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# lec3 - Inference m E P(AlB,E.95.94.29.001 Instructor...

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Unformatted text preview: Inference m E P(AlB,E) .95 .94 .29 .001 . . Instructor: Arindam Banerjee E F F .01 CSci 5512: Artificial Intelligence II map-1H a mama January 21, 2012 How can we compute P(blj, m)? Instructor: Arindam Banerjee Instructor: Arindam Banerjee — Overview: Inference Tasks Inference by Enumeration 0 Simple query can be answered using Bayes rule a From Bayes Rule P(b, j, m) P(J', m) a Each marginal can be obtained from the joint distribution P(in, m) = 0 Simple Queries: Compute posterior marginals P(blj,—um) o Conjunctive Queries: Compute joint marginals P(b,-ue|j,-um) 0 Optimal Decisions: Compute P(outcome|action, evidence) P(b,j,m) = ZZPU’a 5,14,], m) — E A 0 Value of Information: Which evidence to seek next? I _ . Pom) = ZZZP(B,E,A,J,m) o Explanation: Why do I need a new starter motor? B E A o Each term can be written as product of conditionals P(b, 5,14,], m) = P(b)P(E)P(AIba E)P(JIA)P(mIA) o The complexity of the simple approach is 0(n2") Instructor: Arindam Banerjee Instructor: Arindam Banerjee 0 Complexity can be improved by a simple observation P(bU, m) = P01 m) 22 P(b, mm m) ’ E A _ 1 _ POEM) = .1 P(b) 219(5) ZP(A|E,b)P(le)P(mIA) Po’m) E A Z Z P(b)P(E)P(AIE, b)P(j|A)P(m|A) E A 0 Complexity is 0(2") 0 Some computations are repeated . Enumeration Tree for P(bU, m) o P(j|a)P(m|a) and P(j|-ua)P(m|-ua) are computed twice Instructor: Arindam Banerjee Instructor: Arindam Banerjee — Burglary Network Variable Elimination o Queries can be written as sum-of—products 0 Main idea o Sum over products to eliminate variables 0 Storage required to prevent repeat computations Earthquake m o Burglary network P(bli,m) 1 T .90 .0 = . Pb PE PAb,EP'APmA Puma; g3 ;<IA will”) 0 Have a factor for every variable Instructor: Arindam Banerjee Instructor: Arindam Banerjee Factors for Variable Elimination Example: Pointwise Product of Factors 0 Factor for M: fm(A) = [P(m|a) P(m|-Ia)] 0 Factor for J: fJ-(A) = [P(j|a) P(j|-Ia)] E B 0 Similarly, factors fA(B, E,A),fE(E),fB(B) T T 0.3 T T 0.2 T T T 0.3 x 0.2 o Summing out to eliminate variables T F 0-7 T F 0-8 T T F 0-3 X 0-8 F T 0.9 F T 0.6 T F T 0.7 X 0.6 fAjm(B, E) = Ems, E,A)fj(A)fm(A) F F 0.1 F F o 4 T F F 0.7 x 0.4 A F T T 0.9 X 0.2 f___ B = f Ef_. B,E F T F 0.9X0.8 EAJm( ) g E( )Ajm( ) F F T 0.1 X 1 F F F 0.1 x 0.4 Paw, m) — P0,m)fB(B)f,—E,—.jm(3) Instructor: Arindam Banerjee Instructor: Arindam Banerjee — Variable Elimination: Basic Operations Complexity of Exact Inference 0 Sineg connected networks (or polytrees) a Any two nodes are connected by at most one path 0 Time and space cost of variable elimination are 0(dkn) o Summing out a variable from a product of factors :- Pointwise products of (a pair of) factors 0 Sum out a variable from a product of factors 0 Pointwise product of factors 0 Multiply connected networks :- f1(a, b) x f2(b, c) = f(a, b, c) a Can reduce 3SAT to exact inference => NP-hard o In general 0 Equivalent to counting 3SAT models => #P—complete f1(X1,...,Xj,y1,...,yk) Xf2(y1,...,yk,zl,...,z,) 0.5 0.5 0.5 0.5 = f(x1,...,)g,y1,...,yk,21,...,z,) 0 Assuming f1,...,f,- do not depend onX 1_ Ava V C Zﬁx..-ka = ﬁx..-Xf}x<Zﬁ-+lx...xfk) 2.CVDV‘H\ X X 3.Bva—D fix...xf}xf)—( Instructor: Arindam Banerjee Instructor: Arindam Banerjee Inference Problems: Big Picture 0 Broadly the following types of problems Compute likelihood of observations Compute marginals P(xA) on subset A of nodes Compute conditionals P(xA|xB) on disjoint subsets A, B Compute mode argmaxx P(x) 0 First 3 problems are similar 0 Involves marginalization over sum—of—products 0 Last problem is fundamentally different a Entails maximization rather than marginalization 0 There is an important connection between the problems Instructor: Arindam Banerjee ...
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lec3 - Inference m E P(AlB,E.95.94.29.001 Instructor...

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