1. UIUC CS 497: Section EA Lecture #8 Reasoning in Artificial Intelligence Professor: Eyal Amir Spring Semester 2004 (Based on slides by Lise Getoor (UMD))

2. Last Time Approximate Inference with Probabilistic Graphical Models Monte Carlo techniques Markov Chain Monte Carlo

3. Today Probabilistic Relational Models (PRMs) PRMs w/ Attribute Uncertainty PRMs w/ Link Uncertainty

4. Patterns in Structured Data Patient Treatment Strain Contact

5. Bayesian Networks nodes = random variables edges = direct probabilistic influence Network structure encodes independence assumptions: XRay conditionally independent of Pneumonia given Infiltrates XRay Lung Infiltrates Sputum Smear TuberculosisPneumonia

6. Bayesian Networks XRay Lung Infiltrates Sputum Smear TuberculosisPneumonia Associated with each node X i there is a conditional probability distribution P(X i |Pa i :  ) — distribution over X i for each assignment to parents 0.8 0.2 p t p 0.6 0.4 0.010.99 0.2 0.8 tp t t p T P P(I |P, T )

7. BN Semantics conditional independencies in BN structure + local probability models full joint distribution over domain = X I S TP

8. Probabilistic Relational Models Combine advantages of FOL & Bayes Nets: –natural domain modeling –generalization over a variety of situations; –compact, natural probability models. Integrate uncertainty with relational model: –properties of domain entities can depend on properties of related entities; –uncertainty over relational structure of domain.

9. Relational Schema Strain Unique Infectivity Infected with Interacted with Describes the types of objects and relations in the databaseClasses Relationships Contact Close-Contact Skin-Test Age Patient Homeless HIV-Result Ethnicity Disease-Site Attributes Contact-Type

10. Probabilistic Relational Model Close-Contact Transmitted Contact-Type Disease Site Strain Unique Infectivity Patient Homeless HIV-Result POB Contact Age           Cont.Contactor.HIV Cont.Close-Contact Cont.Transmitted | P 4.06.0 3.07.0 2.08.0 1.09.0,,,,, tt ft tf ff P(T | H, C) CH

11. Relational Skeleton Fixed relational skeleton  –set of objects in each class –relations between them Uncertainty over assignment of values to attributes PRM defines distr. over instantiations of attributes Strain s1 Patient p2 Patient p1 Contact c3 Contact c2 Contact c1 Strain s2 Patient p3

12. A Portion of the BN P1.Disease Site P1.Homeless P1.HIV-Result P1.POB C1.Close-Contact C1.Transmitted C1.Contact-Type C1.Age C2.Close-Contact C2.Transmitted C2.Contact-Type true falsetrue 4.06.0 3.07.0 2.08.0 1.09.0,,,,, tt ft tf ff P(T | H, C) CH 4.06.0 3.07.0 2.08.0 1.09.0,,,,, tt ft tf ff CH C2.Age

13. PRM: Aggregate Dependencies sum, min, max, avg, mode, count Disease Site Patient Homeless HIV-Result POB Age Close-Contact Transmitted Contact-Type Contact Age.. Patient Jane Doe POB US Homeless no HIV-Result negative Age ??? Disease Site pulmonary A. Contact #5077 Contact-Type coworker Close-Contact no Age middle-aged Transmitted false Contact #5076 Contact-Type spouse Close-Contact yes Age middle-aged Transmitted true Contact #5075 Contact-Type friend Close-Contact no Age middle-aged Transmitted false mode

14. PRM Semantics Attributes Objects probability distribution over completions I : PRM relational skeleton  += Strain Patient Contact Strain s1 Patient p1 Patient p2 Contact c3 Contact c2 Contact c1 Strain s2 Patient p3

15. Legal Models author-of PRM defines a coherent probability model over a skeleton  if the dependencies between object attributes is acyclic How do we guarantee that a PRM is acyclic for every skeleton? Researcher Prof. Gump Reputation high Paper P1 Accepted yes Paper P2 Accepted yes sum

16. Attribute Stratification PRM dependency structure S dependency graph Paper.Accecpted Researcher.Reputation if Researcher.Reputation depends directly on Paper.Accepted dependency graph acyclic  acyclic for any  Attribute stratification: Algorithm more flexible; allows certain cycles along guaranteed acyclic relations

17. Blood Type M-chromosome P-chromosome Person Result Contaminated Blood Test Blood Type M-chromosome P-chromosome Person Blood Type M-chromosome P-chromosome Person (Father) (Mother)

18. Outline Probabilistic Relational Models (PRMs) »PRMs w/ Attribute Uncertainty PRMs w/ Link Uncertainty

19. Attribute Uncertainty Topic TheoryAI Agent Theory papers Cornell Scientific Paper Topic TheoryAI Attributes of object Attributes of linked objects Attributes of heterogeneous linked objects

