Probabilistic Entity-Relationship Models, PRMs, and Plate ModelsHistory/MotivationOutlineER ModelER Model -- ExampleER Model generates attributesPER Model -- ExamplePER Model generates Bayes netConstraints on arc classesMore on constraintsSlide 11Local distribution classesCaveatPER model, plate model, & PRMModeling issuesRestricted relationship: ExampleRestricted, Self, and Uncertain Relationship: ExampleIn the paper… (Google -> Heckerman -> Papers)Probabilistic Entity-Relationship Models, PRMs, and Plate ModelsDavid Heckerman, Chris Meek, and Daphne KollerSlides from SRL 2004 talkHistory/Motivation•Began with: Plates (stats) ~ PRMs (ML)•Found it to be important to distinguish between entities and relationships•Discovered the ER model (e.g., Ullman and Widom, Ch 2)•Created probabilistic version of ER model: PER model–PER Model is more expressive than Plate Model or PRM and helps to show their connections–PER Model provides a strong link to the db community by virtue of being built on top of ER ModelOutline•Entity-Relationship (ER) Model•Probabilistic Entity-Relationship (PER) Model•Connections to plate model, PRM•Modeling issuesER Model•An abstract representation of data•The creation of an ER model is often the first step in the process of constructing a relational database.•Often constructed before any data has arrived (much like we construct models before collecting data).ER Model -- ExampleA university database maintains records on students and their IQs, courses and their difficulty, and the courses taken by students and the grades they receive.CourseStudentTakesDiffGradeIQRelationshipclassEntityclassesAttribute classesCourse entities:CS107, Stats10, …Student entities:John, Mary, …Takes relations:(John,CS107), …Attributes:John.IQ, CS107.Diff…ER Model generates attributesCourseStudentTakesDiffGradeIQTakesStudent Coursejohn cs107mary cs107mary stat10StudentjohnmaryCoursecs107stat10+ER Model Skeleton=>cs107.DiffT(mary.stat10).Gmary.IQjohn.IQstat10.DiffT(mary,cs107).GT(john,cs107).GAttributesPER Model -- ExampleContinuing the university database example, a student's grade in a course depends both on the student's IQ and on the difficulty of the course.CourseStudentTakesDiffGradeIQArc classesNot shown:Local distributionclass for gradePER Model generates Bayes netCourseStudentTakesDiffGradeIQTakesStudent Coursejohn cs107mary cs107mary stat10StudentjohnmaryCoursecs107stat10+PER Model Skeleton=>cs107.DiffT(mary.stat10).Gmary.IQjohn.IQstat10.DiffT(mary,cs107).GT(john,cs107).GAttributesConstraints on arc classesCourseStudentTakesDiffGradeIQTakesStudent Coursejohn cs107mary cs107mary stat10StudentjohnmaryCoursecs107stat10+ER Model Skeleton=>cs107.DiffT(mary.stat10).Gmary.IQjohn.IQstat10.DiffT(mary,cs107).GT(john,cs107).GAttributesde]course[Graf]course[Dif ade]student[Gr]student[IQ More on constraintsA database contains diseases and symptoms for a given patient. Both diseases and symptoms have labels from a common set of categories (e.g., cardiovascular, neuro, urinary). The possible causes of a symptom are diseases that have at least one category in common with that symptom.DiseaseSymptomPresentPresent),(),(21csRcdRc CategoryR1R2More on constraintsA constraint on the arc class from X.A to Y.B in a PER model is any first-order expression involving entities and relationship classes in the PER model such that the expression is bound when the tail and head entities are taken to be constants. To determinewhether to draw an arc from x.A to y.B, we evaluate thefirst-order expression using the tail and head entities of theputative arc. (It must evaluate to true or false.) We draw the arcfrom x.A to y.B only if the expression is true. DiseaseSymptomPresentPresent),(),(21csRcdRc CategoryR1R2Local distribution classesE.g., Noisy ORDiseaseSymptomPresentPresent),(),(21csRcdRc CategoryR1R2CaveatTypically, a PER model is not based on the ER model of a databasePER model, plate model, & PRMCourse DiffStudent IQTakes Course Student GradeDiffGradeIQCourseStudentTakesde]course[Graf]course[Dif ade]student[Gr]student[IQ de]course[Graf]course[Dif ade]student[Gr]student[IQ CourseStudentTakesDiffGradeIQde]course[Graf]course[Dif ade]student[Gr]student[IQ PER model Plate model PRMModeling issues•Restricted relationships•Self relationships•Probabilistic relationshipsRestricted relationship: ExampleHierarchical model: A binary outcome O is measured on patients in multiple hospitals. Each patient is treated in exactly one hospital. It is believed that outcomes in any given hospital h are i.i.d. given binomial parameter h.; and that these binomial parameters are themselves i.i.d. across hospitals given hyperparameters a.h1. hm.p11.O pm1.O… ……HospitalPatientInOOpn.11Opmmn.),(In phRestricted, Self, and Uncertain Relationship:ExampleA student's grade in a course depends on whether an advisor of the student is a friend of a teacher of the course.ProfessorCourseStudentTeachesTakesDiffGradeIQF(p,pf)FriendAdvisesFull),(Advises),(Teachesspcpf][G][D cc ]G[][IQ ss In the paper…(Google -> Heckerman -> Papers)•Formal definitions and theorems•Precise differences between PER models, plate models, and PRMs•Undirected PER models•PER models for asymmetric independence•Many more
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