We introduce a general Bayesian framework for graph matching grounded in a new theory of exchangeable random permutations. Leveraging the cycle representation of permutations and the literature on exchangeable random partitions, we define, characterize, and study the structural and predictive properties of these probabilistic objects. A novel sequential metaphor, the position-aware generalized Chinese restaurant process, provides a constructive foundation for this theory and supports practical algorithmic design. Exchangeable random permutations offer flexible priors for a wide range of inferential problems centered on permutations. As an application, we develop a Bayesian model for graph matching that integrates a correlated stochastic block model with our novel class of priors. The cycle structure of the matching is linked to latent node partitions that explain connectivity patterns, an assumption consistent with the homogeneity requirement underlying the graph matching task itself. Posterior inference is performed through a node-wise blocked Gibbs sampler directly enabled by the proposed sequential construction. To summarize posterior uncertainty, we introduce perSALSO, an adaptation of SALSO to the permutation domain that provides principled point estimation and interpretable posterior summaries. Together, these contributions establish a unified probabilistic framework for modeling, inference, and uncertainty quantification over permutations.

翻译：本文提出了一种基于可交换随机置换新理论的通用贝叶斯图匹配框架。通过利用置换的循环表示和可交换随机划分的相关文献，我们定义、刻画并研究了这些概率对象的结构特性与预测性质。一种新颖的序列化隐喻——位置感知广义中国餐馆过程——为该理论提供了构造性基础，并支持实际算法设计。可交换随机置换为以置换为核心的一系列推断问题提供了灵活的先验分布。作为应用，我们开发了一种贝叶斯图匹配模型，该模型将相关随机块模型与我们提出的新型先验分布类相结合。匹配的循环结构与解释连接模式的隐节点划分相关联，这一假设与图匹配任务本身所要求的同质性条件保持一致。后验推断通过节点分块吉布斯采样器实现，该采样器直接由所提出的序列化构造方法支持。为总结后验不确定性，我们引入了perSALSO方法——这是SALSO在置换域的适配版本，能够提供原则性的点估计和可解释的后验摘要。这些贡献共同建立了一个统一的概率框架，用于对置换进行建模、推断和不确定性量化。