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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 model for a domain D. Poole and A.

  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 model for a domain � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 1

  2. 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 � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 2

  3. Relational Probabilistic Models Often we want random variables for combinations of individual in populations 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 � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 3

  4. Example: Predicting Relations 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 Students s 3 and s 4 have the same averages, on courses with the same averages. Why should we make different predictions? How can we make predictions when the values of properties Student and Course are individuals? � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 4

  5. From Relations to Belief Networks Gr(s1, c1) I(s1) D(c1) Gr(s2, c1) I(s2) Gr(s1, c2) D(c2) Gr(s2, c3) Gr(s3, c2) D(c3) I(s3) Gr(s3, c4) D(c4) Gr(s4, c3) I(s4) Gr(s4, c4) � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 5

  6. From Relations to Belief Networks Gr(s1, c1) I(s1) D(c1) Gr(s2, c1) I ( S ) D ( C ) Gr ( S , C ) A B C I(s2) Gr(s1, c2) true true 0 . 5 0 . 4 0 . 1 D(c2) Gr(s2, c3) true false 0 . 9 0 . 09 0 . 01 0 . 01 0 . 1 0 . 9 false true Gr(s3, c2) D(c3) I(s3) false false 0 . 1 0 . 4 0 . 5 Gr(s3, c4) P ( I ( S )) = 0 . 5 D(c4) Gr(s4, c3) P ( D ( C )) = 0 . 5 I(s4) Gr(s4, c4) “parameter sharing” � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 6

  7. Plate Notation S D(C) I(S) Gr(S,C) C S is a logical variable representing students C is a logical variable representing courses the set of all individuals of some type is called a population I ( S ), Gr ( S , C ), D ( C ) are parametrized random variables � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 7

  8. Plate Notation S D(C) I(S) Gr(S,C) C S is a logical variable representing students C is a logical variable representing courses the set of all individuals of some type is called a population I ( S ), Gr ( S , C ), D ( C ) are parametrized random variables for every student s , there is a random variable I ( s ) for every course c , there is a random variable D ( c ) for every student s and course c pair there is a random variable Gr ( s , c ) all instances share the same structure and parameters � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 8

  9. Plate Notation for Learning Parameters 휽 휽 tosses t1, t2…tn ... H(T) H(t1) H(t2) H(tn) T T is a logical variable representing tosses of a thumb tack H ( t ) is a Boolean variable that is true if toss t is heads. θ is a random variable representing the probability of heads. Range of θ is { 0 . 0 , 0 . 01 , 0 . 02 , . . . , 0 . 99 , 1 . 0 } or interval [0 , 1]. P ( H ( t i )= true | θ = p ) = � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 9

  10. Plate Notation for Learning Parameters 휽 휽 tosses t1, t2…tn ... H(T) H(t1) H(t2) H(tn) T T is a logical variable representing tosses of a thumb tack H ( t ) is a Boolean variable that is true if toss t is heads. θ is a random variable representing the probability of heads. Range of θ is { 0 . 0 , 0 . 01 , 0 . 02 , . . . , 0 . 99 , 1 . 0 } or interval [0 , 1]. P ( H ( t i )= true | θ = p ) = p H ( t i ) is independent of H ( t j ) (for i � = j ) given θ : i.i.d. or independent and identically distributed. � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 10

  11. Parametrized belief networks Allow random variables to be parametrized. interested ( X ) Parameters correspond to logical variables. X Parameters can be drawn as plates. Each logical variable is typed with a population. X : person A population is a set of individuals. Each population has a size. | person | = 1000000 Parametrized belief network means its grounding: an instance of each random variable for each assignment of an individual to a logical variable. interested ( p 1 ) . . . interested ( p 1000000 ) Instances are independent (but can have common ancestors and descendants). � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 11

  12. Parametrized Bayesian networks / Plates Parametrized Bayes Net: r(X) Bayes Net X ... + r(i1) r(ik) Individuals : i1,...,ik � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 12

  13. Parametrized Bayesian networks / Plates (2) q q ... r(X) r(i1) r(ik) s(X) ... X s(i1) s(ik) t t Individuals : i1,...,ik � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 13

  14. Creating Dependencies Instances of plates are independent, except by common parents or children. q q Common .... r(X) Parents r(i1) r(ik) X Observed .... r(X) Children r(i1) r(ik) X q q � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 14

  15. Overlapping plates Person young genre likes Movie Relations: likes ( P , M ), young ( P ), genre ( M ) likes is Boolean, young is Boolean, genre has range { action , romance , family } � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 15

  16. Overlapping plates Person g(r) g(t) y(s) young l(s,r) l(s,t) y(c) genre likes l(c,r) l(c,t) y(k) 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 people: sam (s), chris (c), kim (k) Two movies: rango (r), terminator (t) � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 16

  17. Overlapping plates Person young genre likes Movie Relations: likes ( P , M ), young ( P ), genre ( M ) likes is Boolean, young is Boolean, genre has range { action , romance , family } If there are 1000 people and 100 movies, Grounding contains: random variables � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 17

  18. Overlapping plates Person young genre likes Movie Relations: likes ( P , M ), young ( P ), genre ( M ) likes is Boolean, young is Boolean, genre has range { action , romance , family } If there are 1000 people and 100 movies, Grounding contains: 100,000 likes + 1,000 age + 100 genre = 101,100 random variables How many numbers need to be specified to define the probabilities required? � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 18

  19. Overlapping plates Person young genre likes Movie Relations: likes ( P , M ), young ( P ), genre ( M ) likes is Boolean, young is Boolean, genre has range { action , romance , family } If there are 1000 people and 100 movies, Grounding contains: 100,000 likes + 1,000 age + 100 genre = 101,100 random variables How many numbers need to be specified to define the probabilities required? 1 for young , 2 for genre , 6 for likes = 9 total. � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 19

  20. 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. There can be more structure about the individuals. . . � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 20

  21. Example: Aggregation x Has_gun(x) Has_motive(x,y) Has_opportunity(x,y) Shot(x,y) Someone_shot(y) y � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 21

  22. Exercise #1 For the relational probabilistic model: a b c X Suppose the the population of X is n and all variables are Boolean. (a) How many random variables are in the grounding? (b) How many numbers need to be specified for a tabular representation of the conditional probabilities? � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 22

  23. Exercise #2 For the relational probabilistic model: a c b d X Suppose the the population of X is n and all variables are Boolean. (a) Which of the conditional probabilities cannot be defined as a table? (b) How many random variables are in the grounding? (c) How many numbers need to be specified for a tabular representation of those conditional probabilities that can be defined using a table? (Assume an aggregator is an “or” which uses no numbers). � D. Poole and A. Mackworth 2010 c Artificial Intelligence, Lecture 14.3, Page 23

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