Statistical Inference In High Dimensional Graphical Models

With the continuous development of data collection techniques,the dimensionality of the data we are dealing with is getting larger and larger.Thanks to the sparsity of estimation,the LASSO method is widely used in high-dimensional models.However,from a statistical point of view,it is more meaningful to give p values to the selected variables than to select them directly.Therefore,it is necessary to establish hypothesis testing in high dimensions.In recent years,the de-biased LASSO method has been implemented for statistical inference of linear models in high dimensions.Probabilistic graphical model is a widely used tool in many fields such as finance and biology.It can characterize the abstract probability structure among variables through graphs,so it is necessary to recover the structure of graphical models.We consider combining the de-biased LASSO method with probabilistic graphical models in high dimensions to construct statistical inference of the graph structure.First,we propose an efficient algorithm to compute the precision matrix which greatly blackuce the computational cost.Second,we consider quantile regression,which characterizes the conditional independence of variables by conditional quantile function,and is more robust than mean regression for non-Gaussian graphical models.Finally,we establish a holistic hypothesis test on edges,which is different from the previous methods of simply splicing the results of multiple quantile points,and thus can apply the commonly used methods to control the false discovery rate of graph structure.