EL HALLAZGO DE OBJETO

The cliquesWi of equation 2.33 must follow a series of conditions. The procedure

of Fig. 2.6 obtains the set of cliques with the required properties. 3The material of this section was taken from the book by Neapolitan (1990).

1. Delete the direction of the arcs, i.e., the DAG is converted to an undirected graph.

2. Moralize the graph. 3. Triangulate the graph.

4. Order the nodes according to a criterion called the maximum cardinality search.

5. Determine the cliques of the triangulated graph. 6. Order the cliques according to their highest labelled

vertices to obtain an ordering of the cliques with the running intersection property.

Figure 2.6: Procedure to convert a network in a tree of cliques.

These steps are better explained with the aid of an example taken from the book by Neapolitan (1990). Figure 2.7 presents the original Bayesian network.

A B C D E F H G

Figure 2.7: Original multiply connected network.

Notice that it is multiply connected since there is more than one path between node F and H. This network requires, as all the Bayesian networks, the prior probability of the roots and the conditional probability matrices of the other nodes given their parents. The rst step in the procedure of Fig. 2.6 is trivial, i.e., only delete the direction of the arcs. The second step, the moralization, is obtained when the pairs of parents of all nodes (if they exist) aremarried. This is

done with the addition of an arc between these parent nodes. Figure 2.8 presents the moral DAG which is obtained by adding the arc between nodes B and E (parents of C), and the arc between C and G (parents of H).

A B C D E F H G

Figure 2.8: Undirected moralized graph.

Next, the triangulation step takes place. An undirected graph is called trian- gulated if every simple cycle of length strictly greater that 3 possesses a chord. In the original network, after the moralization, the nodes [F;E;C;G] form a simple cycle of size 4. Thus, in order to triangularize the undirected graph, the arc between E and G is added. Figure 2.9 shows the triangularized graph. This

A B C D E F H G 1 2 3 4 5 6 7 8

Figure 2.9: Triangulated and ordered undirected graph.

gure also show the ordering step indicated in the procedure of Fig. 2.6 which is now explained. An order of the nodes, according to a criterion known as the maximum cardinality search, is obtained as follows. First, 1 is assigned to an

arbitrary node. To number the next node, select a node that is adjacent to the largest numbered node, breaking ties arbitrary. In Fig. 2.9, number 1 was as- signed to node A. Number 2 must be assigned to node B since it is the only adjacent node to A (two nodes are adjacent if there is an arc between them). Now, number 3 has to be assigned to one of the adjacent nodes to B, i.e., C or E. Node E was chosen arbitrarily. Number 4 was assigned to node C (could be F), and so on until all the nodes are numbered. The next step, determining the cliques of the triangulated graph, is now described. A clique is a subset of nodes which is complete, i.e., every pair of nodes of the clique is adjacent. Also, the subset must be maximal, i.e., there is no other complete set which is a subset. In the triangulated graph of Fig. 2.9, the following cliques are found: fA;Bg, fB;E;Cg, fE;G;Fg, fC;Dg, fE;C;Gg, and fC;G;Hg. Notice that the sub- set fC;E;G;Hg is a complete subset but it is not maximal since fE;C;Gg is a complete subset.

Finally, the ordering of the cliquesis required. An ordering [Clq1;Clq2;:::;Clqp] of the cliques has the running intersection property if for everyj > 1 there exists an i < j such that Cj \(C 1 [C 2 [:::[Cj 1) Ci: (2.34)

In the example, an ordering of the cliques is the following: Clq1 =

fA;Cg, Clq2 = fC;D;Fg, Clq 3 = fD;E;Fg, Clq 4 = fB;D;Eg, Clq 5 = fF;E;Hg, and Clq6 =

fF;Gg. This ordering has the running intersection property. For example: Clq4 \(Clq 1 [Clq 2 [Clq 3) = fD;EgClq 3 (2.35) Clq5 \(Clq 1 [Clq 2 [Clq 3 cupClq4) = fE;FgClq 3

dened. Si =Clqi\(Clq 1 [Clq 2 [:::[Clqi 1) Ri =Clqi Si:

These parameters will be used in the propagation of probabilities and in the denition of the structure. As an example, S4 =

fE;F;Gg\fA;B;C;E;Gg = fE;Gg and R 4= fE;F;Gg S 4 = fFg.

Once the set of ordered cliques has been obtained, the next step is the deni- tion of the structure of the tree of cliques. The rst clique is the root of the tree. Now, for the rest of the nodes, i.e., for each i such that 2ip, there exists at least one j < i such that

Si =Clqi\(Clq 1 [Clq 2 [:::[Clqi 1) Clqj: (2.36) ThenClqjis a parent ofClqi. In case of more than one possible parent, the choice

is arbitrary. Figure 2.10 shows the nal modication of the Bayesian network of Clq1

Clq2

Clq3

Clq4 Clq5 Clq6

Figure 2.10: Resultant tree of cliques.

Fig. 2.7 into the tree of cliques. The next section briey describes the algorithm for probability propagation in this tree of cliques.