Structural Equation Model With Bayesian B-Splines

On the basis of previous studies,this paper further improves the structural equation model.We estimate the structural equation of the SEM by using B-splines,making the modified SEM can describe the more complicated relations between the latent variables.Besides,we regard the number of B-splines knots as a random variable and get the optimal number of knots with Bayesian averaging method,which ensures the objectivity of knots selection.Finally,MCMC methods are developed to estimate the unknown parameters.The main contents of this paper are as follows:The first part is a brief introduction to the theoretical basis of the research,which is necessary for the follow-up study.This section mainly deals with the structural equation model,Bayesian theory,spline function basis and MCMC algorithm content.In terms of the second part,it is the construction of the general nonparametric SEM model.Firstly,the measurement equations are respectively applied to dealing with two different data types.Then,the structural equations of SEM are estimated by using the B-splines of indefinite nodes.Based on the determination of the number of splines and the position of the nodes,the transformation method ensures that the spline node interval can cover the range of the observed variables to ensure the validity of the model.The third part gives the model's restrictive conditions.The second part of the construction of the general meaning of the SEM,but in its framework,the new SEM parameters are not fully defined.This module makes some reasonable constraints on the value of the relevant parameters to ensure that the newly constructed SEM can be identified.As far as the fourth part is concerned,it is the Bayesian interpretation of the unknown parameters of the SEM.This section gives a reasonable prior distribution of the relevant parameters,mainly for the selection of the number of nodes in the prior distribution,and the distribution of the conditions of the parameters in the case of the number of nodes given.The fifth part presents the MCMC algorithm for estimating the unknown parameters of the new SEM.Since the number of nodes is treated as a random variable,the dimension of the model is constantly changing.In order to estimate the specific value of unknown parameters,this paper uses the MCMC algorithm of reversible jump to sample and gives the specific sampling steps.The final paper points out the shortcomings of this model and the direction in which it can continue to be studied;and in the appendix,a demonstration program of the MCMC sampling principle is given.