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An important issue in many cities is related to crime events, and spatio–temporal Bayesian approach leads to identifying crime patterns and hotspots. In Bayesian analysis of spatio–temporal crime data, there is no closed form for posterior distribution because of its non-Gaussian distribution and existence of latent variables. In this case, we face different challenges such as high dimensional parameters, extensive simulation and time-consuming computation in applying MCMC methods. In this paper, we use INLA to analyze crime data in Colombia. The advantages of this method can be the estimation of criminal events at a specific time and location and exploring unusual patterns in places.

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This paper presents a nonparametric Bayesian method for estimating nonstationary covariance structures in big spatial datasets. The approach extends the Vecchia approximation and assumes conditional independence among ordered data points, leading to a sparse precision matrix and sparse Cholesky decomposition. This enables modeling an $n$-dimensional Gaussian process as a sequence of Bayesian linear regressions. Data ordering via maximum minimum distance improves model performance. Applying the grouping algorithm to ordered data removes weak dependencies and defines a block-sparse covariance structure, significantly reducing computational burden and enhancing accuracy. Simulations and real data analysis show that posterior samples from the proposed method yield narrower uncertainty intervals than those from ungrouped approaches.