Large Sample Property of The Bayes Factor in a Spline Semiparametric Regression Model

In this paper, we consider semiparametric regression model where the mean function of this model has two part, the first is the parametric part is assumed to be linear function of p-dimensional covariates and nonparametric ( second part ) is assumed to be a smooth penalized spline. By using a convenient connection between penalized splines and mixed models, we can representation semiparametric regression model as mixed model. In this model, we investigate the large sample property of the Bayes factor for testing the polynomial component of spline model against the fully spline semiparametric alternative model. Under some conditions on the prior and design matrix, we identify the analytic form of the Bayes factor and show that the Bayes factor is consistent. Keywords: Mixed Models, Semiparametric Regression Model, Penalized Spline, Bayesian Model, , Marginal Distribution, Prior Distribution, Posterior Distribution, Bayes Factor, Consistent