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摘要翻译：
研究了以正态分布的Dirichlet混合为先验分布的Bayesian密度估计的后验分布的收敛速度。真密度假定为两次连续可微。带宽被赋予先验序列，先验序列是通过将单个先验按适当的顺序缩放得到的。为了解决这个问题，我们通过考虑先验概率满足可和条件的参数空间的可数覆盖以及Hellinger度量熵上的某些独立界，导出了一个新的一般速率定理。我们通过计算Hellinger（括号）熵数的界、正态混合逼近光滑密度的误差和先验密度的集中率来应用这个关于后验收敛速度的新的一般定理。对于积分均方误差$n^{-2/5}$到一个对数因子，后验的最佳收敛速度与众所周知的频率速率是等价的。
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英文标题：
《Posterior convergence rates of Dirichlet mixtures at smooth densities》
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作者：
Subhashis Ghosal, Aad van der Vaart
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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 rates of convergence of the posterior distribution for Bayesian density estimation with Dirichlet mixtures of normal distributions as the prior. The true density is assumed to be twice continuously differentiable. The bandwidth is given a sequence of priors which is obtained by scaling a single prior by an appropriate order. In order to handle this problem, we derive a new general rate theorem by considering a countable covering of the parameter space whose prior probabilities satisfy a summability condition together with certain individual bounds on the Hellinger metric entropy. We apply this new general theorem on posterior convergence rates by computing bounds for Hellinger (bracketing) entropy numbers for the involved class of densities, the error in the approximation of a smooth density by normal mixtures and the concentration rate of the prior. The best obtainable rate of convergence of the posterior turns out to be equivalent to the well-known frequentist rate for integrated mean squared error $n^{-2/5}$ up to a logarithmic factor.
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PDF链接：
https://arxiv.org/pdf/708.1885
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