Approximate Counting of Graphical Models via MCMC Revisited

Sonntag, Dag; Peña, José M.; Gómez-Olmedo, Manuel

doi:10.1002/int.21704

Cited by 14 publications

(19 citation statements)

References 17 publications

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“…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.…”

Section: Expressivenessmentioning

confidence: 99%

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: Expressivenessmentioning

confidence: 99%

Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

Sonntag¹

2016

Linköping Studies in Science and Technology. Dissertations

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“…For AMP CGs we have two different unique representations, the largest deflagged graphs [17] and the AMP essential graphs [3] while for MVR CGs we have the essential MVR CGs [19]. All of these have been proven to be unique for the interpretation and Markov equivalence class they represent [3,9,17,19]. Definition 1.…”

Section: Unique Representationsmentioning

confidence: 99%

“…I(G * ) = I(H) and A←B is in H. Definition 4. Essential MVR CG [19] A graph G * is said to be the essential MVR CG of a MVR CG G if it has the same skeleton as G and contains all and only the arrowheads common to every MVR CG in the Markov equivalence class of G.…”

Section: Unique Representationsmentioning

confidence: 99%

“…Instead these graphs can contain three types of edges, undirected, directed and bidirected, and although the separation criterion defined for these graphs is close to that of MVR CGs [19], this is of course unfortunate. It can however be shown that no unique representation that is a MVR CG can exist for a Markov equivalence class of MVR CGs unless we assume some ordering of the nodes.…”

Section: Unique Representationsmentioning

confidence: 99%

“…This in turn means that we must go outside the class of MVR CGs to have a unique graph representing this Markov equivalence class. To be able to identify if a graph is of a certain unique representation all the representations have been characterized [3,17,19,21]. In addition to this there exist algorithms that, given a CG in a certain interpretation, outputs the unique representation of that interpretation.…”

Section: Unique Representationsmentioning

confidence: 99%

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Chain graph interpretations and their relations revisited

Sonntag

Peña

2015

International Journal of Approximate Reasoning

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In this paper we study how different theoretical concepts of Bayesian networks have been extended to chain graphs. Today there exist mainly three different interpretations of chain graphs in the literature. These are the Lauritzen-Wermuth-Frydenberg, the Andersson-Madigan-Perlman and the multivariate regression interpretations. The different chain graph interpretations have been studied independently and over time different theoretical concepts have been extended from Bayesian networks to also work for the different chain graph interpretations. This has however led to confusion regarding what concepts exist for what interpretation.In this article we do therefore study some of these concepts and how they have been extended to chain graphs as well as what results have been achieved so far. More importantly we do also identify when the concepts have not been extended and contribute within these areas. Specifically we study the following theoretical concepts: Unique representations of independence models, the split and merging operators, the conditions for when an independence model representable by one chain graph interpretation can be represented by another chain graph interpretation and finally the extension of Meek's conjecture to chain graphs. With our new results we give a coherent overview of how each of these concepts is extended for each of the different chain graph interpretations.

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Order-Independent Structure Learning of Multivariate Regression Chain Graphs

Javidian

Valtorta

Jamshidi

2019

Lecture Notes in Computer Science

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This paper deals with multivariate regression chain graphs (MVR CGs), which were introduced by Cox and Wermuth [3,4] to represent linear causal models with correlated errors. We consider the PC-like algorithm for structure learning of MVR CGs, which is a constraint-based method proposed by Sonntag and Peña in [18]. We show that the PC-like algorithm is order-dependent, in the sense that the output can depend on the order in which the variables are given. This order-dependence is a minor issue in low-dimensional settings. However, it can be very pronounced in high-dimensional settings, where it can lead to highly variable results. We propose two modifications of the PC-like algorithm that remove part or all of this order-dependence. Simulations under a variety of settings demonstrate the competitive performance of our algorithms in comparison with the original PC-like algorithm in low-dimensional settings and improved performance in high-dimensional settings.

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Approximate Counting of Graphical Models via MCMC Revisited

Cited by 14 publications

References 17 publications

Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

Chain graph interpretations and their relations revisited

Order-Independent Structure Learning of Multivariate Regression Chain Graphs

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