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 https://github.com/pengzhaostat/MLG_MFM.

Basketball shot location data provide valuable summary information regarding players to coaches, sports analysts, fans, statisticians, as well as players themselves. Represented by spatial points, such data are naturally analyzed with spatial point process models. We present a novel nonparametric Bayesian method for learning the underlying intensity surface built upon a combination of Dirichlet process and Markov random field. Our method has the advantage of effectively encouraging local spatial homogeneity when estimating a globally heterogeneous intensity surface. Posterior inferences are performed with an efficient Markov chain Monte Carlo (MCMC) algorithm. Simulation studies show that the inferences are accurate and the method is superior compared to a wide range of competing methods. Application to the shot location data of $20$ representative NBA players in the 2017-2018 regular season offers interesting insights about the shooting patterns of these players. A comparison against the competing method shows that the proposed method can effectively incorporate spatial contiguity into the estimation of intensity surfaces.

The computational limitations of enormously large datasets make statistical inference with big data challenging. In this paper, we propose a Bayesian approach that uses nonuniform subsampled data to derive the pseudo-posterior distribution of the parameters of interest Using the proposed subsampling strategies, we investigate a Jeffery’s prior. In addition, we develop a theoretical framework for analyzing the posterior convergence rate of the pseudo-posterior in subsampling, which relaxes the conditions of corresponding theoretical results from sample survey literature. Extensive simulation studies are carried out to confirm the valid estimation and computational efficiency of our proposed method. Finally, we analyze PUMS Data as illustrations of the effectiveness of our approach.

Bayesian methods face unprecedented challenges in the era of big data, as the evaluation of likelihood in each iteration is computationally intensive. To deal with this bottleneck, recent literature focus mostly on speeding up Markov chain Monte Carlo (MCMC). Model selection, which is an important topic, has not received much attention. In the Bayesian context, deviance-based criteria, such as the deviance information criterion (DIC), are well-known for model selection purposes. In this article, we introduce the subsampled DIC and the subsampled information criterion IC in the big data context. Extensive simulation studies are conducted to evaluate the empirical performance of the proposed criterion. The usage of our proposed criterion is further illustrated with an analysis of the Covertype dataset.