标题： 一般高斯隐树模型的代数检验 显示英文标题

标题： Algebraic tests of general Gaussian latent tree models

摘要： 我们考虑一般的高斯隐树模型，其中观测变量不限于树的叶节点。 扩展相关最近的工作，我们给出了任何此类模型的协方差矩阵集合的完整半代数描述。 换句话说，我们找到了多项式约束，这些约束可以表征一个矩阵何时是给定隐树模型中分布的协方差矩阵。 然而，利用这些约束来检验给定的此类模型通常会因约束数量庞大以及个别多项式的奇点而变得复杂，这些奇点可能会使相关概率分布的标准近似失效。 以星形树为例，我们提出了一种新的检验方法，通过牺牲一些统计估计效率来避免奇点问题，并通过高维随机向量和的最大值的高斯近似方面的最新进展来处理大量约束的情况。 我们的检验方法无需最大化此类模型可能具有多模态的似然函数，并适用于变量数量较多的模型。 这些观点在数值实验中得到了说明。 显示英文摘要

摘要： We consider general Gaussian latent tree models in which the observed variables are not restricted to be leaves of the tree. Extending related recent work, we give a full semi-algebraic description of the set of covariance matrices of any such model. In other words, we find polynomial constraints that characterize when a matrix is the covariance matrix of a distribution in a given latent tree model. However, leveraging these constraints to test a given such model is often complicated by the number of constraints being large and by singularities of individual polynomials, which may invalidate standard approximations to relevant probability distributions. Illustrating with the star tree, we propose a new testing methodology that circumvents singularity issues by trading off some statistical estimation efficiency and handles cases with many constraints through recent advances on Gaussian approximation for maxima of sums of high-dimensional random vectors. Our test avoids the need to maximize the possibly multimodal likelihood function of such models and is applicable to models with larger number of variables. These points are illustrated in numerical experiments.