Several Problems On Gaussian Graphical Models Based On Bayesian Method

In recent years,the Gaussian graphical models have been widely used to characterize conditional independence relationships among variables in many socio-scientific domains including economics,genomics and social networks.The general Gaussian graphical model is mainly aimed at simple data,but in some specific applications,the data has complex structural information,such as grouping information,clustering information and time-varying information.Since the Bayesian methods can better utilize the prior information,we adopts the Bayesian method to learn the Gaussian graphical model with structured information.Compared with the traditional sampling method,we uses EM(expectation maximization)and variational EM method for the posterior inference,which has a faster convergence speed.The main contributions are as follows:(1)A novel Bayesian approach is proposed to learn Gaussian graph models incorporating grouping information.We first introduce a spike and slab prior NormalExponential-Gamma(NEG)structural prior for the diagonal elements in the precision matrix.Model can induce the global shrinkage and the group shrinkage.A deterministic expectation maximization method can be used for posterior inference.Second,we add a hierarchical prior distribution to learn the overlapping block structures.Then,a variational EM method is proposed for posterior inference.Simulation results show that the proposed method is able to estimate the sparse graphical structure with a smaller estimation bias than the existing alternative methods.Finally,we use two sets of data,stock prices and gene expression,to show the application of our method.(2)We proposed a new Bayesian method for clustering the high-dimensional data and learning sparse multiple graphical models simultaneously.Different from most previous multiple graphs learning methods which assume that the cluster information is known in advance,we impose a multi-distribution prior for the cluster labels.Firstly,a joint spike-and-slab graphical lasso prior is imposed for the precision matrices,which can induce a sparsity and homogeneity of the heterogeneous graphical models across all clusters adaptively.Additionally,by imposing a structural Markov random field prior,the proposed method can also cluster the network-linked data without the independence assumption of the samples.Then a fast expectation maximization algorithm is utilized for the posterior inference.The proposed model can get a significant improvement both in clustering error and graphical selection precision than the existing alternative methods.The simulations and real data analysis are shown to demonstrate the performance of our method.(3)A new Bayesian method is proposed to learn a time-varying Gaussian graph model.First,by introducing a mixture of spike and slab Gaussian priors and graph Laplacian priors,to obtain sparsity and adaptability.By the introduction of homogeneity indicator variables,the model is able to discover the abrupt points of structure in the graphical model.Then we use the expectation conditional maximization algorithm for the posterior inference.In the E step we use variational inference to approximate the expectation.In the CM step,the optimization problem can be transformed into a linear state space model and solved by Kalman filter method.Finally,the performance of the model is illustrated by experimental analysis and real data.