Ising和Gaussian模型中双样本结构变化检测的下界
Lower Bounds for Two-Sample Structural Change Detection in Ising and Gaussian Models
摘要：
变化检测问题是确定两个马尔可夫随机场的马尔可夫网络结构是否彼此不同，给定两组从各自的底层分布中提取的样本。我们研究的样本大小和变化检测的可靠性之间的权衡，作为一个极大极小风险衡量，为重要的情况下的伊辛模型和高斯马尔可夫随机场限制的模型，具有网络结构的$p$节点和度最多$d$，并获得信息理论的下界可靠的变化检测这些模型。我们表明，对于伊辛模型，$Ω\left(\frac{d^2}{(\log d)^2}\log p\right)$样本需要从每个数据集检测甚至sparterly可能的变化，并为高斯，$Ω\left( γ^{-2} \log(p)\right)$样本需要从每个数据集检测变化，其中$γ$是最小的比例的非对角项的分布的精度矩阵。这些界限相比，相应的结构学习的结果，并密切配合他们在温和的条件下的模型参数。因此，我们的变化检测界继承了部分紧密性的结构学习计划在以前的文献中，表明在某些参数制度，朴素的结构学习为基础的方法来改变检测是最小最大最优常数的因素。
英文摘要：
The change detection problem is to determine if the Markov network structures of two Markov random fields differ from one another given two sets of samples drawn from the respective underlying distributions. We study the trade-off between the sample sizes and the reliability of change detection, measured as a minimax risk, for the important cases of the Ising models and the Gaussian Markov random fields restricted to the models which have network structures with $p$ nodes and degree at most $d$, and obtain information-theoretic lower bounds for reliable change detection over these models. We show that for the Ising model, $Ω\left(\frac{d^2}{(\log d)^2}\log p\right)$ samples are required from each dataset to detect even the sparsest possible changes, and that for the Gaussian, $Ω\left( γ^{-2} \log(p)\right)$ samples are required from each dataset to detect change, where $γ$ is the smallest ratio of off-diagonal to diagonal terms in the precision matrices of the distributions. These bounds are compared to the corresponding results in structure learning, and closely match them under mild conditions on the model parameters. Thus, our change detection bounds inherit partial tightness from the structure learning schemes in previous literature, demonstrating that in certain parameter regimes, the naive structure learning based approach to change detection is minimax optimal up to constant factors.
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