Structural Learning Of Directed Acyclic Graphs With Latent Variables By The Separation Tree

The directed acyclic graph(DAG)has a great advantage in the processing of high dimensional data,and it is widely applied in machine learning and artificial intelligence,etc.DAG is one of the important tools to find the causality between variables.Therefore it is particularly important to accurately find the causality between variables from the observed data in real life.In DAG,latent variables can be included to avoid the wrong discovery of causality.Spirtes(2000)introduced two algorithms that consider the latent variable – the CI algorithm and the FCI algorithm.Because the searching of the unshielded triple in the CI algorithm is too complicated and complex,the CI algorithm becomes infeasible when the number of variables is large.The FCI algorithm solved this problem by applying the PC algorithm,but the efficiency of the subsequent steps for finding the conditional independent sets in dealing with high dimensional variables is too low.Colombo and Maathuis(2012)proposed the improved FCI algorithm,i.e.,RFCI,which reduces the complexity of the algorithm,and the feasibility in sparse high dimensional case has been proved.In this paper,we will improve the RFCI algorithm based on the separation tree algorithm to improve the accuracy of the algorithm.First,we introduce the knowledge of FCI,RFCI,and the separation tree algorithm.The basis of the improvement and the concrete steps are given.Then we compare two algorithms and give numerical simulation study.After a comparative analysis,we find that in most cases,the improved algorithm is more accurate.At the same time,a corresponding explanation for the poor results of structural learning is given when the sample size is smaller and even the sample size is equal to the number of variables.Finally,we carry out simulation studies on real DAGs,then we explain the results.