A Bayesian Framework for Persistent Homology
A Bayesian Framework for Persistent Homology
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DOI：
10.1137/19m1268719
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发表时间：
2020-01-01
影响因子：
3.6
通讯作者：
Oballe, Christopher
中科院分区：
文献类型：
--
作者：
Maroulas, Vasileios;Nasrin, Farzana;Oballe, Christopher
持久性图提供了一种总结数据集中潜在的拓扑和几何属性的方法。虽然已经开发了几种方法，在统计推断中使用持久性图表，但仍然缺乏完整的贝叶斯处理。本文以点过程理论为基础，提出了一个基于替代似然参数的广义贝叶斯推理框架。本质上，我们将持久性图建模为具有先验强度的泊松点过程，并通过采用标记点过程理论的技术来计算后验强度。然后，我们提出了一个家庭的共轭先验强度通过高斯混合，以获得一个封闭的形式的后验强度。最后，我们证明了这种广义贝叶斯框架的效用与分类问题，在材料科学中使用贝叶斯因子。
Persistence diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that use persistence diagrams in statistical inference, a full Bayesian treatment remains absent. This paper, relying on the theory of point processes, presents a generalized Bayesian framework for inference with persistence diagrams relying on a substitution likelihood argument. In essence, we model persistence diagrams as Poisson point processes with prior intensities and compute posterior intensities by adopting techniques from the theory of marked point processes. We then propose a family of conjugate prior intensities via Gaussian mixtures to obtain a closed form of the posterior intensity. Finally, we demonstrate the utility of this generalized Bayesian framework with a classification problem in materials science using Bayes factors.