In presence of high dimensional cavariates, variable selection is an important technique for any further data analysis. Bayesian analysis can reach the aim of model selection based on shrinkage priors. First I would explain Bayesian variable selection technique through three methods, which have been demonstrated giving plausible performance when working on high dimensional model selection problems. I also compared these methods based on both simulation results and real data application. Further I extend the method based on Dirichlet-Laplace prior from normal means problem to linear regression model, and show the minimax contraction rate still holds under mild conditions.While most developments in Bayesian model selection literature are based on local prior on regression parameters, Johnson and Rossell(2012, 2013) proposed a nonlocal prior distribution for model selection. Enlightened by this idea, I applied nonlocal prior while performing spike and slab variable selection method. I used a point mass density for spike prior, while applied nonlocal prior as slab density, this setting could make overlap between spike and slab prior very little, which could achieve variable selection result efficiently. Following I proved the consistency for variable selection of proposed method. At last, I extended nonlocal prior model selection method from Johnson and Rossell(2012,2013) to logistic regression and to generalized linear models. Laplace approximations are used in implementation process due to complicated likelihood. Also, convergence rate is derived under some regularity conditions. The selection based on a nonlocal prior eliminates unnecessary variables and recommends a simple model. This method is validated by simulation study and illustrated by real data example.
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