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Selecting high-dimensional mixed graphical models using minimal AIC or BIC forests.

Edwards D, de Abreu GC, Labouriau R - BMC Bioinformatics (2010)

Bottom Line: In the genetics of gene expression context the method identifies a network approximating the joint distribution of the DNA markers and the gene expression levels.Trees and forests are unrealistically simple models for biological systems, but can provide useful insights.Uses include the following: identification of distinct connected components, which can be analysed separately (dimension reduction); identification of neighbourhoods for more detailed analyses; as initial models for search algorithms with a larger search space, for example decomposable models or Bayesian networks; and identification of interesting features, such as hub nodes.

View Article: PubMed Central - HTML - PubMed

Affiliation: Institute of Genetics and Biotechnology, Faculty of Agricultural Sciences, Aarhus University, Aarhus, Denmark. David.Edwards@agrsci.dk

ABSTRACT

Background: Chow and Liu showed that the maximum likelihood tree for multivariate discrete distributions may be found using a maximum weight spanning tree algorithm, for example Kruskal's algorithm. The efficiency of the algorithm makes it tractable for high-dimensional problems.

Results: We extend Chow and Liu's approach in two ways: first, to find the forest optimizing a penalized likelihood criterion, for example AIC or BIC, and second, to handle data with both discrete and Gaussian variables. We apply the approach to three datasets: two from gene expression studies and the third from a genetics of gene expression study. The minimal BIC forest supplements a conventional analysis of differential expression by providing a tentative network for the differentially expressed genes. In the genetics of gene expression context the method identifies a network approximating the joint distribution of the DNA markers and the gene expression levels.

Conclusions: The approach is generally useful as a preliminary step towards understanding the overall dependence structure of high-dimensional discrete and/or continuous data. Trees and forests are unrealistically simple models for biological systems, but can provide useful insights. Uses include the following: identification of distinct connected components, which can be analysed separately (dimension reduction); identification of neighbourhoods for more detailed analyses; as initial models for search algorithms with a larger search space, for example decomposable models or Bayesian networks; and identification of interesting features, such as hub nodes.

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Markov equivalence. The first three graphs are DAGs, the fourth is undirected. All four graphs represent the same conditional independence relation: that Xa and Xc are conditionally independent given Xb. They are called Markov equivalent.
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Figure 3: Markov equivalence. The first three graphs are DAGs, the fourth is undirected. All four graphs represent the same conditional independence relation: that Xa and Xc are conditionally independent given Xb. They are called Markov equivalent.

Mentions: We describe here a perspective on the forbidden path restriction that gives useful insight. Graphical models encode sets of conditional independence relations, and if two graphical models encode the same set of conditional independence relations they are termed Markov equivalent [25,26]. For example, each graph in Figure 3 represents the conditional independence of Xa and Xc given Xb. Sample data from the joint distribution of Xa, Xb and Xc supply information on which conditional independence relations hold and which do not, but cannot distinguish between the four graphs. To do this would require intervention in the system, for example by perturbing Xa to see whether the distribution of Xb is altered. For this reason algorithms to identify Bayesian networks from sample data [27,28] can only do this up to Markov equivalence.


Selecting high-dimensional mixed graphical models using minimal AIC or BIC forests.

Edwards D, de Abreu GC, Labouriau R - BMC Bioinformatics (2010)

Markov equivalence. The first three graphs are DAGs, the fourth is undirected. All four graphs represent the same conditional independence relation: that Xa and Xc are conditionally independent given Xb. They are called Markov equivalent.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2823705&req=5

Figure 3: Markov equivalence. The first three graphs are DAGs, the fourth is undirected. All four graphs represent the same conditional independence relation: that Xa and Xc are conditionally independent given Xb. They are called Markov equivalent.
Mentions: We describe here a perspective on the forbidden path restriction that gives useful insight. Graphical models encode sets of conditional independence relations, and if two graphical models encode the same set of conditional independence relations they are termed Markov equivalent [25,26]. For example, each graph in Figure 3 represents the conditional independence of Xa and Xc given Xb. Sample data from the joint distribution of Xa, Xb and Xc supply information on which conditional independence relations hold and which do not, but cannot distinguish between the four graphs. To do this would require intervention in the system, for example by perturbing Xa to see whether the distribution of Xb is altered. For this reason algorithms to identify Bayesian networks from sample data [27,28] can only do this up to Markov equivalence.

Bottom Line: In the genetics of gene expression context the method identifies a network approximating the joint distribution of the DNA markers and the gene expression levels.Trees and forests are unrealistically simple models for biological systems, but can provide useful insights.Uses include the following: identification of distinct connected components, which can be analysed separately (dimension reduction); identification of neighbourhoods for more detailed analyses; as initial models for search algorithms with a larger search space, for example decomposable models or Bayesian networks; and identification of interesting features, such as hub nodes.

View Article: PubMed Central - HTML - PubMed

Affiliation: Institute of Genetics and Biotechnology, Faculty of Agricultural Sciences, Aarhus University, Aarhus, Denmark. David.Edwards@agrsci.dk

ABSTRACT

Background: Chow and Liu showed that the maximum likelihood tree for multivariate discrete distributions may be found using a maximum weight spanning tree algorithm, for example Kruskal's algorithm. The efficiency of the algorithm makes it tractable for high-dimensional problems.

Results: We extend Chow and Liu's approach in two ways: first, to find the forest optimizing a penalized likelihood criterion, for example AIC or BIC, and second, to handle data with both discrete and Gaussian variables. We apply the approach to three datasets: two from gene expression studies and the third from a genetics of gene expression study. The minimal BIC forest supplements a conventional analysis of differential expression by providing a tentative network for the differentially expressed genes. In the genetics of gene expression context the method identifies a network approximating the joint distribution of the DNA markers and the gene expression levels.

Conclusions: The approach is generally useful as a preliminary step towards understanding the overall dependence structure of high-dimensional discrete and/or continuous data. Trees and forests are unrealistically simple models for biological systems, but can provide useful insights. Uses include the following: identification of distinct connected components, which can be analysed separately (dimension reduction); identification of neighbourhoods for more detailed analyses; as initial models for search algorithms with a larger search space, for example decomposable models or Bayesian networks; and identification of interesting features, such as hub nodes.

Show MeSH
Related in: MedlinePlus