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lec2 - CSci 5512 Artificial Intelligence II Instructor...

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Unformatted text preview: CSci 5512: Artificial Intelligence II Instructor: Arindam Banerjee January 23, 2012 El Q I III M b p Q Instructor: Arindam Bancrycc Example Topology of network encodes conditional independence assertions Toothache @ 0 Weather is independent of the other variables 0 Toothache, Catch are conditionally independent given Cavity El Q I III M b p Q Instructor: Arindam Bancrycc — Bayesian networks A simple, graphical notation for conditional independence assertions and hence for compact specification of full joint distributions 0 Syntax o A set of nodes, one per variable a A directed, acyclic graph (link implies direct influence) 0 A conditional distribution for each node given its parents 0 Conditional distributions 0 For each X,-, P(X,-|Parents(X,-)) o In the form of a conditional probability table (CPT) 0 Distribution of X,- for each combination of parent values El 5' ' _= ORG“ Instructor: Arindam Bancrycc Example I 'm at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn't call. Sometimes it's set off by minor earthquakes. Is there a burglar? 0 Variables: Burglar, Earthquake, Alarm, JohnCalls, MaryCalls 0 Network topology reflects ”causal" knowledge o A burglar can set the alarm off 0 An earthquake can set the alarm off a The alarm can cause Mary to call o The alarm can cause John to call El Q I III M b p Q Instructor: Arindam Bancrycc Example (Contd.) A _(J|A) @un @E- 0'0“ .0- El Q I III M b p Q Instructor: Arindam Bancrjcc Global semantics 0 Full joint distribution 0 Can be written as product of local conditionals 0 Example: PU, m,ambfle) = P(fib)P(fie)P(alfibafie)PUl3)P(mla) 0 Example: PUfim, 3, bafie) = P(b)P(ne)P(alb,fie)P(jl8)P(nMIa) 0 Can we compute P(b|j,-um)? El Q I III M b p Q Instructor: Arindam Bancrjcc ;@ @fifii) o A CPT for Boolean X,- with k Boolean parents o 2" rows for the combinations of parent values 0 Each row requires one number 0 Each variable has no more than k parents 0 The complete network requires 0(n - 2") numbers - Grows linearly with n 0 Full joint distribution requires 0(2") 0 Example: Burglary network 0 Full joint distribution requires 25 — 1 = 31 numbers o Bayes net requires 10 numbers El .5: — —= Instructor: Arindam Bancrjcc MI 0 p Q Local semantics Each node is conditionally independent of its nondescendants given its parents El Q I III M b p Q Instructor: Arindam Bancrjcc Markov blanket Conditional Independence (Contd.) 0 o Gib/Q Q69 ®\'®/@ @ 9 (a) (b) (c) (d) Which BNs support x J. y|z o For (a)-(b), z is not a collider, so x J. y|z Each node is conditionally independent of all others given its Markov blanket, i.e., parents + children + children '5 parents 0 For (c), z is a collider, so x and y are conditionally dependent o For (d), w is a collider, and z is a descendent of w, so x and y are conditionally dependent El Q I III M b p ’9 El Q I III M b p Q Instructor: Arindam Bancrycc Instructor: Arindam Bancrycc Conditional Independence in EMS d—connection, d—separation @ a 6 ® a a e a 0 Definition (d-connection): X, Y,Z be disjoint sets of vertices in a directed graph G. X, Y is d—connected by Z iff El an a 9 9 @ undirected path U between some x e X,y E Y such that for every collider C on U (a) (b) (c) (d) 0 Either C or a descendent of C is in Z I No non—collider on U is in Z 0 Otherwise X and Y are d-separated by Z 0 If Z d-separates X and Y, then X J_ Y|Z for all distributions represented by the graph Which BNs support x1 J. X2|X3 o For (a), X1,X2 are dependent, X3 is a collider o For (b)-(d), X1 J. X2|X3 El Q I III M b p ’9 El Q I III M b p Q Instructor: Arindam Bancrycc Instructor: Arindam Bancrycc Conditional Independence (Contd.) Examples 0 For (a), a J. e o For (b) a and e are dependent given b; c and e are unconditionally dependent b; but a, e are dependent given {b, d} Instructor: Arindam Bancrycc Constructing Bayesian networks 0 Choose an ordering of variables X1, . . . ,X,, oFori=lton 0 Add X,- to the network 0 Select parents from X1, . . . 7X,-_1 such that P(X,-|Parents(X,-)) = P(XiIX1, - - - ,Xi—1) This choice of parents guarantees global semantics P(X1,...,X,,) n H P(X,-|X1, . . . ,X,-_1) i=1 H P(X,-|Parents(X,-)) i=1 El Q I III M b p Q Instructor: Arindam Bancrycc Conditional Independence: More Examples 0 (a) (b) o For (a), Is a J. c|e? Is a J. e|b? Is a J. e|c? o For (b), Is a J. e|d? Is a J. e|c? Is a J. c|b Instructor: Arindam Bancrycc Example: Car diagnosis Initial evidence: car won't start Testable variables (green), ”broken, so fix it" variables (orange) Hidden variables (gray) ensure sparse structure, reduce parameters Instructor: Arindam Bancrycc Example: Car insurance «w V-/.‘\ V @ \‘\\ 'mgflm EV ’V'z“\ m El Q I III M b p Q Instructor: Armdam Bancucc Inference F F .01 How can we compute P(bU,—um)? El Q I III M b p Q Instructor: Armdam Bancucc ...
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