Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses
Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses
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DOI：
10.1214/13-aos1162
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发表时间：
2012-12
期刊：
影响因子：
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通讯作者：
Po-Ling Loh;M. Wainwright
中科院分区：
文献类型：
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作者：
Po-Ling Loh;M. Wainwright
我们研究了离散图模型的结构与广义协方差矩阵逆的支撑度之间的关系。我们证明了对于某些图结构，图的顶点上指示变量的逆协差阵的支持度反映了图的条件独立性结构。我们的工作扩展了以前仅在多变量高斯图形模型的背景下建立的结果，从而解决了关于非高斯分布的逆协差阵的重要性的公开问题。该证明综合运用了指数族几何、连接树理论和凸分析的思想。这些总体水平的结果对已知的和新的图形选择方法有不同的影响，包括一种用于丢失或损坏观测的结构估计的新方法。我们为这些方法提供了非渐近保证，并通过模拟说明了这些预测的精确性。
We investigate the relationship between the structure of a discrete graphical model and the support of the inverse of a generalized covariance matrix. We show that for certain graph structures, the support of the inverse covariance matrix of indicator variables on the vertices of a graph reflects the conditional independence structure of the graph. Our work extends results that have previously been established only in the context of multivariate Gaussian graphical models, thereby addressing an open question about the significance of the inverse covariance matrix of a non-Gaussian distribution. The proof exploits a combination of ideas from the geometry of exponential families, junction tree theory and convex analysis. These population-level results have various consequences for graph selection methods, both known and novel, including a novel method for structure estimation for missing or corrupted observations. We provide nonasymptotic guarantees for such methods and illustrate the sharpness of these predictions via simulations.