The Nature And Application Of Ancestor Diagram

Probabilistic graphical model combines graph theory and probability theory to study conditional independence,among which directed acyclic graph is the most widely studied.But when there are potential variables and selection bias in the system,directed acyclic graph can not define independence.In order to overcome the limitation of directed acyclic graph,this paper studies the properties of ancestral graph.When there are potential variables and selection bias,the corresponding graphical model should be marginalized and conditional.Unlike directed acyclic graph,ancestral graph is stable in this case,Ancestral graph provide a class of graph that can encode conditional independence relation that arise in DAG models with latent and selection variables.Different ancestral graphs may be Markov equivalent in the sense that they entail the same conditional independent relations.In this paper,based on the model of directed acyclic graphs,we give an algorithm for constructing maximal ancestral graphs,discuss the Markov properties of ancestral graphs,and present a set of orientation rules that construct the Markov equivalence class representative given a member of the equivalence class.Intervention on graphical model is an important method to learn causal structure.The intervention of directed acyclic graphs has been widely studied.In this paper,we use the relationship between ancestral graphs and directed acyclic graphs to extend the intervention property of directed acyclic graphs to ancestral graphs.Based on the observational Markov properties of ancestral graph,we show the change of graph structure if some variables were forced to take certain value,and study the interventional Markov properties of ancestral graph.