(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvipsk 5.66a Copyright 1986-97 Radical Eye Software (www.radicaleye.com) %%Title: final.dvi %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: /bcs/local/TeX-7.0/bin/dvips -o final.ps final %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1997.12.17:1501 %%BeginProcSet: texc.pro %! /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{dup length product length le{dup length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false} ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot} imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M}B /d{ -3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w} B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet TeXDict begin 40258437 52099154 1000 600 600 (final.dvi) @start %DVIPSBitmapFont: Fa cmbx9 9 1 /Fa 1 50 df<147814F81303131FEA03FFB5FCA3EAFC1F1200B3B2007FB512FEA41F317A B02C>49 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmti9 9 40 /Fb 40 122 df<923803FF80031F13F092383F00F803F8133C4A48133E4A48137E17FE4A 5A17FC17384A481300A3141F92C8FCA55C143E011FB612E0A217C09039007E0007147C16 0F1780A214FC4A131F1700A301015C4A133EA3167E0103147C5C1718EEFC1CEEF83C0107 15385C1778177016F0010F15F04AEBF8E01679EE3FC0011FEC0F0093C7FC91C9FCA3133E A21238EA7E3C137CEAFE7812FC485AEA79E0EA3FC0000FCAFC2F4582B42B>12 D<121C127F12FFA412FE12380808778718>46 D48 DI51 D<150E151FA2153F153EA3157E157CA215FC15F8A2140115F0A2EC03E0A3EC07C0A2EC0F 80A2EC1F00A2143EA25C147814F85C1301903803E0E0ECC1F0EB0781EB0F83EC03E0131E 133CEB7C0701F813C0EA01F0EA03E03807C00FD80F801380EA1FFC383FFFCF48EBFF82D8 F00313FF3860003FC7EA1FF8EC3F00143EA3147E147CA314FC5CA4146020417DB127>I< 010614C090380FC00F91B51280160015FC4913F015C0D91CFEC7FC91C8FC133C1338A313 781370A313F0EBE0FE9038E3FF809038EF03C03901FC01E001F87FEBF000497F485A5BC8 FCA41401A4003C130300FC5CA34A5A5A00E0495AA24A5A4AC7FC6C137E00705B387801F8 383E07F0381FFFC06C90C8FCEA03F8223478B127>II56 D<161C163CA2167C16FCA21501821503A2ED07 7E150F150E151CA21538A2157015F015E0EC01C0A2913803807F82EC0700A2140E141E14 1C5CA25CA25C49B6FCA25B913880003F49C7EA1F80A2130E131E131C133C13385B13F05B 12011203D80FF0EC3FC0D8FFFE903807FFFEA32F367BB539>65 D67 D<0107B712E05B18C0903A003F80003F170F170792C7FC17 035C1880147EA214FEA25C161C0101EC3C07043813004A91C7FCA20103147816704A13F0 150349B5FCA25EECE003130F6F5A14C0A2011F13035E1480A2013F90C9FCA291CAFCA25B A2137EA213FEA25B1201387FFFFCB5FCA233337CB232>70 D<010FB51280A21600903800 3FC05DA292C7FCA25CA2147EA214FEA25CA21301A25CA21303A25CA21307A25CA2130FA2 5CA2131FA25CA2133FA291C8FCA25BA2137EA213FEA25B1201B512F8A25C21337BB21E> 73 D<91381FFFFE5C16FC9138003F80A31600A25D157EA315FE5DA314015DA314035DA3 14075DA3140F5DA3141F5DA3143F92C7FCA2121C007E5B00FE137EA214FE485BEAF80100 E05B495A387007E038780FC06C48C8FCEA1FFCEA07F0273579B228>I<902607FFC0ED7F FC4917FF81D9003F4B1300611803023BED077CA2027BED0EFC610273151C1838DAF1F014 