有向信息图
Directed Information Graphs
摘要：
我们提出了一个图形模型表示网络的随机过程，最小生成模型图。它基于联合分布随时间的简化因子分解。我们表明，在适当的条件下，它是唯一的和一致的另一种类型的图形模型，有向信息图，这是基于一个推广的格兰杰因果关系。我们演示了如何有向信息量化格兰杰因果关系在一个特定的顺序预测设置。我们还开发了有效的方法来估计的拓扑结构的数据，估计联合统计。一种算法假设度的上界，并使用必要的最小维统计。在上界无效的情况下，所得到的图仍然是最佳近似。另一种算法使用接近最小维统计时，没有边界是已知的，但分布满足一定的标准。类似于无向图模型的结构学习算法如何使用互信息估计，这些算法使用有向信息估计。我们刻画了两个插入式有向信息估计量的样本复杂性，并得到了置信区间。对于点估计不可靠的情况，我们提出了一种算法，它使用置信区间来确定对估计误差具有鲁棒性的最佳近似。最后，我们通过对Twitter网络的合成数据和真实数据的分析，证明了所提出的算法的有效性。在后一种情况下，我们仅通过分析推文次数来确定哪些新闻来源影响网络中的用户。
英文摘要：
We propose a graphical model for representing networks of stochastic processes, the minimal generative model graph. It is based on reduced factorizations of the joint distribution over time. We show that under appropriate conditions, it is unique and consistent with another type of graphical model, the directed information graph, which is based on a generalization of Granger causality. We demonstrate how directed information quantifies Granger causality in a particular sequential prediction setting. We also develop efficient methods to estimate the topological structure from data that obviate estimating the joint statistics. One algorithm assumes upper-bounds on the degrees and uses the minimal dimension statistics necessary. In the event that the upper-bounds are not valid, the resulting graph is nonetheless an optimal approximation. Another algorithm uses near-minimal dimension statistics when no bounds are known but the distribution satisfies a certain criterion. Analogous to how structure learning algorithms for undirected graphical models use mutual information estimates, these algorithms use directed information estimates. We characterize the sample-complexity of two plug-in directed information estimators and obtain confidence intervals. For the setting when point estimates are unreliable, we propose an algorithm that uses confidence intervals to identify the best approximation that is robust to estimation error. Lastly, we demonstrate the effectiveness of the proposed algorithms through analysis of both synthetic data and real data from the Twitter network. In the latter case, we identify which news sources influence users in the network by merely analyzing tweet times.
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