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Cmput 651 - Learning Undirected Models {14,17}/11/2008 !"#$%$&'&()&*+,"%-.&*%'+/#01'(+234-5)+6789: ;1%"<&<=+><0&"1*)10+/#01'( /%)).1?+@"#?< A8BC8DEF88FGHHI J1%0&<=:+K#''1"LM"&104%<+3.N+8O 8 P-%*1+#Q+)#-&*( P14%<)&*( S<Q1"1<*1 ;1%"<&<= R&(*"1)1 R&"1*)10 ><R&"1*)10 3#<)&<5#5( G

Cmput 651 - Learning Undirected Models {14,17}/11/2008 ;1%"<&<=+/%"T#U+<1)( *#4-'1)1 -%")&%' 0%)% 0%)% T<#?< <#)+%(+1%(X+ .%"0 %(+Q#"+@%X1(+ ()"5*)5"1 <1)( 5<T<#?< .%"0 U1"X+.%"0 ()"5*)5"1 \.&(+'1*)5"1 [ ;1%"<&<=+/%"T#U+<1)(+&(+1V-1<(&U1 P F ( X ) = 1 /%"T#U+Q%*)#"&W%)&#<: � φ i ( scope ( φ i )) Z φ i ∈ F L+-%")&)&#<+Q5<*)&#<+*#5-'1(+%''+).1+Q%*)#"( *%<<#)+(1-%"%)1+-%"%41)1"+1()&4%)&#<+&<)#+'#*%'+="#5-( L+<#+*'#(10+Q#"4+(#'5)&#<( L+1U1<+Q#"+4%VN+'&T1'&.##0+?F+*#4-'1)1+0%)% 2"1*%''+).&(+&(+1%(X+Q#"+@%X1(+<1)(9 L+'1%"<&<=+$%(10+#<+&)1"%)&#< L+&<Q1"1<*1+"1Y5&"10+&<+1%*.+()1-+LZ+1V-1<(&U1 L+$5)+*#<U1V+2Q#"+*#4-'1)1+0%)%9 L+()"5*)5"1+'1%"<&<=+%'(#+1V-1<(&U1 B

Cmput 651 - Learning Undirected Models {14,17}/11/2008 ]5)'&<1 !"#"$%&%#'(%"#)*)+ P)"5*)5"1+'1%"<&<= 7 ;&T1'&.##0+%<0+).1+-%")&)&#<+Q5<*)&#<+ 2%'(#+(11+KM+8ONGN89 φ 1 [ A, B ] ^ @ 3 φ 2 [ B, C ] � 1 � ;#=+'&T1'&.##0: ln P ( a, b, c ) = ln Z φ 1 [ a, b ] φ 2 [ b, c ] = ln φ 1 [ a, b ] + ln φ 2 [ b, c ] − ln Z ;#=+'&T1'&.##0+Q#"+0%)%+ ! +?&).+/+&<()%<*1(: 3#5<)(+#Q+U%"&#5(+%((&=<41<)(+&<+ ! 6

Cmput 651 - Learning Undirected Models {14,17}/11/2008 ;&T1'&.##0+%<0+).1+-%")&)&#<+Q5<*)&#<+ 2%'(#+(11+KM+8ONGN89 23#<)&<510+Q"#4+'%()+('&019 ;#=+'&T1'&.##0+Q#"+0%)%+ ! +?&).+/+&<()%<*1(: 3#5<)&<=+)1"4(+&<U#'U1+#<'X+%+(&<='1+Q%*)#" !%")&)&#<+Q5<*)&#<+)1"4+&<U#'U1(+^;;+Q%*)#"(: D /%V&454+'&T1'&.##0+1()&4%)&#<+ 2%'(#+(11+KM+8ONGN89 @%X1(+<1)(:+1()&4%)1+*#<0&)&#<%'+0&()"&$5)&#<(+ (1-%"%)1'X+Q#"+1%*.+<#01 LZ+'&T1'&.##0+01*#4-#(%$'1 U(N /%"T#U+<1)(:+-%")&)&#<+Q5<*)&#<+&<U#'U1(+%''+Q%*)#"( *.%<=&<=+_ 8 +*#5'0+*.%<=1+#-)&4%'+U%'51+Q#"+_ G I