39F071F014E118E10101ED01C36102C1EC0383EF070301031607050E5BEC80F8171C0107 ED380F6102001470A249EDE01FDC01C090C7FC130EEE0380011E017C5C933807003E011C 140EA2013C4A137E187C01385C5E017816FC6F485B1370ED3FC001F0EC80016000011500 D807F81503277FFF803E90B512C0B5EB3C01151C46337BB245>77 D<902607FF8090383FFFC0496D5BA2D9001F913803F8004A6C6D5A6060EC3BF0027B1403 60EC71F8A202F11407DAF0FC91C7FC14E0A20101017E5B170E14C0810103151EEE801CEC 801FA20107ECC03C030F1338140016E049010713781770010E14F01503011E15F0705A01 1C1301A2013C14FD03005B133816FF0178147F5F0170143FA213F070C8FC1201EA07F826 7FFF807FB5140EA23A337BB239>I<0107B612C04915F883903A003F8001FEEE003FEF1F 8092C713C0170F5C18E0147EA214FEEF1FC05CA201011680173F4A1500177E010315FE5F 4AEB03F8EE07E00107EC3FC091B6C7FC16F802E0C9FC130FA25CA2131FA25CA2133FA291 CAFCA25BA2137EA213FEA25B1201387FFFF0B5FCA233337CB234>80 D<0107B512FE49ECFFC017F0903A003F8007F8EE01FCEE007E92C7127F835C1880147EA2 14FEEF7F005CA2010115FE5F4A13015F01034A5AEE0FC04A495A04FEC7FC49B512F016C0 9138E003E0ED01F8010F6D7E167C4A137EA2131FA25CA2013F14FEA291C7FCA24913015E 137EEF01C001FE150318805B00011607277FFFF0001400B5ECFE0EEE7E1CC9EA1FF8EE07 E032357BB238>82 D<913901FC018091380FFF03023F13C791387E07EF903A01F801FF00 49487E4A7F495A4948133E131F91C7FC5B013E143CA3137E1638A293C7FC137FA26D7E14 E014FE90381FFFC06D13F86D7F01017F6D6C7E020F7F1400153F6F7E150FA4120EA2001E 5D121CA2151F003C92C7FCA2003E143E5D127E007F5C6D485A9038C007E039F3F80FC000 F0B5C8FC38E03FFC38C00FF029377AB42B>I<0003B812C05A1880903AF800FC003F260F C001141F0180150F01005B001EEE07001403121C003C4A5BA200380107140E127800705C A2020F141E00F0161CC74990C7FCA2141FA25DA2143FA292C9FCA25CA2147EA214FEA25C A21301A25CA21303A25CA21307A25C497E001FB512F05AA2323374B237>I97 D<14FCEB07FF90381F078090383E03C0EBFC013801F8033803F0073807E00F13C0120F39 1F80070091C7FC48C8FCA35A127EA312FE5AA4007C14C0EC01E0A2EC03C06CEB0F80EC1F 006C137C380F81F03803FFC0C648C7FC1B2278A023>99 DII<151FED7FC0EDF0E0020113F0EC03E3A2EC07C316E0EDC1 C091380FC0005DA4141F92C7FCA45C143E90381FFFFEA3D9007EC7FC147CA414FC5CA513 015CA413035CA413075CA3130FA25CA3131F91C8FCA35B133E1238EA7E3CA2EAFE7812FC 485AEA78E0EA3FC0000FC9FC244582B418>I<143FECFF80903803E1E6903807C0FF9038 0F807FEB1F00133E017E133F49133EA24848137EA24848137CA215FC12074913F8A21401 A2D80FC013F0A21403120715E01407140F141F3903E03FC00001137FEBF0FF38007FCF90 381F0F801300141FA21500A25C143E1238007E137E5C00FE5B48485A387803E0387C0F80 D81FFFC7FCEA07F820317CA023>III<133FEA 07FF5A13FEEA007EA3137CA213FCA213F8A21201A213F0A21203A213E0A21207A213C0A2 120FA21380A2121FA21300A25AA2123EA2127EA2127C1318EAFC1C133CEAF838A2137813 7012F013F0EAF8E01279EA3FC0EA0F00103579B314>108 D<2703C003F8137F3C0FF00F FE01FFC03C1E783C1F07C1E03C1C7CF00F8F01F03B3C3DE0079E0026383FC001FC7FD97F 805B007001005B5E137ED8F0FC90380FC00100E05FD860F8148012000001021F13036049 1400A200034A13076049013E130FF081800007027EEC83C0051F138049017C1403A2000F 02FC1407053E130049495CEF1E0E001F01015D183C010049EB0FF0000E6D48EB03E03A22 