructions related to the P\'olya tree prior whose optimality was only proven in the case $\alpha\leq 1$. Also, we introduce an adaptive version of this new prior in the sense that it does not require the knowledge of $\alpha$ to be defined and attain optimality.
DOI：
10.1080/10618600.2019.1677243
发表时间：
2020-04
影响因子：
2.4
作者：
M. Pratola;H. Chipman;Edward I. George;R. McCulloch
通讯作者：
M. Pratola;H. Chipman;Edward I. George;R. McCulloch
影响因子：
4.4
作者：
Liu, Yingchun;Shao, Zhen;Yuan, Guo-Cheng
通讯作者：
Yuan, Guo-Cheng