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CS498-EA Reasoning in AI Lecture #9

CS498-EA Reasoning in AI Lecture #9. Instructor: Eyal Amir Fall Semester 2011. Previously. First-Order Logic Syntax: Well-Founded Formulas Semantics: Models, Satisfaction, Entailment Models of FOL: how many, sometimes unexpected Resolution in FOL Resolution rule Unification Clausal form

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CS498-EA Reasoning in AI Lecture #9

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  1. CS498-EAReasoning in AILecture #9 Instructor: Eyal Amir Fall Semester 2011

  2. Previously • First-Order Logic • Syntax: Well-Founded Formulas • Semantics: Models, Satisfaction, Entailment • Models of FOL: how many, sometimes unexpected • Resolution in FOL • Resolution rule • Unification • Clausal form • COMPLETENESS, SOUNDNESS

  3. Pop Quiz (5 min) • Question: how many non-equivalent propositional theories of n variables are there? • After you’re done: Discussion

  4. Today • Probabilistic graphical models • Bayesian Networks • Markov Random Fields • Next week: Treewidth methods: • Variable elimination • Clique tree algorithm

  5. Summary of Our Goal

  6. Probability • A sample space Omega (O) is a set of outcomes of a random experiment • A probability P is a function from a sigma-field (e.g., all measurable subsets) A on O (the events) to [0,1]. • A random variable X is a function X:OR such that for all B Borel set in R, X-1(B) is in A.

  7. Independent Random Variables • Two variables X and Y are independent if • P(X = x|Y = y) = P(X = x) for all values x,y • That is, learning the values of Y does not change prediction of X • If X and Y are independent then • P(X,Y) = P(X|Y)P(Y) = P(X)P(Y) • In general, if X1,…,Xp are independent, then P(X1,…,Xp)= P(X1)...P(Xp) • Requires O(n) parameters

  8. Conditional Independence • Unfortunately, most of random variables of interest are not independent of each other • A more suitable notion is that of conditional independence • Two variables X and Y are conditionally independent given Z if • P(X = x|Y = y,Z=z) = P(X = x|Z=z) for all values x,y,z • That is, learning the values of Y does not change prediction of X once we know the value of Z • notation: I ( X , Y | Z )

  9. Marge Homer Lisa Maggie Bart Example: Family trees Noisy stochastic process: Example: Pedigree • A node represents an individual’sgenotype • Modeling assumptions: • Ancestors can effect descendants' genotype only by passing genetic materials through intermediate generations

  10. Y1 Y2 X Non-descendent Markov Assumption Ancestor Parent • We now make this independence assumption more precise for directed acyclic graphs (DAGs) • Each random variable X, is independent of its non-descendents, given its parents Pa(X) • Formally,I (X, NonDesc(X) | Pa(X)) Non-descendent Descendent

  11. Burglary Earthquake Radio Alarm Call Markov Assumption Example • In this example: • I ( E, B ) • I ( B, {E, R} ) • I ( R, {A, B, C} | E ) • I ( A, R | B,E ) • I ( C, {B, E, R} | A)

  12. X Y X Y I-Maps • A DAG G is an I-Map of a distribution P if all Markov assumptions implied by G are satisfied by P (Assuming G and P both use the same set of random variables) Examples:

  13. X Y Factorization • Given that G is an I-Map of P, can we simplify the representation of P? • Example: • Since I(X,Y), we have that P(X|Y) = P(X) • Applying the chain ruleP(X,Y) = P(X|Y) P(Y) = P(X) P(Y) • Thus, we have a simpler representation of P(X,Y)

  14. THE END

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