摘要翻译：
研究了周期Sobolev类对数密度估计问题中后验分布的收敛速度。后验期望密度提供了一种非参数估计方法，在Hellinger损失下，如果后验分布在一定的均匀性类别上达到最优收敛速度，则可获得最优的极小极大收敛速度。对数密度的三角级数展开式系数的先验，可以导出感兴趣密度类的先验。我们证明了当p已知时，如果先验方差足够快地消失，则高斯先验的后验分布达到最优率。对于混合正态分布，假设指数族维数上的混合权以指数递减序列为界。为了避免使用无限的基，我们发展先验在一个与样本大小相关的截断点处截断序列。当光滑度未知时，由光滑度参数索引的正规先验的有限混合产生最优率，该参数也被赋予先验。推导了一种速率自适应估计器。
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英文标题：
《Convergence rates for Bayesian density estimation of
  infinite-dimensional exponential families》
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作者：
Catia Scricciolo
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
  We study the rate of convergence of posterior distributions in density estimation problems for log-densities in periodic Sobolev classes characterized by a smoothness parameter p. The posterior expected density provides a nonparametric estimation procedure attaining the optimal minimax rate of convergence under Hellinger loss if the posterior distribution achieves the optimal rate over certain uniformity classes. A prior on the density class of interest is induced by a prior on the coefficients of the trigonometric series expansion of the log-density. We show that when p is known, the posterior distribution of a Gaussian prior achieves the optimal rate provided the prior variances die off sufficiently rapidly. For a mixture of normal distributions, the mixing weights on the dimension of the exponential family are assumed to be bounded below by an exponentially decreasing sequence. To avoid the use of infinite bases, we develop priors that cut off the series at a sample-size-dependent truncation point. When the degree of smoothness is unknown, a finite mixture of normal priors indexed by the smoothness parameter, which is also assigned a prior, produces the best rate. A rate-adaptive estimator is derived.
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PDF链接：
https://arxiv.org/pdf/708.0175