We develop simple methods for constructing parameter priors for model choice among Directed Acyclic Graphical (DAG) models. In particular, we introduce several assumptions that permit the construction of parameter priors for a large number of DAG models from a small set of assessments. We then present a method for directly computing the marginal likelihood of every DAG model given a random sample with no missing observations. We apply this methodology to Gaussian DAG models which consist of a recursive set of linear regression models. We show that the only parameter prior for complete Gaussian DAG models that satisfies our assumptions is the normal-Wishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let $W$ be an $n \times n$, $n \ge 3$, positive-definite symmetric matrix of random variables and $f(W)$ be a pdf of $W$. Then, f$(W)$ is a Wishart distribution if and only if $W_{11} - W_{12} W_{22}^{-1} W'_{12}$ is independent of $\{W_{12},W_{22}\}$ for every block partitioning $W_{11},W_{12}, W'_{12}, W_{22}$ of $W$. Similar characterizations of the normal and normal-Wishart distributions are provided as well.
翻译:我们为在定向自行车图形模型(DAG)中进行模型选择而构建参数前程的简单方法。 特别是, 我们引入了几种假设, 允许从一小套评估中为大量DAG模型构建参数前前程。 然后, 我们提出一种方法, 直接计算每个DAG模型的边际可能性, 随机抽样, 没有缺失观察。 我们将这种方法应用于高西亚DAG模型, 其中包括一套循环的线性回归模型。 我们显示, 完整的高西亚DAG模型之前唯一符合我们假设的参数是正常- Wishart分布。 我们的分析基于Wishart分布的以下新定性: 让W$成为美元\timenn, $n\ge 3$, 正- 确定参数矩阵中随机变量的正- 12美元, $(W) 为 pdf 。 然后, f(W)$是一个W值的 Wart分布, 只有当 $*11}W\\\\-1}W'12} W'ZQQQQ} =正常分区的独立。 WQQQQQ* =QQQQ* = = =QQQQQQQQQQQQ=每个分区独立的分配。 然后。 然后, 美元。 然后, 美元。 然后, =W* = = = = = = = = = = = = = = = = = = = =