Relational Probabilistic Models

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1 Relational Probabilistic Models flat or modular or hierarchical explicit states or features or individuals and relations static or finite stage or indefinite stage or infinite stage fully observable or partially observable deterministic or stochastic dynamics goals or complex preferences single agent or multiple agents knowledge is given or knowledge is learned perfect rationality or bounded rationality c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 1

2 Relational Probabilistic Models Often we want a random variable for each individual in a population build a probabilistic model before knowing the individuals learn the model for one set of individuals apply the model to new individuals allow complex relationships between individuals c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 2

3 Predicting students errors x 2 x 1 + y 2 y 1 z 3 z 2 z 1 c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 3

4 Predicting students errors Knows_Carry Knows_Add x 2 x 1 + y 2 y 1 z 3 z 2 z 1 C 2 X 1 Y 1 C 1 X 0 Y 0 Z 2 Z 1 Z 0 c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 4

5 Predicting students errors Knows_Carry Knows_Add x 2 x 1 + y 2 y 1 z 3 z 2 z 1 C 2 X 1 Y 1 C 1 X 0 Y 0 Z 2 Z 1 Z 0 What if there were multiple digits c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 5

6 Predicting students errors Knows_Carry Knows_Add x 2 x 1 + y 2 y 1 z 3 z 2 z 1 C 2 X 1 Y 1 C 1 X 0 Y 0 Z 2 Z 1 Z 0 What if there were multiple digits, problems c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 6

7 Predicting students errors Knows_Carry Knows_Add x 2 x 1 + y 2 y 1 z 3 z 2 z 1 C 2 X 1 Y 1 C 1 X 0 Y 0 Z 2 Z 1 Z 0 What if there were multiple digits, problems, students c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 7

8 Predicting students errors Knows_Carry Knows_Add x 2 x 1 + y 2 y 1 z 3 z 2 z 1 C 2 X 1 Y 1 C 1 X 0 Y 0 Z 2 Z 1 Z 0 What if there were multiple digits, problems, students, times? c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 8

9 Predicting students errors Knows_Carry Knows_Add x 2 x 1 + y 2 y 1 z 3 z 2 z 1 C 2 X 1 Y 1 C 1 X 0 Y 0 Z 2 Z 1 Z 0 What if there were multiple digits, problems, students, times? How can we build a model before we know the individuals? c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 9

10 Multi-digit addition with parametrized BNs / plates S,T knows_carry(s,t) knows_add(s,t) x jx x 2 x 1 + y jz y 2 y 1 z jz z 2 z 1 D,P x(d,p) y(d,p) c(d,p,s,t) z(d,p,s,t) Parametrized Random Variables: x(d, P), y(d, P), knows carry(s, T ), knows add(s, T ), c(d, P, S, T ), z(d, P, S, T ) for digit D, problem P, student S, time T. There is a random variable for each assignment of a value to D and a value to P in x(d, P).... c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 10

11 Parametrized Bayesian networks / Plates Parametrized Bayes Net: r(x) + Individuals: i 1,...,i k X Bayes Net... r(i 1 ) r(i k ) c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 11

12 Parametrized Bayesian networks / Plates (2) r(x) q q r(i 1 )... r(i k ) s(x) t Individuals: i 1,...,i k X s(i ) 1... s(i ) k t c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 12

13 Creating Dependencies Instances of plates are independent, except by common parents or children. q q Common Parents r(x) X r(i 1 )... r(i k ) Observed Children r(x) X r(i 1 )... r(i k ) q q c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 13

14 Overlapping plates Person young likes genre Movie Relations: likes(p, M), young(p), genre(m) likes is Boolean, young is Boolean, genre has range {action, romance, family} c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 14

15 Overlapping plates Person y(s) g(r) g(t) young likes genre y(c) y(k) l(s,r) l(c,r) l(s,t) l(c,t) Movie l(k,r) l(k,t) Relations: likes(p, M), young(p), genre(m) likes is Boolean, young is Boolean, genre has range {action, romance, family} Three students: sam (s), chris (c), kim (k) Two movies: rango (r), terminator (t) c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 15

16 Representing Conditional Probabilities P(likes(P, M) young(p), genre(m)) parameter sharing individuals share probability parameters. P(happy(X ) friend(x, Y ), mean(y )) needs aggregation happy(a) depends on an unbounded number of parents. the carry of one digit depends on carry of the previous digit probability that two authors collaborate depends on whether they have a paper authored together c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 16

