List of Figures
I.1 The graph associated with the inference problem of Exercise I.1.
A.1 A tree of diagnoses for stubborn cars.
2.4 Hypergraph, primal graph and dual graph.
2.5 Possibilities of knowledgebase representation.
2.6 Bayesian network of a medical example.
2.7 A digital circuit of a binary adder.
2.8 The result table of a full adder circuit.
2.9 Connecting two 1-bit-adders produces a 2-bit-adder.
2.10 Input and output of a discrete Fourier transform.
2.11 Function board of a discrete Fourier transform.
3.1 Finalization of the graphical fusion process.
3.2 Extending a labeled tree to a join tree.
3.3 The bucket-tree of Example 3.5.
3.4 The join tree obtained from the bucket-tree of Figure 3.3
3.5 The join tree for Example 1.2.
3.6 Different elimination sequences produce different join trees.
3.7 A primal graph and its induced graph.
3.8 A covering join tree of an inference problem.
3.9 The fusion algorithm as message-passing scheme.
3.10 A covering join tree of Instance 2.1.
3.11 A covering join tree of Instance 2.1.
3.12 A graph with two possible triangulations.
3.13 The triangulated primal graph of Instance 2.1.
3.14 Join graph and derived join tree.
3.15 A complete run of the collect algorithm.
3.16 Join tree of a discrete Fourier transform.
4.1 Changing the root of a join tree.
4.2 Mailboxes in the Shenoy-Shafer architecture.
4.3 Join tree to illustrate the Shenoy-Shafer architecture.
4.4 Using non-binary join trees creates redundant combinations.
4.5 Larges treewidth due to non-binary join trees.
4.6 The performance gain due to the identity element 1.
4.7 The performance gain due to the identity element 2.
4.8 Illustration of the super-cluster approach.
4.9 Separator in the HUGIN architecture.
4.10 The path between node 1 and node 6 in the join tree of Figure 4.3.
4.11 A graphical summary of local computation architectures.
D.1 Embedding a separative valuation algebra into a union of groups.
D.2 Decomposing a regular valuation algebra into a union of groups.
5.1 Semiring valuation algebras with neutral and null elements.
E.1 Embedding a separative semigroup into a union of groups.
E.2 Decomposing a regular semigroup into a union of groups.
E.3 Embedding a cancellative semigroup into a group.
E.4 Semiring valuation algebras and division-related properties.
6.1 The connectivity path problem.
6.2 The shortest distance problem.
6.3 The maximum capacity problem.
6.4 The maximum reliability problem.
6.5 Determining the language of an automaton.
6.6 Path sets in cycled graphs may be infinite.
6.7 The adjacency matrix of a directed, weighted graph.
6.8 An interpretation of equation (6.32).
6.9 The path counting problem.
6.10 Interpreting computations in Markov chains as path problems.
6.11 The computational graph of a function.
8.1 A binary, unreliable, memoryless communication channel.
8.2 The join tree that belongs to the node factors of Example 8.4.
8.3 A serial connection of unreliable, memoryless channels.
8.4 A parallel connection of unreliable, memoryless channels.
9.1 Zero-pattern of a linear equation system.
9.2 The join tree for Figure 9.1.
9.3 The join tree induced by the variable elimination in Example 9.1.
9.4 Recursive buildup of lower triangular matrices.
9.5 The join tree of Example 9.3.
9.6 The graph representing the non-zero pattern of Example 9.4.
9.7 The join tree of the triangulated graph in Figure 9.6.
9.8 The join tree covering the equations of the linear dynamic system.
9.9 A path problem with values from a partially ordered semiring.
10.1 A causal model for the wholesale price of a car.
10.2 The graph of the time-discrete dynamic system.
10.3 A covering join tree for the system (10.6).