Hierarchical spatio-temporal models are used for modeling space-time responses and temporally and spatially correlations of the data is considered via Gaussian latent random field with Matérn covariance function. The most important interest in these models is estimation of the model parameters and the latent variables, and is predict of the response variables at new locations and times. In this paper, to analyze these models, the Bayesian approach is presented. Because of the complexity of the posterior distributions and the full conditional distributions of these models and the use of Monte Carlo samples in a Bayesian analysis, the computation time is too long. For solving this problem, Gaussian latent random field with Matern covariance function are represented as a Gaussian Markov Random Field (GMRF) through the Stochastic Partial Differential Equations (SPDE) approach. Approximatin Baysian method and Integrated Nested Laplace Approximation (INLA) are used to obtain an approximation of the posterior distributions and to inference about the model. Finally, the presented methods are applied to a case study on rainfall data observed in the weather stations of Semnan in 2013.

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.