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Chp17 - Copy - 17 Undirected Graphical Models 17.1...

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17 Undirected Graphical Models 17.1 Introduction A graph consists of a set of vertices (nodes), along with a set of edges join- ing some pairs of the vertices. In graphical models, each vertex represents a random variable, and the graph gives a visual way of understanding the joint distribution of the entire set of random variables. They can be use- ful for either unsupervised or supervised learning. In an undirected graph , the edges have no directional arrows. We restrict our discussion to undi- rected graphical models, also known as Markov random fields or Markov networks . In these graphs, the absence of an edge between two vertices has a special meaning: the corresponding random variables are conditionally independent, given the other variables. Figure 17.1 shows an example of a graphical model for a flow-cytometry dataset with p = 11 proteins measured on N = 7466 cells, from Sachs et al. (2003). Each vertex in the graph corresponds to the real-valued ex- pression level of a protein. The network structure was estimated assuming a multivariate Gaussian distribution, using the graphical lasso procedure discussed later in this chapter. Sparse graphs have a relatively small number of edges, and are convenient for interpretation. They are useful in a variety of domains, including ge- nomics and proteomics, where they provide rough models of cell pathways. Much work has been done in defining and understanding the structure of graphical models; see the Bibliographic Notes for references. © Springer Science+Business Media, LLC 2009 T. Hastie et al., The Elements of Statistical Learning, Second Edition, 625 DOI: 10.1007/b94608_17,
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626 17. Undirected Graphical Models Raf Mek Plcg PIP2 PIP3 Erk Akt PKA PKC P38 Jnk FIGURE 17.1. Example of a sparse undirected graph, estimated from a flow cytometry dataset, with p = 11 proteins measured on N = 7466 cells. The net- work structure was estimated using the graphical lasso procedure discussed in this chapter. As we will see, the edges in a graph are parametrized by values or po- tentials that encode the strength of the conditional dependence between the random variables at the corresponding vertices. The main challenges in working with graphical models are model selection (choosing the structure of the graph), estimation of the edge parameters from data, and compu- tation of marginal vertex probabilities and expectations, from their joint distribution. The last two tasks are sometimes called learning and inference in the computer science literature. We do not attempt a comprehensive treatment of this interesting area. Instead, we introduce some basic concepts, and then discuss a few sim- ple methods for estimation of the parameters and structure of undirected graphical models; methods that relate to the techniques already discussed in this book. The estimation approaches that we present for continuous and discrete-valued vertices are different, so we treat them separately. Sec- tions 17.3.1 and 17.3.2 may be of particular interest, as they describe new, regression-based procedures for estimating graphical models.
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