[Paper Review] Towards Characterizing Markov Equivalence Classes for Directed Acyclic Graphs with Latent Variables

Ayesha R. Ali, Thomas S. Richardson+2 more|arXiv (Cornell University)|Jul 4, 2012

Bayesian Modeling and Causal Inference7 references6 citations

TL;DR

This paper introduces a sound and complete set of orientation rules to construct the Markov equivalence class representative for ancestral graphs, which model directed acyclic graphs (DAGs) with latent and selection variables. The method ensures that when the equivalence class contains a DAG, the representative is the essential graph, enabling consistent causal structure learning under unobserved confounding.

ABSTRACT

It is well known that there may be many causal explanations that are consistent with a given set of data. Recent work has been done to represent the common aspects of these explanations into one representation. In this paper, we address what is less well known: how do the relationships common to every causal explanation among the observed variables of some DAG process change in the presence of latent variables? Ancestral graphs provide a class of graphs that can encode conditional independence relations that arise in DAG models with latent and selection variables. In this paper we present a set of orientation rules that construct the Markov equivalence class representative for ancestral graphs, given a member of the equivalence class. These rules are sound and complete. We also show that when the equivalence class includes a DAG, the equivalence class representative is the essential graph for the said DAG

Motivation & Objective

• To characterize Markov equivalence classes for DAGs with latent variables, which are not fully captured by standard Markov equivalence in standard DAGs.
• To extend the concept of essential graphs to ancestral graphs that include latent and selection variables.
• To develop orientation rules that can uniquely identify the equivalence class representative for ancestral graphs.
• To ensure that when the equivalence class contains a DAG, the representative is the essential graph of that DAG.

Proposed method

• The authors define a set of orientation rules that operate on a given ancestral graph to determine the unique representative of its Markov equivalence class.
• The rules are based on conditional independence relations encoded in ancestral graphs, leveraging their ability to represent dependencies from latent and selection variables.
• The method uses the structure of the ancestral graph to infer the maximal edge orientations consistent with the independence model.
• The rules are proven to be sound and complete, meaning they always produce the correct equivalence class representative.
• The approach generalizes the concept of essential graphs from DAGs to ancestral graphs, preserving the key property that the representative captures all edge orientations common to all graphs in the equivalence class.
• The construction ensures that the representative is unique and fully determines the skeleton and v-structures common to all members of the equivalence class.

Experimental results

Research questions

• RQ1 How can Markov equivalence classes be characterized for DAG models when latent variables are present?
• RQ2 What are the necessary and sufficient conditions for two ancestral graphs to be Markov equivalent?
• RQ3 Can a unique representative be constructed for each Markov equivalence class of ancestral graphs?
• RQ4 Under what conditions does the equivalence class representative coincide with the essential graph of a DAG in the class?
• RQ5 How do the orientation rules for ancestral graphs differ from those used in standard DAGs?

Key findings

• The proposed orientation rules are both sound and complete for constructing the Markov equivalence class representative of an ancestral graph.
• When the equivalence class contains a DAG, the representative produced is the essential graph of that DAG, ensuring consistency with established results in the literature.
• The method successfully extends the concept of essential graphs to ancestral graphs, which model latent and selection variables.
• The representative graph uniquely encodes all edge orientations and v-structures common to all graphs in the equivalence class.
• The approach provides a systematic way to learn causal structures from observational data with unobserved confounders.
• The results demonstrate that ancestral graphs can serve as a unified framework for representing Markov equivalence classes in the presence of latent variables.