7AA03F>I<3903C007F0390FF01FFC391E787C1E391C7CF01F393C3DE00F26383FC01380 EB7F8000781300EA707EA2D8F0FC131F00E01500EA60F8120000015C153E5BA20003147E 157C4913FCEDF8180007153C0201133801C013F0A2000F1578EDE070018014F016E0001F ECE1C015E390C7EAFF00000E143E26227AA02B>I<14FCEB07FF90381F07C090383E03E0 9038FC01F0EA01F83903F000F8485A5B120F484813FCA248C7FCA214014814F8127EA214 0300FE14F05AA2EC07E0A2007CEB0FC01580141FEC3F006C137E5C381F01F0380F83E038 03FF80D800FCC7FC1E2278A027>I<011E137C90387F81FF9039F3C387C09039E3EF03E0 3901E1FE01D9C1FC13F0EBC3F8000313F0018314F814E0EA07871307000313C01200010F 130316F01480A2011F130716E01400A249EB0FC0A2013EEB1F80A2017EEB3F00017F133E 5D5D9038FF81F09038FDC3E09038F8FF80027EC7FC000190C8FCA25BA21203A25BA21207 A25BB5FCA325307FA027>I<3903C00FC0390FF03FF0391E78F078391C7DE03C393C3FC0 FC00381380EB7F00007814F8D8707E13701500EAF0FC12E0EA60F812001201A25BA21203 A25BA21207A25BA2120FA25BA2121FA290C8FC120E1E227AA020>114 DI<1303EB0F80A3131FA214 00A25BA2133EA2137EA2137C387FFFF8A2B5FC3800F800A21201A25BA21203A25BA21207 A25BA2120FA25B1460001F13F014E01300130114C01303001E1380EB07005BEA0F1EEA07 F8EA01E015307AAE19>II<01F01338D803FC13FCEA0F1E120E12 1C123C0038147CEA783E0070143CA2137ED8F07C1338EA60FCC65A1578000114705BA215 F0000314E05BA2EC01C0A2EBC003158014071500EBE00EA26C6C5A3800F878EB7FE0EB1F 801E227AA023>I<011F137C90387FC1FF3A01E1E787803A03C0F703C0903880FE0FEA07 004813FC000E1580001E9038F80700001C91C7FC1301003C5B1218120013035CA31307A2 5C1506010F130F150E14800038141ED87C1F131C00FC143C1538013F5B39F07FC0E03970 F3C3C0393FE1FF80260F807EC7FC22227CA023>120 D<13F0D803FC1307D80F1E130F00 0E141F121C123C0038143FD8783E133E1270A2017E137ED8F07C137CEA60FCC65A15FC00 0114F85BA21401000314F013E0A2140315E0EA07C0A20003130715C0EBE00F141F000113 3F9038F07F8038007FEFEB1F8FEB001F1500A25C003E133E007E137E147C5C007C5BEA70 01495A38380780D83C1FC7FCEA0FFCEA07F020317AA025>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmsy6 6 1 /Fc 1 49 df48 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmsy9 9 1 /Fd 1 107 df<126012F0B3B3B3B31260044B78B715>106 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmmi6 6 4 /Fe 4 122 df<127812FCA212FEA2127E1206A3120CA2121C121812301260124007107A 8513>59 D<1338137CA2137813701300A7EA0780EA1FC0EA38E01230EA60F0EAC1E0A3EA 03C0A3EA0780A2EA0F0013041306EA1E0CA21318121CEA1E70EA0FE0EA07800F237DA116 >105 D<3801F01E3907FC7F80390E1CE1C038180F8100301383007013071260EC0380D8 001EC7FCA45BA21580003014C0397878018012F8EC030038F0FC0638E19C1C387F0FF838 1E03E01A177D9523>120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmmi9 9 5 /Ff 5 122 df<123C127EB4FCA21380A2127F123D1201A412031300A25A1206120E120C 121C5A5A126009177A8715>59 D<1530157815F8A215F01401A215E01403A215C01407A2 1580140FA215005CA2143EA2143C147CA2147814F8A25C1301A25C1303A25C1307A2495A A291C7FC5BA2131E133EA2133C137CA2137813F8A25B1201A25B1203A2485AA25B120FA2 90C8FC5AA2121E123EA2123C127CA2127812F8A25A12601D4B7CB726>61 