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Title: Statistical inference in large Ising graphical models via quadratic programming Authors:  Zhao Ren - University of Pittsburgh (United States) [presenting]
Cun-Hui Zhang - Rutgers University (United States)
Harrison Zhou - Yale University (United States)
Sai Li - University of Pennsylvania (United States)
Abstract: The high dimensional graphical model, a powerful tool for studying conditional dependency relationship of random variables, has attracted great attention in recent years. A statistical inference of each edge for large Ising graphical models is investigated. Significant progress has been achieved recently in computing confidence intervals and $p$-values for each edge. The key role in these new inferential methods is played by a linear projection method to de-bias an initial regularized estimator. Major drawback of this approach in Ising models is that an extra sparsity assumption on the linear projection coefficient besides the sparsity of the graph itself is required, which cannot be checked in practice. In addition, efficiency is often compromised by the usage of sample splitting in these methods. We propose a novel estimator of each edge via quadratic programming and show that our estimator is asymptotically normal without the above mentioned extra sparsity condition. Our proof applies a novel low dimensional maximum likelihood method for the de-bias procedure and a data swap technique to avoid loss of efficiency. We further show that whenever the extra sparsity condition is satisfied, our estimator is adaptively efficient and achieves the Fisher information. Otherwise, we still provide a restricted Fisher information as a lower bound.