Beta分布是定义在连续变量上的一种分布，它的概率密度函数为

其中a > 0, b > 0。

下图为Beta分布在参数a和b取不同值时的概率密度曲线：

Beta分布是二项式分布的共轭先验（conjugate prior），因此经常被用于二项式分布的先验分布

由于Beta分布的概率密度曲线形状比较多样，而Gaussian分布只有钟形一种形状，所以Beta分布也经常被用于连续时间的趋势分析，可参考下面两篇文献：

@INPROCEEDINGS{WM06,
  author = {Wang, Xuerui and McCallum, Andrew},
  title = {Topics over Time: A Non-Markov Continuous-Time Model of Topical Trends},
  booktitle = KDD2006,
  year = {2006},
  pages = {424--433},
  location = {Philadelphia, Pennsylvania, USA},
  publisher = {ACM},
  address = {New York, NY, USA},
  isbn = {1-59593-339-5},
  abstract = {This paper presents an LDA-style topic model that captures not only
 the low-dimensional structure of data, but also how the structure
 changes over time. Unlike other recent work that relies on Markov
 assumptions or discretization of time, here each topic is associated
 with a continuous distribution over timestamps, and for each generated
 document, the mixture distribution over topics is influenced by both
 word co-occurrences and the document's timestamp. Thus, the meaning
 of a particular topic can be relied upon as constant, but the topics'
 occurrence and correlations change significantly over time.We present
 results on nine months of personal email, 17 years of NIPS research
 papers and over 200 years of presidential state-of-the-union addresses,
 showing improved topics, better timestamp prediction, and interpretable
 trends.},

  keywords = {Graphical Models; Temporal Analysis; Topic Modeling}

}
@INPROCEEDINGS{KH10,
  author = {Kawamae, Noriaki and Higashinaka, Ryuichiro},
  title = {Trend Detection Model},
  booktitle = WWW2010,
  year = {2010},
  editor = {Rappa, Michael and Jones, Paul and Freire, Juliana and Chakrabarti,
 Soumen},
  pages = {1129--1130},
  location = {Raleigh, North Carolina, USA},
  publisher = {ACM},
  address = {New York, NY, USA},
  isbn = {978-1-60558-799-8},
  abstract = {This paper presents a topic model that detects topic distributions
 over time. Our proposed model, Trend Detection Model (TDM) introduces
 a latent trend class variable into each document. The trend class
 has a probability distribution over topics and a continuous distribution
 over time. Experiments using our data set show that TDM is useful
 as a generative model in the analysis of the evolution of trends.},
  keywords = {Topic Model; Trend Model; Dynamics Topic Model; Latent Variable Modeling;
}

无论是哪种使用方式，经常会遇到Beta分布的参数a和b的估计的问题

一种比较简单的方法是基于矩估计的

令为样本均值

令为样本方差

则利用矩估计Beta分布的参数为

 

参考资料：