Pattern Recognition and Machine Learning

8.1. Bayesian Networks 361 Figure 8.1 A directed graphical model representing the joint probabil- ity distribution over three va ...

362 8. GRAPHICAL MODELS Figure 8.2 Example of a directed acyclic graph describing the joint distribution over variablesx 1 ,..., ...

8.1. Bayesian Networks 363 Figure 8.3 Directed graphical model representing the joint distribution (8.6) corresponding to the Ba ...

364 8. GRAPHICAL MODELS Figure 8.5 This shows the same model as in Figure 8.4 but with the deterministic parameters shown explic ...

8.1. Bayesian Networks 365 Figure 8.7 The polynomial regression model, corresponding to Figure 8.6, showing also a new input val ...

366 8. GRAPHICAL MODELS Figure 8.8 A graphical model representing the process by which images of objects are created, in which t ...

8.1. Bayesian Networks 367 Figure 8.9 (a) This fully-connected graph describes a general distribu- tion over twoK-state discrete ...

368 8. GRAPHICAL MODELS Figure 8.10 This chain ofMdiscrete nodes, each havingKstates, requires the specification ofK−1+ (M−1)K(K ...

8.1. Bayesian Networks 369 Figure 8.11 An extension of the model of Figure 8.10 to include Dirich- let priors over the param- et ...

370 8. GRAPHICAL MODELS 8.1.4 Linear-Gaussian models In the previous section, we saw how to construct joint probability distribu ...

8.1. Bayesian Networks 371 Figure 8.14 A directed graph over three Gaussian variables, with one missing link. x 1 x 2 x 3 Thus w ...

372 8. GRAPHICAL MODELS We can readily extend the linear-Gaussian graphical model to the case in which the nodes of the graph re ...

8.2. Conditional Independence 373 Figure 8.15 The first of three examples of graphs over three variables a,b, andcused to discus ...

374 8. GRAPHICAL MODELS Figure 8.16 As in Figure 8.15 but where we have conditioned on the value of variablec. c ab where∅denote ...

8.2. Conditional Independence 375 Figure 8.18 As in Figure 8.17 but now conditioning on nodec. ac b which in general does not fa ...

376 8. GRAPHICAL MODELS Figure 8.20 As in Figure 8.19 but conditioning on the value of node c. In this graph, the act of conditi ...

8.2. Conditional Independence 377 G BF G BF G BF Figure 8.21 An example of a 3-node graph used to illustrate the phenomenon of ‘ ...

378 8. GRAPHICAL MODELS and sop(F=0|G=0)>p(F=0). Thus observing that the gauge reads empty makes it more likely that the tank ...

8.2. Conditional Independence 379 Figure 8.22 Illustration of the con- cept of d-separation. See the text for details. f e b a c ...

380 8. GRAPHICAL MODELS Figure 8.24 A graphical representation of the ‘naive Bayes’ model for classification. Conditioned on the ...