This paper presents a focused review of Markov random fields (MRFs)--commonly used probabilistic representations of spatial dependence in discrete spatial domains--for categorical data, with an emphasis on models for binary-valued observations or latent variables. We examine core structural properties of these models, including clique factorization, conditional independence, and the role of neighborhood structures. We also discuss the phenomenon of phase transition and its implications for statistical model specification and inference. A central contribution of this review is the use of response functions, a unifying tool we introduce for prior analysis that provides insight into how different formulations of MRFs influence implied marginal and joint distributions. We illustrate these concepts through a case study of direct-data MRF models with covariates, highlighting how different formulations encode dependence. While our focus is on binary fields, the principles outlined here extend naturally to more complex categorical MRFs and we draw connections to these higher-dimensional modeling scenarios. This review provides both theoretical grounding and practical tools for interpreting and extending MRF-based models.

翻译：本文针对分类数据，聚焦评述马尔可夫随机场（MRFs）——一种在离散空间域中表示空间依赖性的常用概率模型——特别关注二值观测或潜变量模型。我们考察了这些模型的核心结构特性，包括团分解、条件独立性以及邻域结构的作用。同时，我们讨论了相变现象及其对统计模型设定与推断的影响。本综述的一个核心贡献是引入了响应函数这一统一工具进行先验分析，它揭示了MRFs的不同形式如何影响隐含的边际分布与联合分布。我们通过一个带协变量的直接数据MRF模型案例研究来阐释这些概念，重点说明不同形式如何编码依赖性。尽管本文聚焦于二值场，但所述原理可自然推广至更复杂的分类MRF模型，文中亦建立了与这些高维建模场景的联系。本综述为解释和拓展基于MRF的模型提供了理论基础与实践工具。