[Paper Review] A Discovery Algorithm for Directed Cyclic Graphs

Thomas Richardson|arXiv (Cornell University)|Feb 13, 2013

Bayesian Modeling and Causal Inference5 references123 citations

TL;DR

This paper presents a polynomial-time discovery algorithm for learning causal structures from observational data in directed cyclic graphs (DCGs), under the assumption of linear non-recursive structural equation models. It identifies the existence or non-existence of causal pathways between variables using conditional independence relations, and is provably correct in the large sample limit under standard assumptions, enabling reliable causal inference in cyclic models commonly found in social sciences.

ABSTRACT

Directed acyclic graphs have been used fruitfully to represent causal strucures (Pearl 1988). However, in the social sciences and elsewhere models are often used which correspond both causally and statistically to directed graphs with directed cycles (Spirtes 1995). Pearl (1993) discussed predicting the effects of intervention in models of this kind, so-called linear non-recursive structural equation models. This raises the question of whether it is possible to make inferences about causal structure with cycles, form sample data. In particular do there exist general, informative, feasible and reliable precedures for inferring causal structure from conditional independence relations among variables in a sample generated by an unknown causal structure? In this paper I present a discovery algorithm that is correct in the large sample limit, given commonly (but often implicitly) made plausible assumptions, and which provides information about the existence or non-existence of causal pathways from one variable to another. The algorithm is polynomial on sparse graphs.

Motivation & Objective

• To address the challenge of inferring causal structure from sample data in models with directed cycles, which are common in social sciences but underexplored in causal discovery.
• To develop a feasible and reliable procedure for identifying causal pathways between variables in directed cyclic graphs using conditional independence relations.
• To extend causal discovery methods beyond acyclic graphs to include cyclic structures, particularly linear non-recursive structural equation models.
• To ensure the algorithm is computationally efficient, with polynomial time complexity on sparse graphs.
• To provide a theoretically grounded method that is correct in the large sample limit under standard assumptions.

Proposed method

• The algorithm uses conditional independence relations among observed variables to infer the structure of a directed cyclic graph.
• It applies a constraint-based approach, similar to those used in acyclic causal discovery, but adapted to handle cycles.
• The method identifies whether a causal pathway exists from one variable to another by analyzing the d-separation properties in the presence of cycles.
• It relies on the assumption of linear non-recursive structural equation models, which allow feedback loops and are common in social science applications.
• The algorithm operates in polynomial time on sparse graphs, making it computationally feasible for large-scale data.
• It leverages the structure of the covariance matrix and conditional independence constraints to infer the existence or absence of directed paths.

Experimental results

Research questions

• RQ1 Can we reliably infer causal structure from observational data in directed cyclic graphs?
• RQ2 Is there a feasible and efficient algorithm to determine the existence of causal pathways between variables in cyclic models?
• RQ3 Can conditional independence relations be used to discover causal structure in linear non-recursive structural equation models with cycles?
• RQ4 What assumptions are necessary for a discovery algorithm to be correct in the large sample limit for cyclic causal models?
• RQ5 How can we extend traditional causal discovery methods, designed for acyclic graphs, to handle cyclic dependencies?

Key findings

• The proposed algorithm is correct in the large sample limit under standard assumptions, such as faithfulness and Gaussianity, for linear non-recursive structural equation models.
• The algorithm can determine whether a causal pathway exists from one variable to another, even in the presence of cycles.
• It operates in polynomial time on sparse graphs, making it computationally efficient for practical use.
• The method successfully identifies the structure of directed cyclic graphs using only conditional independence relations from observational data.
• The algorithm provides a reliable framework for causal inference in models where feedback loops are plausible, such as in social and economic systems.
• The approach extends the applicability of constraint-based causal discovery to cyclic structures, previously limited to acyclic models.