Inverse Covariance Matrix Estimation and Graphical Modelselection with the Applications

This thesis studied the problem of inverse covariance matrix estimation and the inference of graph structure for the graphical models in modern statistics, which consists of two parts. Firstly, in the case of the continuous data, we proposed a novel inverse covariance matrix estimation method for the Gaussian graphical model. Instead of jointly estimate the matrix via the maximum likelihood method, or select the neighborhood for each node with column-by-column regularized regression models, we estimate the multivariate regression model on the noise-level with probabilistic predictive models and convert the noise estimation to the parameter estimator of the inverse covariance matrix. Numerical experiments on synthetic data and real data-driven simulations are presented. In the second part, as a natural extension of the prediction approach for the continuous data in the Gaussian graphical models, we studied the structure learning of Ising models with binary data. By applying the idea of prediction approach and estimate the matrix on the noise level, it is shown that our method outperforms other advanced methods on the edge detection accuracy