A Study of Chain Graph Interpretations

Abstract

Probabilistic graphical models are today one of the most well used architectures for modelling and reasoning about knowledge with uncertainty. The most widely used subclass of these models is Bayesian networks that has found a wide range of applications both in industry and research. Bayesian networks do however have a major limitation which is that only asymmetric relationships, namely cause and effect relationships, can be modelled between its variables. A class of probabilistic graphical models that has tried to solve this shortcoming is chain graphs. It is achieved by including two types of edges in the models, representing both symmetric and asymmetric relationships between the connected variables. This allows for a wider range of independence models to be modelled. Depending on how the second edge is interpreted this has also given rise to different chain graph interpretations.Although chain graphs were first presented in the late eighties the field has been relatively dormant and most research has been focused on Bayesian networks. This was until recently when chain graphs got renewed interest. The research on chain graphs has thereafter extended many of the ideas from Bayesian networks and in this thesis we study what this new surge of research has been focused on and what results have been achieved. Moreover we do also discuss what areas that we think are most important to focus on in further research.

On the Properties of MVR Chain Graphs

“…The following result is often mentioned in the literature (Wermuth and Sadeghi, 2012;Peña, 2015;Sadeghi and Lauritzen, 2014;Sonntag, 2014), but we know of no published proof.…”

“…PC like algorithm for MVR CGs in (Sonntag, 2014;Sonntag and Peña, 2012), Decompositionbased algorithm for MVR CGs in (Javidian and Valtorta, 2019).…”

“…where p(xτ |x pa(τ ) ) = Z −1 (x pa(τ ) ) c∈C φc(xc), where C are the complete sets in the moral graph (τ ∪ pa(τ )) m . IC like algorithm in (Studený, 1997), LCD algorithm in (Ma et al, 2008), CKES algorithm in Sonntag, 2014) AMP CGs (Andersson et al, 1996) , 1998).…”

“…In other words, in an MVR CG any bidirected edge X ↔ Y can be replaced by X ← H → Y to obtain a Bayesian network representing the same independence model over the original variables, i.e. excluding the new variables H. These variables are called hidden, or latent, and have been marginalized away in the CG model (Sonntag, 2014). This causal interpretation of bidirected edges in MVR CGs along with the discussion preceding Theorem 7 provides strong motivation for the importance of MVR CGs.…”

Information Theoretic Causal Effect Quantification

“…From the point of view of the data generating process, a DAG describes a set of relationships, where each variable is generated from the set of its parents and external noise [ 46 , 47 ]. Chain graphs are an extension of DAGs in which undirected edges between nodes are allowed as long as no semi-directed cycles (cycles with directed and undirected edges) emerge [ 48 , 49 ]. The corresponding data generating process consists of two levels.…”