Pattern Recognition and Machine Learning

8.2. Conditional Independence 381 using maximum likelihood assuming that the data are drawn independently from the model. The so ...

382 8. GRAPHICAL MODELS p(x) DF Figure 8.25 We can view a graphical model (in this case a directed graph) as a filter in which a ...

8.3. Markov Random Fields 383 Figure 8.26 The Markov blanket of a nodexicomprises the set of parents, children and co-parents of ...

384 8. GRAPHICAL MODELS Figure 8.27 An example of an undirected graph in which every path from any node in set Ato any node in s ...

8.3. Markov Random Fields 385 Figure 8.28 For an undirected graph, the Markov blanket of a node xiconsists of the set of neighbo ...

386 8. GRAPHICAL MODELS the joint distribution is written as a product ofpotential functionsψC(xC)over the maximal cliques of th ...

8.3. Markov Random Fields 387 choice ofxC). Given this restriction, we can make a precise relationship between factorization and ...

388 8. GRAPHICAL MODELS Figure 8.30 Illustration of image de-noising using a Markov random field. The top row shows the original ...

8.3. Markov Random Fields 389 Figure 8.31 An undirected graphical model representing a Markov random field for image de-noising, ...

390 8. GRAPHICAL MODELS Figure 8.32 (a) Example of a directed graph. (b) The equivalent undirected graph. (a) x 1 x 2 xN− 1 xN ( ...

8.3. Markov Random Fields 391 Figure 8.33 Example of a simple directed graph (a) and the corre- sponding moral graph (b). x 1 x ...

392 8. GRAPHICAL MODELS then drop the arrows on the original links to give the moral graph. Then we initialize all of the clique ...

8.4. Inference in Graphical Models 393 Figure 8.35 A directed graph whose conditional independence properties cannot be expresse ...

394 8. GRAPHICAL MODELS Figure 8.37 A graphical representation of Bayes’ theorem. See the text for details. x y x y x y (a) (b) ...

8.4. Inference in Graphical Models 395 The joint distribution for this graph takes the form p(x)= 1 Z ψ 1 , 2 (x 1 ,x 2 )ψ 2 , 3 ...

396 8. GRAPHICAL MODELS the desired marginal in the form p(xn)= 1 ⎡ Z ⎣ ∑ xn− 1 ψn− 1 ,n(xn− 1 ,xn)··· [ ∑ x 2 ψ 2 , 3 (x 2 ,x 3 ...

8.4. Inference in Graphical Models 397 Figure 8.38 The marginal distribution p(xn)for a nodexnalong the chain is ob- tained by m ...

398 8. GRAPHICAL MODELS Now suppose we wish to evaluate the marginalsp(xn)for every noden ∈ { 1 ,...,N}in the chain. Simply appl ...

8.4. Inference in Graphical Models 399 Figure 8.39 Examples of tree- structured graphs, showing (a) an undirected tree, (b) a di ...

400 8. GRAPHICAL MODELS Figure 8.40 Example of a factor graph, which corresponds to the factorization (8.60). x 1 x 2 x 3 fa fb ...