A characterization of the Dirichlet distribution through global and local parameter independence

• 成果类型: Article
• 署名作者: Geiger, D; Heckerman, D
• 署名单位: Technion Israel Institute of Technology; Microsoft
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1997
• 页码: 1344-1369
• 关键词: graphical structures; influence diagrams; Bayesian networks; belief networks; expert-systems; models; probabilities

摘要

We provide a new characterization of the Dirichlet distribution. Let theta(ij), 1 less than or equal to i less than or equal to k, 1 less than or equal to j less than or equal to n, be positive random variables that sum to unity. Define theta(i). = Sigma(j=1)(n) theta(ij), theta(I), = (theta(i.))(i=1)(k-1), theta(j\1) = theta(ij)/Sigma(j)theta(ij) and theta(J\i) = (theta(j\i))(j=1)(n-1). We prove that if (theta(I)., theta(J\1),..., theta(J\k)) are mutually independent and (theta.(J), theta(I\l),..., theta(I\n)) are mutually independent (where theta.(J) and theta(I\j) are defined analogously), and each parameter set has a strictly positive pdf, then the pdf of theta(ij) is Dirichlet. This characterization implies that under assumptions made by several previous authors for selecting a Bayesian network structure out of a set of candidate structures, a Dirichlet prior on the parameters is inevitable.