S, and its associated parameters, es. The dependency task of making an inference about a new, previously structure defines the parents Pa(X A)for each at- unseen Vote. score. To accor is task. we lever tribute X.A. The parent for an attribute X A can age the ground Bayesian Network [Geto2 induced by be a descriptive attribute within the class X, or it can a PRM. Briefly, a Bayesian Network is constructed be a descriptive attribute in another class y that is from a database using the link structure of the as- eachable through a reference slot chain. For instance, sociated PRMs dependency graph, together with the in the above example, Vote. Score could have the par- parameters that are associated with that dependency ent Vote. ofPersonGender. Note that in most cases graph. For example, for the PRM in Figure 1(a), if we the parent of a given attribute will take on a multiset needed to infer the Score value for a new Vote object of values S in V(XT A). For example, we could dis- we simply construct a ground Bayesian Network using cover a dependency of a Person's age on their rating the appropriate attributes retrieved from the associ- of movies in the Children genre. However, we cannot ated Person and Movie objects; see Figure 1(b). The directly model this dependency since the user's rat- PRMs class-level parameters for the various attributes ings on Children's movies is a multiset of values, say are then tied to the ground Bayesian Network's param- 14, 5, 3, 5, 4. For such a numeric attribute, we may eters, and standard Bayesian Network inference proce- choose to use the Median database aggregate operator dures can be used on the resulting network Geto2 to reduce this multiset to a single value. in this case 4. In this paper we reduce s to a single value using various types of aggregation functions. 3 Hierarchical Probabilistic relational The following definition summarizes the key elements Models of a prm Definition 1(IGeto2J)A probabilistic relational In this section we describe our approach to extend- model(PRM)II for a relational schema s is defined as ing standard PRMs to include class hierarchies. The follows. For each class X E x and each descriptive at- concept of learning PRMs with class hierarchies is tribute AEA(x), we have a set of parents Pa(X A), troduced in (Geto2 and a conditional probability distribution( CPD) that represents Pn(XAPa(XA) 3.1 Motivation 2.1 Applying Standard PRMs to the EachMovie dataset The collaborative filtering problem presents two ma- jor motivations for hPRMs. First, in the above model to exploit. In general, model-based collaborative il- Vote. Score depend onitselfi, t ir tes of related ob- PRMs provide an ideal framework for capturing the Vote. Score can depend on attribu kinds of dependencies a recommender system needs jects, such as Person Age, but not possible to have tering algorithms try to capture high-level patterns in the fact that the class-level PRMs dependency struc data that provide some amount of predictive accuracy. ture must be a directed acyclic graph(DAg) in order For example, in the EachMovie dataset, one may want to guarantee that the instance-level ground Bayesian to capture the pattern that young males tend to rate Network forms a DAG FGKP99, and thus a well Action movies quite highly, and subsequently use this formed probability distribution. Without the ability dependency to make inferences about unknown votes. to have Vote. Score depend probabilistically on itself. PRMs are able to model such patterns as class-level we lose the ability to have a user's rating of an item de- dependencies, which can subsequently be used at an pend on his rating of other items or on other user's rat- instance level to make predictions on unknown ratings. ings on this movie. For example, we may wish to have how will George vote on SoM the user's ratings of Comedies infuence his rating of Action movies, or his rating of a specific Comedy movie In order to use a PRM to make predictions about an influence his ratings of other Comedy movies. Second data.In our experiments we use the prm learn in the above model we are restricted to one depen- ing produce described in FGKPggl, which provides the type of object the rating is for, we may wish to In algorithm for both learning a legal structure for a have a specialized dependency graph to better model PRM and estimating the parameters associated with the dependencies. For example, the dependency grap that PRM. Figure 1(a)shows a sample PRM structure for an Action movie may have Vote. Score depend on learned from the eachMovie dataset Vote. PersonOf. Gender, whereas a Documentary may With the learned PRM in hand, we are left with the depend on Vote. PersonOf AgeS, and its associated parameters, θS . The dependency structure defines the parents P a(X.A) for each at￾tribute X.A. The parent for an attribute X.A can be a descriptive attribute within the class X, or it can be a descriptive attribute in another class Y that is reachable through a reference slot chain. For instance, in the above example, Vote.Score could have the par￾ent Vote.ofPerson.Gender. Note that in most cases the parent of a given attribute will take on a multiset of values S in V (X.