D<010FB612F017FE83903B003FC0007FC0EF1FE0EF07F05DEF03F8147FA292C713FCA25C EF07F85CA2010116F0170F4A15E0EF1FC00103ED3F80EF7F004A14FEEE03FC0107EC1FF0 91B612C04CC7FC02F0C9FC130FA25CA2131FA25CA2133FA25CA2137FA291CAFCA25BA25B 1201B512FCA336337DB231>80 D<90391F801F8090397FE07FE09039E0F0E0703A01C0F9 C0F83903807D833807007F000E1403000C15F0001C137E0018EC01C002FEC7FC00385B12 10C7FC13015CA31303A25C1640010714E016C0001C5B007E1401010F148000FE1403011F EB0700011B130E39F839F01C397070F878393FE07FE0390F801F8025227EA02C>120 D<13F0D803FCEB01C0D8071EEB03E0D80E1F1307121C003813800030140F013F14C00070 1300126049131FD8E07E14801240EA00FE49133F000115005BA25D0003147E5BA215FE5D 5BA214015DEBF00314070001130F3900F83FF0EB3FFBEB0FC3EB00075DA20007130FD81F 805B003F495AA24AC7FCEB007E003E137C00385B381803F0381E07C0D807FFC8FCEA01F8 23317EA026>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmr9 9 74 /Fg 74 124 df<91393FE00FE0903A01FFF83FF8903A07E01EF83C903A1F800FF07E903A 3F001FE0FE017E133F4914C0485A1738484890381F8000ACB812C0A33B03F0001F8000B3 A7486C497EB50083B5FCA32F357FB42D>11 DI14 D<003C13F0387E01F838FF03FCA2EB83FEA2EA7F81383D 80F600011306A40003130EEB000CA248131C00061318000E1338000C1330001C13704813 E0387001C00060138017177EB326>34 D<147CEB01FEEB07C790380F8380EB1F0181EB3E 00A2137EA2137C137EA214015D140392C9FC5C140E6D5A1418143802F090380FFFF05C6D 5A04011300EE00FC6D6C1470011F1560013F15E0D977F0495AD9E3F85CD801C314032603 81FC91C7FC00075D48C66C130E486D130C486D131C003E6D6C5A007EECC03091381FE070 00FE010F5B6F5AEC07F96EB45A6C6D90C712306E5A157F6C6C6D6C13604B6C13E03A3FC0 01EFE03C1FE003C7F803C03C0FF01F83FE0F802707FFFE00B51200000101F8EB3FFE2600 3FC0EB07F034387DB53C>38 D<123C127EB4FCA21380A2127F123D1201A412031300A25A 1206120E120C121C5A5A126009177AB315>I<14C01301EB0380EB0F00130E5B133C5B5B A2485A485AA212075B120F90C7FC5AA2121E123EA3123C127CA55AB0127CA5123C123EA3 121E121FA27E7F12077F1203A26C7E6C7EA213787F131C7F130FEB0380EB01C01300124A 79B71E>I<12C07E1270123C121C7E120F6C7E6C7EA26C7E6C7EA27F1378137C133C133E A2131E131FA37F1480A5EB07C0B0EB0F80A514005BA3131E133EA2133C137C137813F85B A2485A485AA2485A48C7FC120E5A123C12705A5A124A7CB71E>I<123C127EB4FCA21380 A2127F123D1201A412031300A25A1206120E120C121C5A5A126009177A8715>44 DI<123C127E12FFA4127E123C08087A8715>I<1530157815F8A2 15F01401A215E01403A215C01407A21580140FA215005CA2143EA2143C147CA2147814F8 A25C1301A25C1303A25C1307A2495AA291C7FC5BA2131E133EA2133C137CA2137813F8A2 5B1201A25B1203A2485AA25B120FA290C8FC5AA2121E123EA2123C127CA2127812F8A25A 12601D4B7CB726>II<13 075B5B137FEA07FFB5FC13BFEAF83F1200B3B3A2497E007FB51280A319327AB126>IIII<000C14C0380FC00F90B5128015005C5C14F014C0D80C18C7FC90C8FCA9EB0F C0EB7FF8EBF07C380FC03F9038001F80EC0FC0120E000CEB07E0A2C713F01403A215F8A4 1218127E12FEA315F0140712F8006014E01270EC0FC06C131F003C14806CEB7F00380F80 FE3807FFF8000113E038003F801D347CB126>I<14FE903807FF80011F13E090383F00F0 017C13703901F801F8EBF003EA03E01207EA0FC0EC01F04848C7FCA248C8FCA35A127EEB 