Advances On Smoothing Spline Model Based On Bayesian Framework

On one hand,Bayesian smoothing spline model is based on the idea of“a priori-observation-posteriori”of Bayesian method,and usually combines with efficient sam-pling methods such as Markov Chain Monte Carlo(MCMC).On the other hand,it has a wide application in fitting complex functions.Therefore,Bayesian smoothing spline model has become a powerful statistical tool in many fields,such as finance,biostatis-tics and medical fields.The Bayesian smoothing spline model becomes more and more popular.This is due to the developments of the Bayesian methods and the smoothing spline methods.Firstly,Bayesian methods have been developed significantly in recent decades:In the priori selection,the use of objective Bayesian methods avoid the disa-vantages cauded by the subjectivity;The improvement of computing efficiencies and the proposal of many efficient algorithms(such as MCMC algorithm,Gibbs sampling,etc.)improve the speed of a posterior sampling.Secondly,smoothing spline methods have become a powerful statistical tool for fitting complex functions using non-parametric methods,because of their wide application and flexibility,i.e.,using nodes with special conditions to fit smoothing curves or surfaces.In this paper,we introduce the significant development made by the Bayesians in the efficiency of sampling the posterior and objectivity criterion.Also,we give ex-amples to illustrate the advantages of Bayesian in statistical inference and solving the problems of small sample size.We then introduce not only two smoothing spline meth-ods on the framework of frequency which are natural smoothing spline and reproducing kernel Hilbert space method but also two bayesian smppthing spline methods which are Gaussian process prior method(Wahba,1978)and partially normal prior distribution method(Speckman&Sun,2003).Last but not least,we focus on the equivalence of these four classical methods.In this chapter,we mainly prove that the partially normal prior distribution method(Speckman&Sun,2003)is equivalent with the other three methods.Furthermore,we propose two different kinds of multivariate smoothing spline models:One is the heteroskedasticity multivariate smoothing spline model based on the shrinkage inverse Wishart prior distribution(HSIW).In this method,we improve the multivariate smoothing spline model with the inverse Wishart prior distribution by relaxing the homovariance assumption,combining the real-world application scenarios,and considering the heteroscedasticity condition.The other one is multivariate smooth-ing spline with the commutative prior.The use of the commutative prior can not only reduce the calculation dimension,but also it meets many special models,which broad-ens the practical application scenarios.Finally,we analyse a data set which contains the numbers of confirmed,cured,suspected,and dead people in some provinces at the beginning of COVID-19.We use Bayesian methods and non-Bayesian methods such as natural smooth spline to fit the curve of the initial epidemic development trend,and compares them using different evaluation indexes,so as to conclude that using Bayesian methods is better than non-Bayesian methods.In addition,we conduct several Monte carlo simulations to compare the behavior of HSIW and the natural smoothing spline method in fitting models using the mean squared error(MSE)and R squares(R~2).The results demonstrate that HSIW has a good fit and outperforms the natural smoothing spline method.