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.

Spatio-temporal data often exhibit complex dependence structures and skewness, which makes their modeling with classical frameworks, such as Gaussian random fields, either computationally expensive or overly restrictive. In this paper, we introduce a novel Bayesian Neural Field framework for modeling skewed spatio-temporal processes. The proposed approach incorporates spatial and temporal coordinates, along with explanatory variables and prior distributions, allowing flexible representation of dependence and skewness, as well as prediction at new locations and at unseen time points. Parameter inference is performed using variational inference, which offers both computational efficiency and the ability to quantify uncertainty. Simulation results demonstrate that the proposed framework achieves higher accuracy and faster computation compared to standard Monte Carlo methods.