[Paper Review] Asymmetric separation for local independence graphs

Vanessa Didelez|Explore Bristol Research|Jun 27, 2012

Bayesian Modeling and Causal Inference18 references8 citations

TL;DR

This paper introduces delta-separation, a novel asymmetric graph separation criterion for local independence graphs that model continuous-time stochastic processes with directional, time-asymmetric dependencies. By extending semi-graphoid axioms to asymmetric contexts, the framework enables rigorous probabilistic inference from directed, possibly cyclic graphs, offering a formal basis for interpreting causal-like relationships in continuous-time stochastic systems beyond standard conditional independence.

ABSTRACT

Directed possibly cyclic graphs have been proposed by Didelez (2000) and Nodelmann et al. (2002) in order to represent the dynamic dependencies among stochastic processes. These dependencies are based on a generalization of Granger-causality to continuous time, first developed by Schweder (1970) for Markov processes, who called them local dependencies. They deserve special attention as they are asymmetric unlike stochastic (in)dependence. In this paper we focus on their graphical representation and develop a suitable, i.e. asymmetric notion of separation, called delta-separation. The properties of this graph separation as well as of local independence are investigated in detail within a framework of asymmetric (semi)graphoids allowing a deeper insight into what information can be read off these graphs.

Motivation & Objective

• To formalize a notion of separation suitable for directed, possibly cyclic graphs representing local dependencies in continuous-time stochastic processes.
• To address the asymmetry inherent in local independence, which differs fundamentally from symmetric stochastic (in)dependence.
• To develop a graphical separation criterion—delta-separation—that captures time-asymmetric dependencies in dynamic systems.
• To establish a theoretical foundation using asymmetric (semi)graphoids to analyze and interpret local independence graphs.
• To enable reliable probabilistic reasoning and conditional independence queries in systems governed by continuous-time dynamic dependencies.

Proposed method

• Proposes delta-separation as a graph separation criterion tailored for directed, possibly cyclic graphs representing local independence.
• Adapts the axioms of semi-graphoids to an asymmetric setting to model time-ordered dependencies.
• Defines local independence as a generalization of Granger causality in continuous time, based on Schweder's local dependence concept.
• Introduces a framework of asymmetric (semi)graphoids to formalize the properties of delta-separation and local independence.
• Uses the graphical structure to derive conditional independence statements that reflect the asymmetric nature of dynamic dependencies.
• Applies the framework to analyze the structure and interpretability of local independence graphs in continuous-time stochastic processes.

Experimental results

Research questions

• RQ1 How can a graph separation criterion be defined for directed, possibly cyclic graphs that represent local dependencies in continuous-time stochastic processes?
• RQ2 What are the formal properties of separation in asymmetric graphical models where dependencies are inherently directional and time-ordered?
• RQ3 How does delta-separation relate to standard conditional independence and what axioms does it satisfy?
• RQ4 Can an asymmetric analog of the semi-graphoid axioms be formulated to support probabilistic reasoning in dynamic systems?
• RQ5 What information can be reliably read off a local independence graph using the proposed separation criterion?

Key findings

• Delta-separation is introduced as a formal, asymmetric graph separation criterion suitable for local independence graphs with directed, possibly cyclic structures.
• The framework of asymmetric (semi)graphoids is established, providing a sound foundation for reasoning about local independence in continuous time.
• Local independence is shown to be inherently asymmetric, differing fundamentally from symmetric stochastic (in)dependence, and thus requiring specialized separation criteria.
• The paper demonstrates that delta-separation satisfies key properties analogous to those of standard graph separation, adapted to the asymmetric context.
• The approach enables the interpretation of conditional independence statements in dynamic systems where time-ordered dependencies dominate over symmetric associations.
• The results provide a rigorous graphical and probabilistic foundation for modeling and inferring causal-like relationships in continuous-time stochastic processes.