17 Creating Dependencies: Exploit Domain Structure r(x) s(x) X r(i 1 ) r(i 2 ) r(i 3 ) r(i 4 ) s(i 1 ) s(i 2 ) s(i 3 )... c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 17

18 Creating Dependencies: Relational Structure author(a,p) P A' collaborators(a,a') A author(a i,p j ) author(a k,p j ) a i A a k A a i a k p j P collaborators(a i,a k ) c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 18

19 Independent Choice Logic A language for first-order probabilistic models. Idea : combine logic and probability, where all uncertainty in handled in terms of Bayesian decision theory, and a logic program specifies consequences of choices. Parametrized random variables are represented as logical atoms, and plates correspond to logical variables. c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 19

20 Independent Choice Logic An alternative is a set of ground atomic formulas. C, the choice space is a set of disjoint alternatives. F, the facts is a logic program that gives consequences of choices. P 0 a probability distribution over alternatives: A C a A P 0 (a) = 1. c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 20

21 Meaningless Example C = {{c 1, c 2, c 3 }, {b 1, b 2 }} F = { f c 1 b 1, f c 3 b 2, d c 1, d c 2 b 1, e f, e d} P 0 (c 1 ) = 0.5 P 0 (c 2 ) = 0.3 P 0 (c 3 ) = 0.2 P 0 (b 1 ) = 0.9 P 0 (b 2 ) = 0.1 c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 21

22 Semantics of ICL There is a possible world for each selection of one element from each alternative. The logic program together with the selected atoms specifies what is true in each possible world. The elements of different alternatives are independent. c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 22

23 Meaningless Example: Semantics F = { f c 1 b 1, f c 3 b 2, d c 1, d c 2 b 1, e f, e d} P 0 (c 1 ) = 0.5 P 0 (c 2 ) = 0.3 P 0 (c 3 ) = 0.2 P 0 (b 1 ) = 0.9 P 0 (b 2 ) = 0.1 selection {}}{ logic program {}}{ w 1 = c 1 b 1 f d e P(w 1 ) = 0.45 w 2 = c 2 b 1 f d e P(w 2 ) = 0.27 w 3 = c 3 b 1 f d e P(w 3 ) = 0.18 w 4 = c 1 b 2 f d e P(w 4 ) = 0.05 w 5 = c 2 b 2 f d e P(w 5 ) = 0.03 w 6 = c 3 b 2 f d e P(w 6 ) = 0.02 P(e) = = 0.77 c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 23

24 Belief Networks, Decision trees and ICL rules Ta There is a local mapping from belief networks into ICL. Al Le Re Fi Sm prob ta : prob fire : alarm ta fire atf. alarm ta fire antf. alarm ta fire atnf. alarm ta fire antnf. prob atf : 0.5. prob antf : prob atnf : prob antnf : smoke fire sf. prob sf : 0.9. smoke fire snf. prob snf : c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 24

25 Belief Networks, Decision trees and ICL rules Rules can represent decision tree with probabilities: 0.3 f C 0.5 A t B D P(e A,B,C,D) e a b h 1. P 0 (h 1 ) = 0.7 e a b h 2. P 0 (h 2 ) = 0.2 e a c d h 3. P 0 (h 3 ) = 0.9 e a c d h 4. P 0 (h 4 ) = 0.5 e a c h 5. P 0 (h 5 ) = 0.3 c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 25

26 Movie Ratings Person young likes genre Movie prob young(p) : 0.4. prob genre(m, action) : 0.4, genre(m, romance) : 0.3, genre(m, family) : 0.4. likes(p, M) young(p) genre(m, G) ly(p, M, G). likes(p, M) young(p) genre(m, G) lny(p, M, G). prob ly(p, M, action) : 0.7. prob ly(p, M, romance) : 0.3. prob ly(p, M, family) : 0.8. prob lny(p, M, action) : 0.2. prob lny(p, M, romance) : 0.9. prob lny(p, M, family) : 0.3. c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 26

27 Example: Multi-digit addition S,T knows_carry(s,t) knows_add(s,t) x jx x 2 x 1 + y jz y 2 y 1 z jz z 2 z 1 D,P x(d,p) y(d,p) c(d,p,s,t) z(d,p,s,t) c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 27

