Enumeration of labelled chain graphs and labelled essential directed acyclic graphs

Steinsky, Bertran

doi:10.1016/s0012-365x(02)00838-5

Cited by 28 publications

(30 citation statements)

References 4 publications

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“…These numbers follow directly from the number of 26 CGs per CG model, shown in Table 5.3, and the equations for calculating the number of CG structures for a given number of nodes defined by Steinsky [35]. We can here see that the AMP and MVR CG interpretations can represent approximately the same number of independence models, while the LWF CGs only can represent approximately 65% of this number since LWF CG Markov equivalence classes are larger on average.…”

Section: Ratios Of Cg Models Representable As Subclassesmentioning

confidence: 89%

“…where #CGmodels represents the number of CG models of a certain interpretation and so on. The ratio #CGs #DAGs can then be found using the iterative equations by Robinsson [25] and Steinsky [35] while #DAGs #DAGmodels has been approximated in previous studies for DAG models [18]. Finally, we can also get the ratio #DAGmodels #CGmodels from [18].…”

Section: Ratios Of Cg Models Representable As Subclassesmentioning

confidence: 99%

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Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

Sonntag¹

2016

Linköping Studies in Science and Technology. Dissertations

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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....

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Section: Ratios Of Cg Models Representable As Subclassesmentioning

confidence: 89%

Section: Ratios Of Cg Models Representable As Subclassesmentioning

confidence: 99%

Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

Sonntag¹

2016

Linköping Studies in Science and Technology. Dissertations

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“…Because the number of possible graphs increases super-exponentially as the number of nodes increases (Steinsky, 2003), it is impractical to enumerate all the possibilities and sum them up. For certain prior distributions, given the order of nodes, Friedman and Koller (Friedman & Koller, 2003) in 2003 derived a formula that can calculate the exact posterior probability of a structure feature with the computational complexity bounded by O(N D in +1 ), where N is the number of nodes and D in is the upper bound of node in-degrees.…”

Section: Structure-learning Methods With Error Controlledmentioning

confidence: 99%

“…Because modern exploratory research usually involves investigation of a large number of candidate models, scalability has become a highly desirable feature for group analysis. For example, if the connectivity between ten brain regions is studied with Bayesian networks, then a group-analysis method should be able to handle the diversity of about 3.1 × 10 17 (Steinsky, 2003) possible network structures.…”

Section: Desirable Features Of Graph Analysis Methodsmentioning

confidence: 99%

Graphical Models of Functional MRI Data for Assessing Brain Connectivity

Li¹,

JaneWang²,

McKeown³

2012

Neuroimaging - Cognitive and Clinical Neuroscience

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“…This ratio can be calculated using the equation #CGs #CGmodels = #CGs #DAGs * #DAGs #DAGmodels * #DAGmodels #CGmodels (1) where #CGmodels represents the number of CG models of a certain interpretation and so on. The ratio #CGs #DAGs can then be found using the iterative equations by Robinsson [14] and Steinsky [18] while #DAGs #DAGmodels can be found in previous studies for DAG models [11]. Finally we can also get the ratio #DAGmodels #CGmodels from Figure 1: The ratios (displayed with a logarithmic scale) of the number of DAG models compared to the number of CG models for different number of nodes for the different CG interpretations.…”

Section: Ratios Of the Number Of Cgs Per Cg Model And Approximate Nummentioning

confidence: 99%

On expressiveness of the chain graph interpretations

Sonntag

Peña

2016

International Journal of Approximate Reasoning

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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.

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Enumeration of labelled chain graphs and labelled essential directed acyclic graphs

Cited by 28 publications

References 4 publications

Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

Graphical Models of Functional MRI Data for Assessing Brain Connectivity

On expressiveness of the chain graph interpretations

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