Cmput 651 - Learning Undirected Models {14,17}/11/2008 /%V&454+'&T1'&.##0+1()&4%)&#<+ 2%'(#+(11+KM+8ONGN89 /%V&454+'&T1'&.##0+(5"Q%*1 /%V+?N"N)N+d 01-1<0(+#< U%'51+#Q+e O ;#=L'&T1'&.##0+ 2%'(#+(11+KM+8ONGNG9 ;#=+'&<1%"+4#01': ?1&=.) &<0&*%)#"+ Q5<*)&#< ;#=L'&T1'&.##0: `a4bc4 ). +0%)%+-#&<) 8H

Cmput 651 - Learning Undirected Models {14,17}/11/2008 ;#=L'&T1'&.##0+ 2%'(#+(11+KM+8ONGNG9 ;#=L'&T1'&.##0: R&U&01+$X+/+ 2<#N+0%)%+-#&<)(9 f4-&"&*%'+1V-1*)%)&#<+#Q+_ & !%")&)&#<+Q5<*)&#<: P54+#U1"+0%)%+-#&<)( 88 ;#=L'&T1'&.##0+ 2%'(#+(11+KM+8ONGN[9 ;&<1%"+&<+ !%")&)&#<+Q5<*)&#<: L+&(+*#<U1V ⇒ +'#=L'&T1'&.##0+&(+*#<*%U1 3#<U1V+Q5<*)&#<: 8G

Cmput 651 - Learning Undirected Models {14,17}/11/2008 ;#=L'&T1'&.##0+ 2%'(#+(11+KM+8ONGN[9 ;#=L'&T1'&.##0+&(+*#<*%U1 L+ ∃ +='#$%'+4%V&454 L+ � +'#*%'+4%V&4% L+='#$%'+4%V&454+4%X+<#)+$1+5<&Y51 L+/%"T#U+<1)+-%"%41)1"&W%)&#<+4%X+$1+"105<0%<) L+&N1N+45')&-'1+"1-"1(1<)%)&#<(+#Q+(%41+0&()"&$5)&#< include simple examples based on diamond net if there’s time 8[ /%V&454+'&T1'&.##0+-%"%41)1"+1()&4%)&#<+ 2KM+8ON[N89 g%<)+)#+Q&<0+-%"%41)1"(+).%)+4%V&4&W1+'#=L'&T1'&.##0 ,"%0&1<)+c+H+%)+4%V&454 KM+!"#-#(&)&#<+8ONGN[ LZ+h+4%V&454: @>\+ � +*'#(10+Q#"4+(#'5)&#<i L+5(1+="%0&1<)+%(*1<)+&<()1%0 8B

Cmput 651 - Learning Undirected Models {14,17}/11/2008 /%V&454+'&T1'&.##0+-%"%41)1"+1()&4%)&#< P.#?+<541"&*%'+1V%4-'1+5(&<=+^L@L3+<1)?#"T 87 /%V&454+'&T1'&.##0+-%"%41)1"+1()&4%)&#<+ 2KM+8ON[N89 g%<)+)#+Q&<0+-%"%41)1"(+).%)+4%V&4&W1+'#=L'&T1'&.##0 ,"%0&1<)+%(*1<) f%(X+)#+*#4-5)1 fV-1*)10+*#5<)(+#U1"+-%"%41)1"+(-%*1 L+"1Y5&"1(+&<Q1"1<*1 L+#<*1+%)+1%*.+()1-+#Q+="%0&1<)+%(*1<) L+)X-&*%''X+1V-1<(&U1 86

Cmput 651 - Learning Undirected Models {14,17}/11/2008 3#<0&)&#<%''XL)"%&<10+4#01'(+ 2%'(#+(11+KM+8ON[NG9 R&(*"&4&<%)&U1+2&<()1%0+#Q+=1<1"%)&U19 3#<0&)&#<%'+"%<0#4+Q&1'0+23JM9 1<*#01( /%V&4&W1+'#=+*#<0&)&#<%'+'&T1'&.##0 L+&(+*#<*%U1 L+5(1+="%0&1<)+%(*1<) 8D 3#<0&)&#<%''XL)"%&<10+4#01'(+ 2%'(#+(11+KM+8ON[NG9 ,"%0&1<)+%(*1<)+#<+'#=+*#<0&)&#<%'+'&T1'&.##0 ,"%0&1<): , - 3#5<)(+#<+0%)%(1) fV-1*)%)&#<+?N"N)N+4#01'+*#<0&)&#<10+#<+4 ). +0%)%+-#&<) LZ+45()+"5<+&<Q1"1<*1+/+)&41(+Q#"+1%*.+="%0&1<)+()1-i jN+1V-1<(&U1 @5)C+*%<+$1+#QQ(1)+$X+(&4-'1"+4#01'+ 2*#4-N+)#+=1<1"%)&U19 8I

