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Unformatted text preview: CSci 5512: Artificial Intelligence II Instructor: Arindam Banerjee January 23, 2012 El
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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
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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
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Q Instructor: Arindam Bancrycc Example (Contd.) A _(JA)
@un @E 0'0“
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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(bj,um)? El
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@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
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Q Local semantics Each node is conditionally independent of its
nondescendants given its parents El
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Q Instructor: Arindam Bancrjcc Markov blanket Conditional Independence (Contd.) 0 o
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(a) (b) (c) (d) Which BNs support x J. yz
o For (a)(b), z is not a collider, so x J. yz 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
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El
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Q Instructor: Arindam Bancrycc Instructor: Arindam Bancrycc Conditional Independence in EMS d—connection, d—separation @ a 6 ® a a e a 0 Definition (dconnection): 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 dseparated by Z 0 If Z dseparates X and Y, then X J_ YZ for all distributions
represented by the graph Which BNs support x1 J. X2X3 o For (a), X1,X2 are dependent, X3 is a collider
o For (b)(d), X1 J. X2X3 El
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El
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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
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Q Instructor: Arindam Bancrycc Conditional Independence: More Examples
0 (a) (b) o For (a), Is a J. ce? Is a J. eb? Is a J. ec?
o For (b), Is a J. ed? Is a J. ec? Is a J. cb 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/.‘\
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'mgﬂm
EV ’V'z“\ m El
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Q Instructor: Armdam Bancucc Inference F F .01 How can we compute P(bU,—um)? El
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