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Unformatted text preview: Inference m E P(AlB,E) .95
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Instructor: Arindam Banerjee E
F F .01 CSci 5512: Artificial Intelligence II map1H 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,uej,um) 0 Optimal Decisions: Compute P(outcomeaction, 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(AE,b)P(le)P(mIA)
Po’m) E A Z Z P(b)P(E)P(AIE, b)P(jA)P(mA)
E A 0 Complexity is 0(2")
0 Some computations are repeated . Enumeration Tree for P(bU, m)
o P(ja)P(ma) and P(jua)P(mua) are computed twice Instructor: Arindam Banerjee Instructor: Arindam Banerjee —
Burglary Network Variable Elimination o Queries can be written as sumof—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(ma) P(mIa)] 0 Factor for J: fJ(A) = [P(ja) P(jIa)] 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 07 T F 08 T T F 03 X 08
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 => NPhard
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(xAxB) 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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This note was uploaded on 02/07/2012 for the course CSCI 5512 taught by Professor Staff during the Spring '08 term at Minnesota.
 Spring '08
 Staff

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