Course Relationship Analysis Based On Bayesian Network For Mixed Data

Bayesian network is a Directed Acyclic Graph(DAG)model used to describe the dependencies between random variables.Bayesian network research mainly includes two parts of the parameters and structure learning.Structure learning refers to the discovery of dependency between variables and the learning of DAG structure based on observed data,which is essentially a problem of model selection in statistics.In this paper,we consider the Bayesian network structure learning problem with mixed variables containing both ordinal variables and Gaussian variables.In this paper,we first assume that the ordinal variable is truncated by the Gaussian latent variable,and further assume that the joint distribution of the Gaussian latent variable and the observed Gaussian variable is the Gaussian distribution,and the conditional independent relation contained in the distribution can be expressed by DAG,and then propose the potential Gaussian DAG model in the case of mixed variables containing the ordinal variable and the Gaussian variable.In addition,based on structural EM(SEM)algorithm,a Bayesian network structure learning algorithm,namely MSEM algorithm,is proposed in the case of mixed variables including ordinal variables and Gaussian variables.The MSEM algorithm is divided into three steps,namely initialization,E step and M step.Initialization gives the initial value of the algorithm iteration,including the initial DAG and the corresponding parameters.E step calculates the expected log-likelihood of the candidate DAG model under the current parameters.M step is calculated to punish the DAG structure with the optimal log-likelihood expectation,and the corresponding parameters of the optimal DAG structure are obtained.The MSEM algorithm iterates E and M steps until convergence.Since E step involves high dimensional numerical integration,this paper uses MCEM algorithm approximation to calculate E step by sampling from the posterior distribution of latent variable,namely truncated normal distribution.We compare the performance of MSEM algorithm with CPC,MMPC and OPC algorithms in the Bayesian network structure learning problem with mixed data containing ordinal data and Gaussian data through simulation experiments.According to the simulation results,it is found that MSEM algorithm is superior to other algorithms under different variable numbers and different sample sizes.We apply the MSEM algorithm to analyze the scores of 22 courses of students in a certain grade of statistics major in a university,and then explore the internal relationship between each course.