An analysis on covariates selection problem for Gaussian model by Maximum a posteriori criterion using frequentist and Bayesian approaches

Choosing the most suitable fitted model on data is one of the common challenges in statistical modeling. Maximum a posteriori (MAP) criterion is a method used in both frequentist and Bayesian approaches. Additionally, the utility of the model is used as a tool to compare the performances of methods. In this paper, the MAP method is applied for the Gaussian model and its performance is compared to that of frequentist approach. Also, an analytical form of utility estimation is proposed. Besides, using simulation studies, it is shown that the Gaussian model has better performance, based on both utility and mean of squared errors (MSE) criteria, when it is used by the Bayesian approach. However, both frequentist and Bayesian approaches avoid over-fitting by increasing the sample size. Also, by increasing correlation among covariates, MSE increases, while the tendency of choosing fewer covariates is raised. Eventually, the study on a real dataset is shown that in both frequentist and Bayesian approaches, MSE of selected models decreases when the size of sample increases.