随机生长树的历史推断
Inference on the History of a Randomly Growing Tree
摘要：
传染病在人类社区中的传播或社交媒体上假新闻的扩散可以被建模为一个随机增长的树形图。随机增长过程的历史往往是不可观察的，但包含重要的信息，如感染源。我们考虑的问题，统计推断方面的潜在的历史只使用一个单一的快照的最终树。我们的方法是将随机标签应用于观察到的未标记树，并分析生长过程的分布，以最终结果为条件。我们表明，这个条件分布是听话的形状交换条件下，我们在这里介绍，这个条件是满足许多流行的随机生长的树木，如均匀附件，线性优先附件和均匀附件的$D$ -定期树模型。对于根的推理下的形状交换，我们提出了O（n log n）时间算法，用于构建置信集的有效频率覆盖范围以及置信集的预期大小的界限。我们还提供了有效的采样算法，扩展我们的方法广泛的一类推理问题。
英文摘要：
The spread of infectious disease in a human community or the proliferation of fake news on social media can be modeled as a randomly growing tree-shaped graph. The history of the random growth process is often unobserved but contains important information such as the source of the infection. We consider the problem of statistical inference on aspects of the latent history using only a single snapshot of the final tree. Our approach is to apply random labels to the observed unlabeled tree and analyze the resulting distribution of the growth process, conditional on the final outcome. We show that this conditional distribution is tractable under a shape-exchangeability condition, which we introduce here, and that this condition is satisfied for many popular models for randomly growing trees such as uniform attachment, linear preferential attachment and uniform attachment on a $D$-regular tree. For inference of the root under shape-exchangeability, we propose O(n log n) time algorithms for constructing confidence sets with valid frequentist coverage as well as bounds on the expected size of the confidence sets. We also provide efficient sampling algorithms that extend our methods to a wide class of inference problems.
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