Investigating relationships between response variables and covariates in areas such as environmental science, geoscience, and public health is an important endeavor. Based on a Bayesian mixture of finite mixtures model, we present a novel spatially clustered coefficients regression model for count value data. The proposed method detects the spatial homogeneity of the Poisson regression coefficients. A Markov random field constrained mixture of finite mixtures prior provides a regularized estimator of the number of clusters of regression coefficients with geographical neighborhood information. As a by-product, we also provide the theoretical properties of our proposed method when the Markov random field is exchangeable. An efficient Markov chain Monte Carlo algorithm is developed by using the multivariate log gamma distribution as a base distribution. Simulation studies are carried out to examine the empirical performance of the proposed method. Additionally, we analyze Georgia's premature death data as an illustration of the effectiveness of our approach. The supplementary materials are provided on GitHub at \url{https://github.com/pengzhaostat/MLG_MFM}.

翻译：探究环境科学、地球科学与公共卫生等领域中响应变量与协变量之间的关系具有重要研究意义。本文基于贝叶斯有限混合混合模型，提出一种适用于计数数据的新型空间聚类系数回归模型。该方法可检测泊松回归系数的空间同质性。通过马尔可夫随机场约束的有限混合混合先验，利用地理邻域信息实现对回归系数聚类数量的正则化估计。作为附带成果，本文还证明了所提方法在马尔可夫随机场具备可交换性时的理论性质。通过采用多元对数伽马分布作为基分布，开发了高效的马尔可夫链蒙特卡洛算法。通过仿真实验检验了所提方法的实证性能，并利用佐治亚州过早死亡数据验证了方法的有效性。补充材料详见GitHub仓库：\url{https://github.com/pengzhaostat/MLG_MFM}。