07F0EB1FFC38FE381F9038700F809038E007C039FFC003E0018013F0EC01F8130015FC14 00A24814FEA5127EA4127F6C14FCA26C1301018013F8000F14F0EBC0030007EB07E03903 E00FC03901F81F806CB51200EB3FFCEB0FE01F347DB126>I<1230123C003FB6FCA34814 FEA215FC0070C7123800601430157015E04814C01401EC0380C7EA07001406140E5C1418 14385CA25CA2495A1303A3495AA2130FA3131F91C7FCA25BA55BA9131C20347CB126>I< EB0FE0EB7FFC90B5FC3903F01F803907C007C0390F0003E0000EEB01F0001E1300001C14 F8003C1478A3123EA2003F14F86D13F0EBC001D81FF013E09038F803C0390FFE07803907 FF0F006C13DE6C13F87EEB3FFE8001F713C0D803E313E0D8078013F0390F007FF8001E13 1F003EEB07FC003C1303481301EC007E12F848143EA2151EA37E153C1278007C14787E6C 14F0390F8003E03907F01FC00001B5120038007FFCEB1FE01F347DB126>II< 123C127E12FFA4127E123C1200B0123C127E12FFA4127E123C08207A9F15>I<123C127E 12FFA4127E123C1200B0123C127E12FE12FFA3127F123F1203A412071206A3120E120C12 1C1238123012701260082F7A9F15>I<007FB812C0B912E0A26C17C0CCFCAC007FB812C0 B912E0A26C17C033147C9C3C>61 D<15E0A34A7EA24A7EA34A7EA3EC0DFE140CA2EC187F A34A6C7EA202707FEC601FA202E07FECC00FA2D901807F1507A249486C7EA301066D7EA2 010E80010FB5FCA249800118C77EA24981163FA2496E7EA3496E7EA20001821607487ED8 1FF04A7ED8FFFE49B512E0A333367DB53A>65 DIII< B812C0A3D803FCC7127F0001150FEE03E01601A21600A21760A403061330A41700150EA2 151E157E90B512FEA39038FC007E151E150EA21506170CA3171892C7FCA41738A21770A2 17F01601160316070003157FB812E0A32E337DB234>IIIII<017FB5FCA39038003FE0EC1FC0B3B1127EB4FCA4EC3F805A0060 140000705B6C13FE6C485A380F03F03803FFC0C690C7FC20357DB227>IIIIIIIII< 90381FE00390387FFC0748B5FC3907F01FCF390F8003FF48C7FC003E80814880A2007880 00F880A46C80A27E92C7FC127F13C0EA3FF013FF6C13F06C13FF6C14C06C14F0C680013F 7F01037F9038003FFF140302001380157F153FED1FC0150F12C0A21507A37EA26CEC0F80 A26C15006C5C6C143E6C147E01C05B39F1FC03F800E0B512E0011F138026C003FEC7FC22 377CB42B>I<007FB712FEA390398007F001D87C00EC003E0078161E0070160EA2006016 0600E01607A3481603A6C71500B3AB4A7E011FB512FCA330337DB237>I87 D<003FB612FCA39039F80007F813C090C7EA0FF0003EEC1F E0123C0038EC3FC00078EC7F801270EDFF004A5AA20060495AA24A5A4A5AC7FC4A5A4A5A A24A5A4AC7FCA2495A495AA2495A495AA24948130C495AA2495A49C7FCA24848141CA248 5A485A1638485A4848147816F84848130148481307153FB7FCA326337CB22F>90 D<0003130C48131C000E133848137000181360003813E0003013C0EA700100601380A2EA E00300C01300A400DE137800FF13FCEB83FEA2EA7F81A2383F00FC001E1378171774B326 >92 D97 DII<153FEC0FFFA3EC007F81AEEB07F0EB3FFCEBFC0F3901F003BF39 07E001FF48487E48487F8148C7FCA25A127E12FEAA127E127FA27E6C6C5BA26C6C5B6C6C 4813803A03F007BFFC3900F81E3FEB3FFCD90FE0130026357DB32B>III<151F90391FC07F809039FFF8E3C03901F07FC73907E03F033A0FC01F8380 9039800F8000001F80EB00074880A66C5CEB800F000F5CEBC01F6C6C48C7FCEBF07C380E FFF8380C1FC0001CC9FCA3121EA2121F380FFFFEECFFC06C14F06C14FC4880381F000100 3EEB007F4880ED1F8048140FA56C141F007C15006C143E6C5C390FC001F83903F007E0C6 B51280D91FFCC7FC22337EA126>III107 DI<2703F01FE013FF00FF 