20. PRMs w/ AU: example Vote Rank Movie IncomeGender Person AgeGenre PRM consists of: Relational Schema Dependency Structure           Vote.Person.Gender, Vote.Person.Age Vote.Movie.Genre, Vote.Rank | P   Local Probability Models

21. Fixed relational skeleton  : –set of objects in each class –relations between them Movie m1 Vote v1 Movie: m1 Person: p1 Person p2 Person p1 Movie m2 Uncertainty over assignment of values to attributes PRM w/ Attribute Uncertainty Vote v2 Movie: m1 Person: p2 Vote v3 Movie: m2 Person: p2 Primary Keys Foreign Keys

22. PRM with Attribute Uncertainty Semantics Attributes Objects Ground BN defining distribution over complete instantiations of attributes I : PRM relational skeleton  += Patient p2 Vote Movie Person Movie Vote Vote Person Movie Vote

23. Issue PRM w/ AU applicable only in domains where we have full knowledge of the relational structure Next we introduce PRMs which allow uncertainty over relational structure…

24. Outline Probabilistic Relational Models (PRMs) PRMs w/ Attribute Uncertainty »PRMs w/ Link Uncertainty

25. Approach Construct probabilistic models of relational structure that capture link uncertainty Two new mechanisms: –Reference uncertainty –Existence uncertainty Advantage: –Applicable with partial knowledge of relational structure

26. Citation Relational Schema Wrote Paper Topic Word1 WordN … Word2 Paper Topic Word1 WordN … Word2 Cites Count Citing Paper Cited Paper Author Institution Research Area

27. Attribute Uncertainty Paper Word1 Topic WordN Wrote Author... Research Area P( WordN | Topic) P( Topic | Paper.Author.Research Area Institution P( Institution | Research Area)

28. Reference Uncertainty Bibliography Scientific Paper ` 1. ----- 2. ----- 3. ----- ? ? ? Document Collection

29. PRM w/ Reference Uncertainty Cites Cited Citing Dependency model for foreign keys Paper Topic Words Paper Topic Words Naïve Approach: multinomial over primary key noncompact limits ability to generalize

30. Reference Uncertainty Example Paper P5 Topic AI Paper P4 Topic AI Paper P3 Topic AI Paper M2 Topic AI Paper P1 Topic Theory Cites Cited Citing Paper P5 Topic AI Paper P3 Topic AI Paper P4 Topic Theory Paper P2 Topic Theory Paper P1 Topic Theory Paper.Topic = AI Paper.Topic = Theory P1 P2 Paper Topic Words P1 P2 3.07.0 P1 P2 1.09.0 Topic 99.001.0 Theory AI

31. PRMs w/ RU Semantics PRM-RU + entity skeleton   probability distribution over full instantiations I Cites Cited Citing Paper Topic Words Paper Topic Words PRM RU Paper P5 Topic AI Paper P4 Topic Theory Paper P2 Topic Theory Paper P3 Topic AI Paper P1 Topic ??? Paper P5 Topic AI Paper P4 Topic Theory Paper P2 Topic Theory Paper P3 Topic AI Paper P1 Topic ??? Reg Cites entity skeleton 

32. Existence Uncertainty Document Collection ? ? ?

33. PRM w/ Existence Uncertainty Cites Dependency model for existence of relationship Paper Topic Words Paper Topic Words Exists

34. Exists Uncertainty Example Cites Paper Topic Words Paper Topic Words Exists Citer.Topic Cited.Topic 0.9950005 Theory FalseTrue AI Theory0.9990001 AI 0.9930008 AI Theory0.9970003

35. PRMs w/ EU Semantics PRM-EU + object skeleton   probability distribution over full instantiations I Paper P5 Topic AI Paper P4 Topic Theory Paper P2 Topic Theory Paper P3 Topic AI Paper P1 Topic ??? Paper P5 Topic AI Paper P4 Topic Theory Paper P2 Topic Theory Paper P3 Topic AI Paper P1 Topic ??? object skeleton  ??? PRM EU Cites Exists Paper Topic Words Paper Topic Words

36. Inference in Unrolled BN Exact Inference in “unrolled” BN –Infeasible for large networks –Structural (Attr/Reference/Exists) Uncertainty creates very large cliques –Use caching (Pfeffer ’00) –FOL-Resolution-style techniques Loopy belief propagation (Pearl, 88; McEliece, 98) –Scales linearly with size of network –Guaranteed to converge only for polytrees –Empirically, often converges in general nets (Murphy’99) Use approx. inference: MCMC (Pasula etal. ’01)

37. MCMC with PRMs Prof1. $$ Prof2. $$ Prof3. $$ Prof1. fame Prof2. fame Prof3. fame Student1. advisor Student1. success

38. MCMC with PRMs Prof1. $$ Prof2. $$ Prof3. $$ Prof2. fame Student1. advisor Student1. success =Prof2 Network structure changed

