Chain Graphs and Gene Networks

Sonntag, Dag; Peña, José M.

doi:10.1007/978-3-319-28007-3_10

Cited by 5 publications

(8 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence the linear equation of a node v j in a CG is v j = β j ⋅pa G (K i )+ j where K i is the component to which v j belongs [31]. Here the β j -vector represents the influence of the parents of the component over the nodes in the component while j represents the noise, or influence, between the nodes in the same component.…”

Section: The Lwf Interpretationmentioning

confidence: 99%

“…The weight of each path is also the product of the weights of its edges [31]. Meanwhile, the noise j in an LWF CG is determined by the associated inverse covariance matrix of that component such that an entry in the inverse covariance matrix for two nodes v j and v m can be non-zero iff there exists an undirected edge v j − v m in G. For example, from Figure 4.1a it follows that the influence from node v 2 onto node v 4 is direct since only one path exists between them.…”

Section: The Lwf Interpretationmentioning

confidence: 99%

“…For a node v j , in an AMP CG G, the associated linear equation is v j = β j ⋅ pa G (v j ) + j and hence the node depends only on its parents and not on the parents of the whole component, as it does in the case of LWF CGs [31]. The noise j is then controlled by the inverse covariance matrix of that component where the corresponding entry in the inverse covariance matrix for two nodes v k and v l can be non-zero iff there is an undirected edge v k − v l in G. Intuitively, a small set of nodes works as an interface between other nodes in the component and its parents.…”

Section: The Amp Interpretationmentioning

confidence: 99%

“…Therefore, the associated linear equation for a node v j can be written as v j = β j ⋅ pa(v j ) + j , where j is dependent on the other nodes in the same component. Unlike AMP CGs, MVR CGs can contain non-zero values in the corresponding covariance matrix (not the inverse covariance matrix as for AMP CGs) only for nodes that are spouses [31]. The intuitive meaning behind MVR CGs is therefore very close to that of AMP CGs, differing only in the noise modelling.…”

Section: The Mvr Interpretationmentioning

confidence: 99%

See 3 more Smart Citations

Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

Sonntag¹

2016

Linköping Studies in Science and Technology. Dissertations

Self Cite

View full text Add to dashboard Cite

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....

show abstract

Section: The Lwf Interpretationmentioning

confidence: 99%

Section: The Lwf Interpretationmentioning

confidence: 99%

Section: The Amp Interpretationmentioning

confidence: 99%

Section: The Mvr Interpretationmentioning

confidence: 99%

See 2 more Smart Citations

Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

Sonntag¹

2016

Linköping Studies in Science and Technology. Dissertations

Self Cite

View full text Add to dashboard Cite

show abstract

“…Note that for AMP CGs the parents only affect the direct children in the chain component, not all the nodes in the chain component as in the case of LWF CGs. An example in medicine (Sonntag & Peña, 2015b) when such a model might be appropriate is when we are modelling pain levels on different areas on the body of a patient. The pain levels can then be seen as correlated "geographically" over the body, and hence be modelled as a Markov network.…”

Section: Introductionmentioning

confidence: 99%

AMP Chain Graphs: Minimal Separators and Structure Learning Algorithms

Javidian

Valtorta²,

Jamshidi³

2020

jair

View full text Add to dashboard Cite

This paper deals with chain graphs (CGs) under the Andersson–Madigan–Perlman (AMP) interpretation. We address the problem of finding a minimal separator in an AMP CG, namely, finding a set Z of nodes that separates a given non-adjacent pair of nodes such that no proper subset of Z separates that pair. We analyze several versions of this problem and offer polynomial time algorithms for each. These include finding a minimal separator from a restricted set of nodes, finding a minimal separator for two given disjoint sets, and testing whether a given separator is minimal. To address the problem of learning the structure of AMP CGs from data, 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. We propose several modifications of the PC-like algorithm that remove part or all of this order-dependence. We also extend the decomposition-based approach for learning Bayesian networks (BNs) to learn AMP CGs, which include BNs as a special case, under the faithfulness assumption. We prove the correctness of our extension using the minimal separator results. Using standard benchmarks and synthetically generated models and data in our experiments demonstrate the competitive performance of our decomposition-based method, called LCD-AMP, in comparison with the (modified versions of) PC-like algorithm. The LCD-AMP algorithm usually outperforms the PC-like algorithm, and our modifications of the PC-like algorithm learn structures that are more similar to the underlying ground truth graphs than the original PC-like algorithm, especially in high-dimensional settings. In particular, we empirically show that the results of both algorithms are more accurate and stabler when the sample size is reasonably large and the underlying graph is sparse

show abstract

On a wider class of prior distributions for graphical models

2023

View full text Add to dashboard Cite

Gaussian graphical models are useful tools for conditional independence structure inference of multivariate random variables. Unfortunately, Bayesian inference of latent graph structures is challenging due to exponential growth of $\mathcal{G}_n$ , the set of all graphs in n vertices. One approach that has been proposed to tackle this problem is to limit search to subsets of $\mathcal{G}_n$ . In this paper we study subsets that are vector subspaces with the cycle space $\mathcal{C}_n$ as the main example. We propose a novel prior on $\mathcal{C}_n$ based on linear combinations of cycle basis elements and present its theoretical properties. Using this prior, we implement a Markov chain Monte Carlo algorithm, and show that (i) posterior edge inclusion estimates computed with our technique are comparable to estimates from the standard technique despite searching a smaller graph space, and (ii) the vector space perspective enables straightforward implementation of MCMC algorithms.

show abstract

Chain Graphs and Gene Networks

Cited by 5 publications

References 0 publications

Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

Chain Graphs : Interpretations, Expressiveness and Learning Algorithms

AMP Chain Graphs: Minimal Separators and Structure Learning Algorithms

On a wider class of prior distributions for graphical models

Contact Info

Product

Resources

About