On expressiveness of the chain graph interpretations

Abstract

In this article we study the expressiveness of the different chain graph interpretations. Chain graphs is a class of probabilistic graphical models that can contain two types of edges, representing different types of relationships between the variables in question. Chain graphs is also a superclass of directed acyclic graphs, i.e. Bayesian networks, and can thereby represent systems more accurately than this less expressive class of models. Today there do however exist several different ways of interpreting chain graphs and what conditional independences they encode, giving rise to different so called chain graph interpretations. Previous research has approximated the number of representable independence models for the Lauritzen-Wermuth-Frydenberg and the multivariate regression chain graph interpretations using an MCMC based approach. In this article we use a similar approach to approximate the number of models representable by the latest chain graph interpretation in research, the Andersson-MadiganPerlman interpretation. Moreover we summarize and compare the different chain graph interpretations with each other. Our results confirm previous results that directed acyclic graphs only can represent a small fraction of the models representable by chain graphs, even for a low number of nodes. The results also show that the Andersson-Madigan-Perlman and multivariate regression interpretations can represent about the same amount of models and twice the amount of models compared to the Lauritzen-Wermuth-Frydenberg interpretation. However, at the same time almost all models representable by the latter interpretation can only be represented by that interpretation while the former two have a large intersection in terms of representable models.

Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

“…If the operators then fulfill the aperiodicity, irreducibility and reversibility criteria, and k transitions are performed before sampling a state, each state has equal probability of being sampled when k → ∞. This approach has been successfully applied to all three CG interpretations [18,32,33] and the results presented in this chapter are based on it. In Section 5.1 we first discuss how good the approximations are and 23 CHAPTER 5.…”

“…CGs contain two types of edges, the directed edge that corresponds to the causal relationship in DAGs and a second type of edge representing a symmetric relationship. This allows CGs to correctly model a much larger set of systems than DAGs [32] in a compact way that is, at the same time, interpretable, efficient to perform inference on and for which efficient learning algorithms exist. CGs were introduced in the late eighties, but there has recently been renewed interest in them as researchers have begun modelling more advanced systems, such as gene networks [2] or financial networks [6].…”

Bayesian inference for latent chain graphs

“…The most common interpretations are the Lauritzen-Wermuth-Frydenberg (LWF) interpretation, the Andersson-Madigan-Perlman interpretation and the multivariate regression (MVR) interpretation. Each interpretation has its own way of determining conditional independences in a CG and each interpretation subsumes another in terms of representable independence models; see [30]. Moreover, [20] discuss causal interpretation of chain graphs.…”

Alternative Markov and Causal Properties for Acyclic Directed Mixed Graphs

“…They have been extensively studied as a formalism to represent probabilistic independence models, because they can model symmetric and asymmetric relationships between random variables. Moreover, they are much more expressive than directed acyclic graphs (DAGs) and undirected graphs (UGs) (Sonntag and Peña, 2016). There are three different interpretations of CGs as independence models: The Lauritzen-Wermuth-Frydenberg (LWF) interpretation (Lauritzen, 1996), the multivariate regression (MVR) interpretation (Cox and Wermuth, 1996), and the Andersson-Madigan-Perlman (AMP) interpretation (Andersson et al, 2001).…”