28 ICL rules for multi-digit addition z(d, P, S, T ) = V x(d, P) = Vx y(d, P) = Vy c(d, P, S, T ) = Vc knows add(s, T ) mistake(d, P, S, T ) V is (Vx + Vy + Vc) div 10. z(d, P, S, T ) = V knows add(s, T ) mistake(d, P, S, T ) selectdig(d, P, S, T ) = V. z(d, P, S, T ) = V knows add(s, T ) selectdig(d, P, S, T ) = V. Alternatives: DPST {nomistake(d, P, S, T ), mistake(d, P, S, T )} DPST {selectdig(d, P, S, T ) = V V {0..9}} c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 28

29 Hidden Variables Student Course Grade s 1 c 1 A s 2 c 1 C s 1 c 2 B s 2 c 3 B s 3 c 2 B s 4 c 3 B s 3 c 4? s 4 c 4? c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 29

30 Hidden Variables Student Course Grade s 1 c 1 A s 2 c 1 C s 1 c 2 B s 2 c 3 B s 3 c 2 B s 4 c 3 B s 3 c 4? s 4 c 4? int(s) diff (C) grade(s, C) A B C true true true false false true false false P(int(S)) = 0.5 P(diff (C)) = 0.5 c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 30

31 Hidden Variables Student Course Grade s 1 c 1 A s 2 c 1 C s 1 c 2 B s 2 c 3 B s 3 c 2 B s 4 c 3 B s 3 c 4? s 4 c 4? int(s) diff (C) grade(s, C) A B C true true true false false true false false P(int(S)) = 0.5 P(diff (C)) = 0.5 P(grade(s3, c4, a) Obs) = 0.491, P(grade(s3, c4, b) Obs) = 0.245, P(grade(s3, c4, c) Obs) = P(grade(s4, c4, a) Obs) = 0.264, P(grade(s4, c4, b) Obs) = 0.245, P(grade(s4, c4, c) Obs) = c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 31

32 Learning Relational Models with Hidden Variables User Item Date Rating Sam Terminator Sam Rango Sam The Holiday Chris The Holiday Netflix: 500,000 users, 17,000 movies, 100,000,000 ratings. c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 32

33 Learning Relational Models with Hidden Variables User Item Date Rating Sam Terminator Sam Rango Sam The Holiday Chris The Holiday Netflix: 500,000 users, 17,000 movies, 100,000,000 ratings. r ui = rating of user u on item i rˆ ui = predicted rating of user u on item i D = set of (u, i, r) tuples in the training set Sum squares error: ( rˆ ui r) 2 (u,i,r) D c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 33

34 Learning Relational Models with Hidden Variables Predict same for all ratings: rˆ ui = µ c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 34

35 Learning Relational Models with Hidden Variables Predict same for all ratings: rˆ ui = µ Adjust for each user and item: rˆ ui = µ + b i + c u c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 35

36 Learning Relational Models with Hidden Variables Predict same for all ratings: rˆ ui = µ Adjust for each user and item: rˆ ui = µ + b i + c u One hidden feature: f i for each item and g u for each user rˆ ui = µ + b i + c u + f i g u c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 36

37 Learning Relational Models with Hidden Variables Predict same for all ratings: rˆ ui = µ Adjust for each user and item: rˆ ui = µ + b i + c u One hidden feature: f i for each item and g u for each user rˆ ui = µ + b i + c u + f i g u k hidden features: rˆ ui = µ + b i + c u + k f ik g ku c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 37

38 Learning Relational Models with Hidden Variables Predict same for all ratings: rˆ ui = µ Adjust for each user and item: rˆ ui = µ + b i + c u One hidden feature: f i for each item and g u for each user rˆ ui = µ + b i + c u + f i g u k hidden features: rˆ ui = µ + b i + c u + k f ik g ku Regularize minimize (u,i) K(µ + b i + c u + k + λ(bi 2 + cu 2 + k f ik g ku r ui ) 2 f 2 ik + g 2 ku ) c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 38

39 Parameter Learning using Gradient Descent µ average rating assign f [i, k], g[k, u] randomly assign b[i], c[u] arbitrarily repeat: for each (u, i, r) D: e µ + b[i] + c[u] + k f [i, k] g[k, u] r b[i] b[i] η e η λ b[i] c[u] c[u] η e η λ c[u] for each feature k: f [i, k] f [i, k] η e g[k, u] η λ f [i, k] g[k, u] g[k, u] η e f [i, k] η λ g[k, u] c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 39

Learning Objectives. c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 1

Learning Objectives. c D. Poole and A. Mackworth 2010 Artificial Intelligence, Lecture 14.3, Page 1 Learning Objectives At the end of the class you should be able to: describe the mapping between relational probabilistic models and their groundings read plate notation build a relational probabilistic