39. Gibbs Sampling with PRMs For each complex attribute A: reference attribute Ref[A], w/finite domain Val[Ref[A]] Reference uncertainty modifies chain of attributes

40. Gibbs Sampling with PRMs For each complex attribute A: reference attribute Ref[A], w/finite domain Val[Ref[A]] Reference uncertainty modifies chain of attributes Gibbs for simple attributes: Use MB Gibbs for complex attributes (RU): –Add reference variables

41. Gibbs Sampling with PRMs Prof1. $$ Prof2. $$ Prof3. $$ Prof2. fame Student1. advisor Student1. success =Prof2 P(P3.f | mb(P3.f))=  P(P3.f|Pa(P3.f))P(P3.$$|P3.f)P(S1.s|S1.a=P2,P1.f,P2.f,P3.f)=  P(P3.f) P(P3.$$ | P3.f) P(S1.s | S1.a=P2,P2.f)=  ’P(P3.f) P(P3.$$ | P3.f) Prof3. fame Constant wrt P3.f Gibbs when reference var does not change

42. M-H Sampling with PRMs Prof1. $$ Prof2. $$ Prof3. $$ Prof2. fame Student1. advisor Student1. success =Prof2 P(s1.a=P3,...X…) q(s1.a=P2,...X…| s1.a=P3,...X…) --------------------------------------------------------------------- = P(s1.a=P2,...X…) q(s1.a=P3,...X…| s1.a=P2,...X…) Prof3. fame Changing a ref. variable P(s1.a=P3,...X…) P(s1.a=P3 | P1.$$,…,Pn.$$) P(s1.s|P3.f, ------------------------ = ----------------------------------------- P(s1.a=P2,...X…) P(s1.a=P3,...X…)

43. M-H Sampling with PRMs Prof1. $$ Prof2. $$ Prof3. $$ Prof2. fame Student1. advisor Student1. success =Prof2 Prof3. fame Changing a ref. variable P(s1.a=P3,...X…) ------------------------ = P(s1.a=P2,...X…) P(s1.a=P3 | P1.$$,…,Pn.$$) P(s1.s | P3.f,S1.a=P3) ------------------------------------------------------------------- P(s1.a=P2 | P1.$$,…,Pn.$$) P(s1.s | P2.f,S1.a=P2)

44. P(s1.a=P3 | P1.$$,…,Pn.$$) P(s1.s | P3.f,S1.a=P3) -------------------------------------------------------------------- = P(s1.a=P2 | P1.$$,…,Pn.$$) P(s1.s | P2.f,S1.a=P2) M-H Sampling with PRMs Prof1. $$ Prof2. $$ Prof3. $$ Prof2. fame Student1. advisor Student1. success =Prof2 Prof3. fame Changing a ref. variable P(s1.a=P3 | P3.$$) P(s1.s | P3.f,S1.a=P3) -------------------------------------------------------- P(s1.a=P2 | P2.$$) P(s1.s | P2.f,S1.a=P2) When aggregation function (e.g.,max, softmax)

45. Conclusions PRMs can represent distribution over attributes from multiple tables PRMs can capture link uncertainty PRMs allow inferences about individuals while taking into account relational structure (they do not make inapproriate independence assuptions)

46. Next Time Dynamic Bayesian Networks

47. THE END

48. Selected Publications “Learning Probabilistic Models of Link Structure”, L. Getoor, N. Friedman, D. Koller and B. Taskar, JMLR 2002. “Probabilistic Models of Text and Link Structure for Hypertext Classification”, L. Getoor, E. Segal, B. Taskar and D. Koller, IJCAI WS ‘Text Learning: Beyond Classification’, 2001. “Selectivity Estimation using Probabilistic Models”, L. Getoor, B. Taskar and D. Koller, SIGMOD-01. “Learning Probabilistic Relational Models”, L. Getoor, N. Friedman, D. Koller, and A. Pfeffer, chapter in Relation Data Mining, eds. S. Dzeroski and N. Lavrac, 2001. –see also N. Friedman, L. Getoor, D. Koller, and A. Pfeffer, IJCAI-99. “Learning Probabilistic Models of Relational Structure”, L. Getoor, N. Friedman, D. Koller, and B. Taskar, ICML-01. “From Instances to Classes in Probabilistic Relational Models”, L. Getoor, D. Koller and N. Friedman, ICML Workshop on Attribute-Value and Relational Learning: Crossing the Boundaries, 2000. Notes from AAAI Workshop on Learning Statistical Models from Relational Data, eds. L.Getoor and D. Jensen, 2000. Notes from IJCAI Workshop on Learning Statistical Models from Relational Data, eds. L.Getoor and D. Jensen, 2003. See http://www.cs.umd.edu/~getoor

49. Queries Full joint distribution specifies answer to any query: P(variable | evidence about others) XRay Lung Infiltrates Sputum Smear TuberculosisPneumonia XRay Sputum Smear