Uncertainty in group recommending Table 3 Stored probability values P(U0),P(U1),P(U2).P(U3),P(U4),P(U5) P(V1|U0,U2) P(u1.l1,1) P(U111.2) 12,2) P(V2|U1,U3,U4)P(V2|l,1,1) P(n12,1,1)P(u212.1,2)P(u,112,2,1)P(u2,2,2,2) P(V3|U2,U4,Us)P(u3.11,1,1)P(u3.1,1,2)P(u3.11,2,1)P(n3.11,2,2) P(u3,12,L,1)P(u3,12.1,2)P(u3,112,2,1)P(u3,12,2,2) P(Gav1,V,v3)P(ga,11l.1,1)P(ga,l1,1,2)P(ga,11,2,1)P(ga,1,2,2) In order to complete the BN-based model, it is necessary to estimate the local probabilities that must be stored in the nodes. In particular, each node Xi has a set of onditional probability distributions, P(ilpa(Xi ))(except root nodes that store mar ginal probability distributions). 4 For each possible configuration pa(X; )of the parent set Pa(Xi), these distributions quantify the effect that the parents have on the node Xi. In our case, these probabilities are used to encode both the strength of the user-user interactions and the processes leading to the final choice or recommendation for the group. In Table 3, we show those probability distributions stored in the example of Fig 3, where for instance P(V2ll, 1, 2)represents P(V2lu1 1, u3, 1, u4.2). The method of assessing the particular values will be discussed in Sect. 4. 3.3 How to predict the group rating: inference Once the Bn is completed, it can be used to perform inference tasks. In our case, we are interested in the prediction of the group's rating for an unobserved item, 1. As evi dence, we will consider how this product was rated in the past. The problem therefore omes down to computing the conditional(posterior) probability distribution for the target group Ga given the evidence, i.e. Pr(Galev). For instance, let us assume that we want to predict the rating given by Ga in Fig. 3 to item 17. If we look at Table 2. the evidences are ev=Uo= 2, U4=1, U5 = 2) and the problem is to compute Since the Bn is a concise representation of a joint distribution, we could propagate the observed evidence through the network towards group variables. This propaga- tion implies a marginalization process(summing out over uninstantiated variables) 4 Throughout this paper we will use upper-case letters to denote variables and lower-case letters to denote the particular instantiation. More specifically, we use vi to denote a general value of variable Vi and vi, j to indicate that Vi takes the jth-value. at we consider that no member of the group has observed the ite therefore the evidences er the values taken by variables in u. In the case of a group member(let say Ui) having also previously rated 1, we shall instantiate both node Ui and Vi to the value of the given ratings. The instantiation of Vi will imply that there is no uncertainty about its rating when the information is combined at a group level. Nevertheless, the computations are more complex in this situation.Uncertainty in group recommending 217 Table 3 Stored probability values P(U0), P(U1), P(U2), P(U3), P(U4), P(U5) P(V1|U0, U2) P(v1,1|1, 1) P(v1,1|1, 2) P(v1,1|2, 1) P(v1,1|2, 2) P(V2|U1, U3, U4) P(V2|1, 1, 1) P(V2|1, 1, 2) P(V2|1, 2, 1) P(V2|1, 2, 2) P(v2,1|2, 1, 1) P(v2,1|2, 1, 2) P(v2,1|2, 2, 1) P(v2,1|2, 2, 2) P(V3|U2, U4, U5) P(v3,1|1, 1, 1) P(v3,1|1, 1, 2) P(v3,1|1, 2, 1) P(v3,1|1, 2, 2) P(v3,1|2, 1, 1) P(v3,1|2, 1, 2) P(v3,1|2, 2, 1) P(v3,1|2, 2, 2) P(Ga|V1, V2, V3) P(ga,1|1, 1, 1) P(ga,1|1, 1, 2) P(ga,1|1, 2, 1) P(ga,1|1, 2, 2) P(ga,1|2, 1, 1) P(ga,1|2, 1, 2) P(ga,1|2, 2, 1) P(ga,1|2, 2, 2) In order to complete the BN-based model, it is necessary to estimate the local probabilities that must be stored in the nodes. In particular, each node Xi has a set of conditional probability distributions, P(xi|pa(Xi)) (except root nodes that store mar￾ginal probability distributions).4 For each possible configuration pa(Xi) of the parent set Pa(Xi), these distributions quantify the effect that the parents have on the node Xi . In our case, these probabilities are used to encode both the strength of the user-user interactions and the processes leading to the final choice or recommendation for the group. In Table 3, we show those probability distributions stored in the example of Fig. 3, where for instance P(V2|1, 1, 2) represents P(V2|u1,1, u3,1, u4,2). The method of assessing the particular values will be discussed in Sect. 4. 3.3 How to predict the group rating: inference Once the BN is completed, it can be used to perform inference tasks. In our case, we are interested in the prediction of the group’s rating for an unobserved item, I. As evi￾dence, we will consider how this product was rated in the past.5 The problem therefore comes down to computing the conditional (posterior) probability distribution for the target group Ga given the evidence, i.e. Pr(Ga|ev). For instance, let us assume that we want to predict the rating given by Ga in Fig. 3 to item I7. If we look at Table 2, the evidences are ev = {U0 = 2, U4 = 1, U5 = 2} and the problem is to compute Pr(Ga = 1| u0,2, u4,1, u5,2). Since the BN is a concise representation of a joint distribution, we could propagate the observed evidence through the network towards group variables. This propaga￾tion implies a marginalization process (summing out over uninstantiated variables). 4 Throughout this paper we will use upper-case letters to denote variables and lower-case letters to denote the particular instantiation. More specifically, we use vi to denote a general value of variable Vi and vi,j to indicate that Vi takes the jth-value. 5 It should be noted that we consider that no member of the group has observed the items beforehand and therefore the evidences are over the values taken by variables in U. In the case of a group member (let us say Ui) having also previously rated I, we shall instantiate both node Ui and Vi to the value of the given ratings. The instantiation of Vi will imply that there is no uncertainty about its rating when the information is combined at a group level. Nevertheless, the computations are more complex in this situation. 123