[Paper Review] An N Time-Slice Dynamic Chain Event Graph

Rodrigo A. Collazo, Jim Q. Smith|arXiv (Cornell University)|Aug 17, 2018

Bayesian Modeling and Causal Inference31 references4 citations

TL;DR

This paper introduces the N Time-Slice Dynamic Chain Event Graph (N T-DCEG), a new subclass of Dynamic Chain Event Graphs designed to model time-homogeneous Markov processes with context-specific conditional independences. By leveraging a finite set of objects connected through an infinite tree structure, the N T-DCEG enables compact, interpretable, and causally interpretable modeling of complex multivariate dynamic processes, with a formal link established to Dynamic Bayesian Networks (DBNs) as a special case.

ABSTRACT

The Dynamic Chain Event Graph (DCEG) is able to depict many classes of discrete random processes exhibiting asymmetries in their developments and context-specific conditional probabilities structures. However, paradoxically, this very generality has so far frustrated its wide application. So in this paper we develop an object-oriented method to fully analyse a particularly useful and feasibly implementable new subclass of these graphical models called the N Time-Slice DCEG (NT-DCEG). After demonstrating a close relationship between an NT-DCEG and a specific class of Markov processes, we discuss how graphical modellers can exploit this connection to gain a deep understanding of their processes. We also show how to read from the topology of this graph context-specific independence statements that can then be checked by domain experts. Our methods are illustrated throughout using examples of dynamic multivariate processes describing inmate radicalisation in a prison.

Motivation & Objective

• To resolve topological ambiguities in the original Dynamic Chain Event Graph (DCEG) framework that hindered rigorous modeling of dynamic processes.
• To develop a new, practical subclass of DCEGs—called the N Time-Slice DCEG (N T-DCEG)—specifically tailored for modeling time-homogeneous Markov processes.
• To provide a methodology for constructing and interpreting N T-DCEGs using finite components networked through an infinite tree structure, enabling scalable modeling of infinite processes.
• To establish a formal connection between N T-DCEGs and Dynamic Bayesian Networks (DBNs), showing that DBNs are a special case of this new model class.
• To enable domain experts to read context-specific independence statements directly from the graph topology, facilitating model validation and causal interpretation.

Proposed method

• Revises the foundational definition of DCEGs to eliminate topological ambiguities present in earlier formulations.
• Introduces the N T-DCEG model class based on a Time-Ordered Graph (TOG) structure defined by a finite event tree $\mathcal{T}_{-1}$ and a repeating structure $\mathcal{T}$, enabling periodic, time-homogeneous modeling.
• Uses a staged tree representation $\mathcal{ST}$ with colored edges to encode conditional probabilities, which are then collapsed into a finite, cyclic DCEG graph via isomorphism reduction.
• Employs graphical and probabilistic isomorphism between subtrees to ensure time-homogeneity and structural consistency across time slices.
• Applies a mapping function $f_t$ to track positions across time slices, ensuring that identical process behaviors are represented by equivalent graph positions.
• Leverages the concept of 'positions' in the CEG to identify and verify context-specific conditional independences through topological equivalence.

Experimental results

Research questions

• RQ1 How can the topological ambiguities in the original DCEG framework be resolved to support rigorous modeling of dynamic processes?
• RQ2 What is the structure and formal definition of a new, practical subclass of DCEGs—specifically the N T-DCEG—that supports time-homogeneous Markov processes?
• RQ3 How can a finite graphical model represent an infinite dynamic process while preserving context-specific conditional independences and time-homogeneity?
• RQ4 In what way is the class of Dynamic Bayesian Networks (DBNs) a special case of the proposed N T-DCEG model?
• RQ5 How can domain experts extract and validate context-specific independence statements directly from the topology of the N T-DCEG?

Key findings

• The N T-DCEG model provides a finite, cyclic graphical representation of an infinite dynamic process by exploiting periodicity and time-homogeneity, enabling scalable modeling.
• The model establishes a formal isomorphism between the staged subtrees of situations that are in the same position, ensuring consistent probabilistic behavior across time slices.
• The paper proves that if two situations in a CEG $\mathbb{C}_T$ are in the same position, then the processes unfolding from them are identical for all $t \geq N-1$, under time-homogeneity and TOG structure.
• The class of Dynamic Bayesian Networks (DBNs) is formally shown to be a special case of the N T-DCEG, validating its generality and compatibility with established models.
• Context-specific conditional independences can be read directly from the graph topology of the N T-DCEG, allowing for explicit, graphical validation by domain experts.
• The model supports causal interpretation and learning, with the ability to refine DBNs using more interpretable, context-specific models without sacrificing interpretability.