Wishart distributions for decomposable graphs
Wishart distributions for decomposable graphs
复制标题
DOI：
10.1214/009053606000001235
复制
发表时间：
2007-06-01
影响因子：
4.5
通讯作者：
Massam, Helene
中科院分区：
文献类型：
--
作者：
Letac, Gerard;Massam, Helene
当考虑关于可分解图G的图形高斯模型N-G马尔可夫时，精度参数感兴趣的参数空间是与G的缺失边对应的固定零的正定矩阵的锥PG。N-G的尺度参数的参数空间是与G的团对应的子矩阵为正定的不完全矩阵的锥Q(G)对偶P-G。本文在锥Q(G)和P-G上构造了两族Wishart分布，即I型和II型Wisharts。它们可以被看作是由david和Lauritzen [Ann]定义的超逆Wishart的概括和超逆Wishart的逆。统计学家。21(1993)1272-1317。我们证明了I型和II型Wishart具有与超和超逆Wishart相似的性质。实际上，II型Wishart的逆形成了图形高斯模型的协方差参数的共轭先验族，并且对于由其团的完美顺序给出的图的每个方向都是强有向超马尔可夫，而I型Wishart是弱超马尔可夫。此外，逆II型Wishart作为共轭族呈现出具有多维形状参数的优势，从而为选择先验提供了灵活性。I型和II型Wishart分布都依赖于多变量形状参数。当且仅当形状参数满足某一特征值属性时，形状参数是可接受的。我们证明了非完全G的可接受形状参数集的维数至少等于1加上G中团的个数。这些族作为共轭族，比传统的Diaconis-Ylvisaker共轭族更丰富，后者的形状参数集都是维数为1。一个不包含三环链作为诱导子图的可分解图称为齐次图。在这种情况下，我们的Wisharts是由Andersson和Wojnar [J]定义的齐次锥上的Wisharts的特殊情况。系统结构。Probab. 17(2004) 781-818]，形状参数集的维数甚至比非齐次情况下更大：它确实等于团的数量加上不同的最小分隔符的数量。使用G是一个三环链的模型，我们通过计算一个7元组积分表明，通常我们不能期望形状参数集的维度大于团的数量加1。
When considering a graphical Gaussian model N-G Markov with respect to a decomposable graph G, the parameter space of interest for the precision parameter is the cone PG of positive definite matrices with fixed zeros corresponding to the missing edges of G. The parameter space for the scale parameter of N-G is the cone Q(G), dual to P-G, of incomplete matrices with submatrices corresponding to the cliques of G being positive definite. In this paper we construct on the cones Q(G) and P-G two families of Wishart distributions, namely the Type I and Type II Wisharts. They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272-1317]. We show that the Type I and II Wisharts have properties similar to those of the hyper and hyper inverse Wishart. Indeed, the inverse of the Type II Wishart forms a conjugate family of priors for the covariance parameter of the graphical Gaussian model and is strong directed hyper Markov for every direction given to the graph by a perfect order of its cliques, while the Type I Wishart is weak hyper Markov. Moreover, the inverse Type II Wishart as a conjugate family presents the advantage of having a multidimensional shape parameter, thus offering flexibility for the choice of a prior. Both Type I and II Wishart distributions depend on multivariate shape parameters. A shape parameter is acceptable if and only if it satisfies a certain eigenvalue property. We show that the sets of acceptable shape parameters for a noncomplete G have dimension equal to at least one plus the number of cliques in G. These families, as conjugate families, are richer than the traditional Diaconis-Ylvisaker conjugate families which all have a shape parameter set of dimension one. A decomposable graph which does not contain a three-link chain as an induced subgraph is said to be homogeneous. In this case, our Wisharts are particular cases of the Wisharts on homogeneous cones as defined by Andersson and Wojnar [J. Theoret. Probab. 17 (2004) 781-818] and the dimension of the shape parameter set is even larger than in the nonhomogeneous case: it is indeed equal to the number of cliques plus the number of distinct minimal separators. Using the model where G is a three-link chain, we show by computing a 7-tuple integral that in general we cannot expect the shape parameter sets to have dimension larger than the number of cliques plus one.