Cmput 651 - Learning Undirected Models {14,17}/11/2008 3#<0&)&#<%''XL)"%&<10+4#01'(+ 2%'(#+(11+KM+fV%4-'1+8ON[N[9 fU&01<*1+<#01(+ ?F 01<(1+&<)1"*#<<1*)&U&)X 2#<'X+-%")'X+(.#?<9 + e 8 e G e [ e B e 7 e 6 R&(*"&4&<%)&U1+4#01'+*#<0&)&#<10+#<+1U&01<*1+<#01( e 8 e G e [ e B e 7 e 6 S<Q1"1<*1+#<+*.%&<+&(+45*.+1%(&1" 8O /%V&454+1<)"#-X+k+4%V&454+'&T1'&.##0+ 2KM+8ON[NB9 + ,&U1<+(#41+0%)%+RC+Q&<0+0&()"&$5)&#<+l+).%)+4%)*.1(+ &)+?&).#5)+%+'#)+#Q+1V)"%+()"5*)5"1+F+%((54-)&#<( f<)"#-X .&=.+U%'51+LZ+'&))'1+()"5*)5"1 1V-1*)%)&#<+*#<()"%&<)( GH

Cmput 651 - Learning Undirected Models {14,17}/11/2008 /%V&454+1<)"#-X+k+4%V&454+'&T1'&.##0+ 2KM+8ON[NB9 + \.1#"14:+/%V+1<)"#-X+(#'5)&#<+lm+ 2)#+-"#$'14+Q"#4+ -"1U&#5(+('&019 +(%)&(Q&1( c+4%V&454+'&T1'&.##0+(#'5)&#<+"1'%)&U1+)#+0%)%+(1)+R &N1N+4%V+'&T1'&.##0+%<0+4%V+1<)"#-X+%"1+"1'%)10 G8 !"&#"(+%<0+"1=5'%"&W%)&#<+ 2%'(#+(11+KM+8ONB9 /%VN+'&T1'&.##0+1()&4%)&#<+-"#<1+)#+#U1"Q&))&<= >(1+-"&#"(+)#+*#<()"%&<+n+-%"%41)1"( Q"#4+'#=+'&<1%"+4#01' *'#(10+Q#"4+'&T1+@%X1(+Q#"45'%+<#)+%U%&'%$'1 5(1+4%VN+%+-#()1"&#"&+2/^!9+&<()1%0+L+4%V&4&W1+ GG

Cmput 651 - Learning Undirected Models {14,17}/11/2008 ,%5((&%<+-"&#"+ 2%'(#+(11+KM+8ONBN89 o G +.X-1"-%"%41)1"+Q#"+U%"&%<*1 ^4#5<)+#Q+"1=5'%"&W%)&#< 3%<+5(1+0&QQ1"1<)+o & \X-&*%''X+5(1+3#U2n & C+n p 9cH+Q#"+&qp &N1N+%((541+n & +&<01-1<01<) !"10&(-#(1(+nr(+)#+$1+(4%'' G[ ;%-'%*&%<+-"&#"+ 2%'(#+(11+KM+8ONBN89 s+.X-1"-%"%41)1" 3%<+5(1+0&QQ1"1<)+s & ^((541+n & +&<01-1<01<) ^'(#+-"10&(-#(1(+nr(+)#+$1+(4%'' GB

Cmput 651 - Learning Undirected Models {14,17}/11/2008 !"&#"(+%<0+"1=5'%"&W%)&#<+ 2%'(#+(11+KM+8ONBN89 !"&#" ;&T1'&.##0 !1<%')X+)1"4+"1=5'%"&W1(+/;f \%T&<=+'#=%"&).4(: t+-1<%')X+)1"4 ,%5((&%<+-1<%')X ; G +"1=5'%"&W%)&#< ;%-'%*&%<+-1<%')X ; 8 +"1=5'%"&W%)&#< G7 ; 8 +U(+; G +"1=5'%"&W%)&#<+ 2%'(#+(11+KM+8ONBN89 ; G +"1=5'%"&W%)&#< '%"=1"+n+U%'51(+-1<%'&W10+4#"1+.1%U&'X ; 8 +"1=5'%"&W%)&#< 5<&Q#"4+-1<%')X LZ+$1))1"+%)+0"&U&<=+n+%''+).1+?%X+)#+W1"# LZ+(-%"(1"+4#01'(+'1%"<10+24#"1+n & cH9 G6