标题： 多元伯努利分布 显示英文标题

标题： Multivariate Bernoulli distribution

摘要： 在本文中，我们考虑将多元伯努利分布作为模型，以估计具有二元节点的图结构。 该分布是在指数族的框架下进行讨论的，并展示了其关于节点独立性的统计特性。 重要的是，该模型不仅可以估计节点的主要效应和成对交互作用，还能够建模更高阶的交互作用，从而允许存在复杂的团效应。 我们将多元伯努利模型与现有的图形推断模型——伊辛模型和多元高斯模型进行了比较，在这些模型中仅考虑了成对交互作用。 另一方面，多元伯努利分布具有一种有趣的性质，即各分量随机变量的独立性和不相关性是等价的。 在多元伯努利分布中，变量子集的边缘分布和条件分布仍然遵循多元伯努利分布。 此外，通过利用规范链接函数，在广义线性模型理论下开发了多元伯努利逻辑回归模型，以包含关于节点、边和团的协变量信息。 我们还在逻辑回归模型中考虑了如LASSO之类的变量选择技术，以在图上施加稀疏结构。 最后，我们讨论将平滑样条ANOVA方法扩展到多元伯努利逻辑回归模型，以实现预测变量非线性效应的估计。 显示英文摘要

摘要： In this paper, we consider the multivariate Bernoulli distribution as a model to estimate the structure of graphs with binary nodes. This distribution is discussed in the framework of the exponential family, and its statistical properties regarding independence of the nodes are demonstrated. Importantly the model can estimate not only the main effects and pairwise interactions among the nodes but also is capable of modeling higher order interactions, allowing for the existence of complex clique effects. We compare the multivariate Bernoulli model with existing graphical inference models - the Ising model and the multivariate Gaussian model, where only the pairwise interactions are considered. On the other hand, the multivariate Bernoulli distribution has an interesting property in that independence and uncorrelatedness of the component random variables are equivalent. Both the marginal and conditional distributions of a subset of variables in the multivariate Bernoulli distribution still follow the multivariate Bernoulli distribution. Furthermore, the multivariate Bernoulli logistic model is developed under generalized linear model theory by utilizing the canonical link function in order to include covariate information on the nodes, edges and cliques. We also consider variable selection techniques such as LASSO in the logistic model to impose sparsity structure on the graph. Finally, we discuss extending the smoothing spline ANOVA approach to the multivariate Bernoulli logistic model to enable estimation of non-linear effects of the predictor variables.