Dirichlet过程及相关随机测度的渐近行为

Asymptotic behavior of Dirichlet process and related random measures

Dirichlet过程是取纯原子概率值的随机测度.它最早由Ferguson (1973)提出,目的是作为Bayes非参数统计的先验分布. Pitman和Yor在20世纪90年代把Dirichlet过程推广到了两参数,从而得到了非常一般的两参数Dirichlet过程或Pitman-Yor过程. Dirichlet过程及其两参数推广,不仅成为Bayes非参数统计领域的基本模型,同时还被广泛应用到生物学、通信、经济学、金融、遗传学、统计物理和机器学习等领域.近年来在各种应用问题的推动下,关于Dirichlet过程的极限行为方面的研究非常活跃.本文将介绍这方面的一些研究结果,主要包括大样本和相关参数趋近边界的极限行为.

Dirichlet process is an atomic-valued random measure. It was introduced by Ferguson(1973) with motivation from Bayesian statistics. Pitman and Yor generalized the process to the two-parameter setting in 1990s.It has since found applications in a wide range of subjects including Bayesian statistics, biology, communication,economy, finance, genetics, and machine learning. Driven by various applications, there has been active study of asymptotic behavior of this class of processes. This paper provides a brief survey on recent work by the author and collaborators.