Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

Abstract

Probabilistic graphical models are currently one of the most commonly used architectures for modelling and reasoning with uncertainty. The most widely used subclass of these models is directed acyclic graphs, also known as Bayesian networks, which are used in a wide range of applications both in research and industry. Directed acyclic graphs do, however, have a major limitation, which is that only asymmetric relationships, namely cause and effect relationships, can be modelled between their variables. A class of probabilistic graphical models that tries to address this shortcoming is chain graphs, which include two types of edges in the models representing both symmetric and asymmetric relationships between the variables. This allows for a wider range of independence models to be modelled and depending on how the second edge is interpreted, we also have different so-called chain graph interpretations.Although chain graphs were first introduced in the late eighties, most research on probabilistic graphical models naturally started in the least complex subclasses, such as directed acyclic graphs and undirected graphs. The field of chain graphs has therefore been relatively dormant. However, due to the maturity of the research field of probabilistic graphical models and the rise of more data-driven approaches to system modelling, chain graphs have recently received renewed interest in research. In this thesis we provide an introduction to chain graphs where we incorporate the progress made in the field. More specifically, we study the three chain graph interpretations that exist in research in terms of their separation criteria, their possible parametrizations and the intuition behind their edges. In addition to this we also compare the expressivity of the interpretations in terms of representable independence models as well as propose new structure learning algorithms to learn chain graph models from data. iii Populärvetenskaplig sammanfattningInom statistik, fysik och datavetenskap har modeller använts genom historien för att förstå och beskriva olika delar av världen. I de system man försökt beskriva har då de relevanta faktorerna ofta representerats som variabler och relationerna mellan dessa variabler har på olika sätt avspeglats i modellerna. Beroende på karakteristiken av systemet, och dess variabler, har olika typer av modeller utvecklats och använts men en vanlig del har varit att använda någon typ av grafisk illustration för att underlätta förståelsen av hur variablerna relaterar till varandra. I denna avhandling behandlar vi en typ av sådan modell kallad sannolikhetsbaserad grafisk modell (probabilistic graphical model) och mer specifikt en underklass av dessa kallade kedje-grafer (chain graphs). I en sannolikhetsbaserad grafisk modell representeras variablerna som noder och relationerna mellan variablerna som olika typer av bågar mellan dessa noder. Variablerna kan vara av olika natur, men oftastär de antingen diskreta, alltså att de kan vara i ett av flera tillstånd, eller kontinuerliga, d.v.s....

Chain graph reduction into power chain graphs

“…MVR CGs have more in common with AMP CGs, as each node depends only on its own parents according to both interpreta tions. Both interpretations can also be modelled with linear equations, being the main difference between MVR CGs and AMP CGs the modelling of noise (Sonntag, 2016). The bidirected edges in MVR CGs are used to model latent (or hidden) variables (Javidian & Valtorta, 2021), being related, then, to DAG1 represented in Figure 1.…”

Identifiability of AMP chain graph models

“…Chain graph models contain both directed and undirected edges and can be used to represent both association and causation in real-world applications [Sonntag, 2016]. The three following interpretations are the best known in the literature: LWF [Lauritzen and Wermuth, 1989b, Wermuth and Lauritzen, 1990, Frydenberg, 1990 which generalizes both Markov random fields and Bayesian networks; AMP [Andersson et al, 2001[Andersson et al, , 2006 which directly extends the DAG Markov property; and MVR Wermuth, 1993, 2014] which originates from viewing undirected edges as representing hidden common causes.…”

AMP Chain Graphs: Minimal Separators and Structure Learning Algorithms

“…In fact, a chain graph is a type of mixed graph, admitting both directed and undirected edges, which contain no partially directed cycles. So, CGs may contain two types of edges, the directed type that corresponds to the causal relationship in DAGs and a second type of edge representing a symmetric relationship (Sonntag, 2016). In particular, X 1 is a direct cause of X 2 only if X 1 → X 2 (i.e., X 1 is a parent of X 2 ), and X 1 is a (possibly indirect) cause of X 2 only if there is a directed path from X 1 to X 2 (i.e., X 1 is an ancestor of X 2 ).…”