τ.A). For example, we could dis￾cover a dependency of a Person’s age on their rating of movies in the Children genre. However, we cannot directly model this dependency since the user’s rat￾ings on Children’s movies is a multiset of values, say {4, 5, 3, 5, 4}. For such a numeric attribute, we may choose to use the Median database aggregate operator to reduce this multiset to a single value, in this case 4. In this paper we reduce S to a single value using various types of aggregation functions. The following definition summarizes the key elements of a PRM: Definition 1 ([Get02]) A probabilistic relational model (PRM) Π for a relational schema S is defined as follows. For each class X ∈ X and each descriptive at￾tribute A ∈ A(X), we have a set of parents Pa(X.A), and a conditional probability distribution (CPD) that represents PΠ(X.A|P a(X.A)) 2.1 Applying Standard PRMs to the EachMovie Dataset PRMs provide an ideal framework for capturing the kinds of dependencies a recommender system needs to exploit. In general, model-based collaborative fil￾tering algorithms try to capture high-level patterns in data that provide some amount of predictive accuracy. For example, in the EachMovie dataset, one may want to capture the pattern that young males tend to rate Action movies quite highly, and subsequently use this dependency to make inferences about unknown votes. PRMs are able to model such patterns as class-level dependencies, which can subsequently be used at an instance level to make predictions on unknown ratings. — i.e., how will George vote on SoM. In order to use a PRM to make predictions about an unknown rating, we must first learn the PRM from data. In our experiments we use the PRM learn￾ing produce described in [FGKP99], which provides an algorithm for both learning a legal structure for a PRM and estimating the parameters associated with that PRM. Figure 1(a) shows a sample PRM structure learned from the EachMovie dataset. With the learned PRM in hand, we are left with the task of making an inference about a new, previously unseen Vote.score. To accomplish this task, we lever￾age the ground Bayesian Network [Get02] induced by a PRM. Briefly, a Bayesian Network is constructed from a database using the link structure of the as￾sociated PRM’s dependency graph, together with the parameters that are associated with that dependency graph. For example, for the PRM in Figure 1(a), if we needed to infer the Score value for a new Vote object, we simply construct a ground Bayesian Network using the appropriate attributes retrieved from the associ￾ated Person and Movie objects; see Figure 1(b). The PRM’s class-level parameters for the various attributes are then tied to the ground Bayesian Network’s param￾eters, and standard Bayesian Network inference proce￾dures can be used on the resulting network [Get02]. 3 Hierarchical Probabilistic Relational Models In this section we describe our approach to extend￾ing standard PRMs to include class hierarchies. The concept of learning PRMs with class hierarchies is in￾troduced in [Get02]. 3.1 Motivation The collaborative filtering problem presents two ma￾jor motivations for hPRMs. First, in the above model, Vote.Score can depend on attributes of related ob￾jects, such as Person.Age, but it is not possible to have Vote.Score depend on itself in any way. This is due to the fact that the class-level PRM’s dependency struc￾ture must be a directed acyclic graph (DAG) in order to guarantee that the instance-level ground Bayesian Network forms a DAG [FGKP99], and thus a well￾formed probability distribution. Without the ability to have Vote.Score depend probabilistically on itself, we lose the ability to have a user’s rating of an item de￾pend on his rating of other items or on other user’s rat￾ings on this movie. For example, we may wish to have the user’s ratings of Comedies influence his rating of Action movies, or his rating of a specific Comedy movie influence his ratings of other Comedy movies. Second, in the above model we are restricted to one depen￾dency graph for Vote.Score; however, depending on the type of object the rating is for, we may wish to have a specialized dependency graph to better model the dependencies. For example, the dependency graph for an Action movie may have Vote.Score depend on Vote.PersonOf.Gender, whereas a Documentary may depend on Vote.PersonOf.Age