216 L. M. de Campos et al. D(Ua, Ub) JI(Ub) where I(U) is the set of items rated by user U in the data set. It should be noted that we are not considering the particular votes, merely whether the users rated an item or 3.2 Modeling the group component As mentioned previously, since groups are usually created by their members, we shall not consider how groups are formed or how they are managed. We shall therefore assume that we know the composition of the groups, and our problem is to study how this information can be represented in the Bn and also how to predict ratings for We propose to identify a group G as a new node in the BN. Since the recommen- dations are made by considering the preferences of its members, we propose that the parents(Pa(G))of the group node(G)will be the set of nodes in V representing its individuals. In this case, we are modeling that the predictions of the grou ill depend on the collaborative predictions obtained for each of its members. Figure 3 illustrates a group Ga with three members: V1, V2, and V3. We use dashed lines to epresent user-group relations since we assume that the composition of the group is known In this paper, we will focus on how different aggregation strategies can be repre sented in our BN-based model. In order to maintain generality (so that the proposed aggregation mechanisms can be applied in more general situations ) we will use the following independence assumption: given that we know the opinion(ratings)of all the group members, group opinion does not change(it is independent)if the state of any other variable in system Xi is known, i.e. I(G, XiIPa(g)), vxi Pa(Gi. It is important to remember that in certain domains this restriction might be very restric tive. For example, it might also be possible to consider other factors that would affect the group rating such as the context. Nevertheless, the study of how to include these factors in the model is beyond the scope of this paper. 3 For example, it might be used to combine multiple classifiers(Kittler et al. 1998; Abellan and Masegosa 2007)where the new cases will be classified by considering all the results obtained by each classifier216 L. M. de Campos et al. Fig. 3 Modeling groups Ga U0 U1 U2 U3 U4 U5 V1 V2 V3 V4 V5 D(Ua, Ub) = |I(Ua) ∩ I(Ub)| |I(Ub)| . where I(U) is the set of items rated by user U in the data set. It should be noted that we are not considering the particular votes, merely whether the users rated an item or not. 3.2 Modeling the group component As mentioned previously, since groups are usually created by their members, we shall not consider how groups are formed or how they are managed. We shall therefore assume that we know the composition of the groups, and our problem is to study how this information can be represented in the BN and also how to predict ratings for groups. We propose to identify a group G as a new node in the BN. Since the recommen￾dations are made by considering the preferences of its members, we propose that the parents (Pa(G)) of the group node (G) will be the set of nodes in V representing its individuals. In this case, we are modeling that the predictions of the group’s ratings will depend on the collaborative predictions obtained for each of its members. Figure 3 illustrates a group Ga with three members: V1, V2, and V3. We use dashed lines to represent user-group relations since we assume that the composition of the group is known. In this paper, we will focus on how different aggregation strategies can be repre￾sented in our BN-based model. In order to maintain generality (so that the proposed aggregation mechanisms can be applied in more general situations3), we will use the following independence assumption: given that we know the opinion (ratings) of all the group members, group opinion does not change (it is independent) if the state of any other variable in system Xi is known, i.e. I(G, Xi|Pa(G)), ∀Xi ∈/ Pa(Gi). It is important to remember that in certain domains this restriction might be very restric￾tive. For example, it might also be possible to consider other factors that would affect the group rating such as the context. Nevertheless, the study of how to include these factors in the model is beyond the scope of this paper. 3 For example, it might be used to combine multiple classifiers (Kittler et al. 1998; Abellán and Masegosa 2007) where the new cases will be classified by considering all the results obtained by each classifier. 123