Stratified Gaussian Graphical Models

Gaussian graphical models represent the backbone of the statistical toolbox for analyzing continuous multivariate systems. However, due to the intrinsic properties of the multivariate normal distribution, use of this model family may hide certain forms of context-specific independence that are natural to consider from an applied perspective. Such independencies have been earlier introduced to generalize discrete graphical models and Bayesian networks into more flexible model families. Here we adapt the idea of context-specific independence to Gaussian graphical models by introducing a stratification of the Euclidean space such that a conditional independence may hold in certain segments but be absent elsewhere. It is shown that the stratified models define a curved exponential family, which retains considerable tractability for parameter estimation and model selection.

高斯图模型（Gaussian graphical models）是分析连续多变量系统的统计工具箱的核心框架。然而，由于多元正态分布的固有属性，该模型族的应用可能会掩盖某些形式的特定上下文独立性（context-specific independence），而从应用视角来看，这类独立性本是值得自然考量的。此前已有研究引入这类独立性的概念，将离散图模型与贝叶斯网络（Bayesian networks）推广为更为灵活的模型族。本文通过对欧几里得空间（Euclidean space）进行分层处理，将特定上下文独立性的理念适配至高斯图模型中，使得条件独立性仅在部分分段中成立，而在其余分段中不成立。研究表明，此类分层模型属于弯曲指数族（curved exponential family），在参数估计与模型选择方面仍具备相当高的可处理性。