查看“︁狄利克雷分布”︁的源代码

{{Probability distribution |
  name       =狄利克雷分布|
  type       =密度|
  pdf_image  =[[File:Dirichlet-3d-panel.png|565px|A panel illustrating probability density functions of a few Dirichlet distributions over a 2-simplex, for the following ''α'' vectors (clockwise, starting from the upper left corner): (1.3, 1.3, 1.3), (3,3,3), (7,7,7), (2,6,11), (14, 9, 5), (6,2,6).]]||
  cdf_image  =|
  parameters =<math>K \geq 2</math> 分类数 ([[整数]])<br /><math>\alpha_1, \ldots, \alpha_K</math> [[concentration parameters]]，<math>\alpha_i > 0</math>|
  support    =<math>x_1, \ldots, x_K</math>，<math>x_i \in (0,1)</math>，<math>\sum_{i=1}^K x_i = 1</math>|
  pdf        =<math>\frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1} </math><br /><math>\mathrm{B}(\boldsymbol\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K \alpha_i\bigr)}</math><br />
<math>\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_K)</math>|
  cdf        =|
  mean       =<math>\operatorname{E}[X_i] = \frac{\alpha_i}{\sum_k \alpha_k}</math><br /><math> \operatorname{E}[\ln X_i] = \psi(\alpha_i)-\psi(\textstyle\sum_k \alpha_k)</math><br />(试看 [[digamma function]])|
  median     =|
  mode       =<math>x_i = \frac{\alpha_i - 1}{\sum_{k=1}^K\alpha_k - K}, \quad \alpha_i > 1. </math>|
  variance   =<math>\operatorname{Var}[X_i] = \frac{\tilde{\alpha}_i(1-\tilde{\alpha}_i)}{\bar{\alpha}+1},</math><br/>
其中<math>\tilde{\alpha}_i = \frac{\alpha_i}{\sum_{i=1}^K\alpha_i}</math><br> 
而且<math>\bar{\alpha} = \sum_{i=1}^K\alpha_i</math><br /><math>\operatorname{Cov}[X_i,X_j] = \frac{-\tilde{\alpha}_i \tilde{\alpha}_j}{\bar{\alpha}+1}~~(i\neq j)</math> |
  skewness   =|
  kurtosis   =|
  entropy    = <math> H(X) = \log \mathrm{B}(\alpha) + (\alpha_0-K)\psi(\alpha_0) - \sum_{j=1}^K (\alpha_j-1)\psi(\alpha_j) </math><br/>|
}}

狄利克雷分布是一组连续多变量概率分布，是多变量普遍化的[[Β分布]]。为了纪念德国数学家[[約翰·彼得·古斯塔夫·勒熱納·狄利克雷]]（Peter Gustav Lejeune Dirichlet）而命名。狄利克雷分布常作为贝叶斯统计的[[先验概率]]。当狄利克雷分布维度趋向无限时，这过程便称为[[狄利克雷过程]]（Dirichlet process）。

狄利克雷分布奠定了狄利克雷过程的基础，被广泛应用于[[自然语言处理]]特别是[[主题模型]]（topic model）的研究。

==概率密度函数==

[[Image:LogDirichletDensity-alpha 0.3 to alpha 2.0.gif|thumb|right|250px|此图展示了当''K''=3、参数''&alpha;''从''&alpha;''=(0.3,&nbsp;0.3,&nbsp;0.3)变化到(2.0,&nbsp;2.0,&nbsp;2.0)时，密度函数取对数后的变化。]]

维度''K''&nbsp;≥&nbsp;2的狄利克雷分布在参数''α''<sub>1</sub>, ..., ''α''<sub>''K''</sub> >&nbsp;0上、基于[[欧几里得空间]]'''R'''<sup>''K-1''</sup>里的[[勒贝格测度]]有个概率密度函数，定义为：

:<math>f(x_1,\dots, x_{K}; \alpha_1,\dots, \alpha_K) = \frac{1}{\mathrm{B}(\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1} </math>

其中<math>\boldsymbol x</math>满足<math>\sum_{i=1}^{K} x_i = 1</math>，同时对于任意<math>i \in \{1,\dots,K\}</math>，都有<math>x_i \ge 0</math>。即<math>\boldsymbol x</math>在(''K''&nbsp;&minus;&nbsp;1)维的[[单纯形]][[开集]]上密度为0。

归一化衡量''B(&alpha;)''是多项[[Β函数]]，可以用[[Γ函数]]（gamma function）表示：

:<math>\mathrm{B}(\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K \alpha_i\bigr)},\qquad\alpha=(\alpha_1,\dots,\alpha_K).</math>

<br />
==参见==
* [[約翰·彼得·古斯塔夫·勒熱納·狄利克雷  ]]
* [[狄利克雷过程]]
* [[隐含狄利克雷分布]]

==參考==
* [https://web.archive.org/web/20121021045524/http://www.ee.washington.edu/research/guptalab/publications/UWEETR-2010-0006.pdf Introduction to the Dirichlet Distribution and Related Processes by Frigyik, Kapila and Gupta]
* [https://web.archive.org/web/20120309151200/http://www.gatsby.ucl.ac.uk/~ywteh/research/npbayes/Teh2010a.pdf Dirichlet processes by Yee Whye Teh]

{{概率分布}}

[[Category:概率与统计]]
[[Category:连续分布]]