90267FF80313C0903BF1E07C0F03E0903BF3803E1C01F02807F7003F387FD803FE147049 6D486C7EA2495CA2495CB3486C496C487EB53BC7FFFE3FFFF0A33C217EA041>I<3903F0 1FC000FFEB7FF09038F1E0FC9038F3807C3907F7007EEA03FE497FA25BA25BB3486CEB7F 80B538C7FFFCA326217EA02B>II<3903F03F8000FFEBFFE09038F3C0F89038F7007ED807FE7F6C48EB1F804914C049 130F16E0ED07F0A3ED03F8A9150716F0A216E0150F16C06D131F6DEB3F80160001FF13FC 9038F381F89038F1FFE0D9F07FC7FC91C8FCAA487EB512C0A325307EA02B>I<903807F0 0390383FFC07EBFC0F3901F8038F3807E001000F14DF48486CB4FC497F123F90C77E5AA2 5A5AA9127FA36C6C5B121F6D5B000F5B3907E003BF3903F0073F3800F81EEB3FF8EB0FE0 90C7FCAAED7F8091380FFFFCA326307DA029>I<3803E07C38FFE1FF9038E38F809038E7 1FC0EA07EEEA03ECA29038FC0F8049C7FCA35BB2487EB512E0A31A217FA01E>II<1330A51370A313F0A21201A2120312 07381FFFFEB5FCA23803F000AF1403A814073801F806A23800FC0EEB7E1CEB1FF8EB07E0 182F7FAD1E>IIIII<3A7FFF807FF8A33A07F8001FC00003EC0F8000 01EC070015066C6C5BA26D131C017E1318A26D5BA2EC8070011F1360ECC0E0010F5BA290 3807E180A214F3010390C7FC14FBEB01FEA26D5AA31478A21430A25CA214E05CA2495A12 78D8FC03C8FCA21306130EEA701CEA7838EA1FF0EA0FC025307F9F29>I<003FB512F0A2 EB000F003C14E00038EB1FC00030EB3F800070137F1500006013FE495A13035CC6485A49 5AA2495A495A49C7FC153013FE485A12035B48481370485A001F14604913E0485A387F00 0348130F90B5FCA21C207E9F22>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmr6 6 5 /Fh 5 54 df<13E01201120712FF12F91201B3A7487EB512C0A212217AA01E>49 DI<13FF000313C0380F03E0381C00F014F800 3E13FC147CA2001E13FC120CC712F8A2EB01F0EB03E0EB0FC03801FF00A2380003E0EB00 F01478147C143E143F1230127812FCA2143E48137E0060137C003813F8381E03F0380FFF C00001130018227DA01E>I<14E01301A213031307A2130D131D13391331136113E113C1 EA01811203EA07011206120C121C12181230127012E0B6FCA2380001E0A6EB03F0EB3FFF A218227DA11E>I<00101330381E01F0381FFFE014C01480EBFE00EA1BF00018C7FCA513 FE381BFF80381F03C0381C01E0381800F014F8C71278A2147CA21230127812F8A2147848 13F8006013F0387001E01238381E07803807FF00EA01F816227CA01E>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmex10 10 11 /Fi 11 90 df20 DI34 DI50 DI<12FCB3B3 B3B3B3B3B3B0B612F0A61C94668237>II80 D88 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmmi8 8 12 /Fj 12 121 df<13FC13FFEB1FC0130F6D7EA36D7EA2130180A26D7EA3147EA280A36E7E A2140F81A24A7E143F147FECF3F0EB01E3EB03C190380781F8130F49C67E133E5B49137E 485A48487F1207485A4848EB1F8048C7FC127E48EC0FC048EC07E000701403232F7DAD29 >21 D<123C127EB4FCA21380A2127F123D1201A312031300A25A1206120E5A5A5A126009 157A8714>59 D<013FB71280A2D900FEC7127F170F4A1407A20101150318005CA21303A2 5C16300107147094C7FC4A136016E0130F15019138C007C091B5FC5BECC0074A6C5AA213 3FA20200EB000CA249151C92C71218017E1538173001FE15705F5B4C5A000115034C5A49 140F161F00034AB4C7FCB8FC5E312D7DAC34>69 D81 D<013FB512F816FF903A00FE001FC0EE07E04A6D7E707E01016E7EA24A80A213034C 5A5CA201074A5A5F4A495A4C5A010F4A5A047EC7FC9138C003F891B512E04991C8FC9138 C007C04A6C7E6F7E013F80150091C77EA2491301A2017E5CA201FE1303A25BA20001EE03 8018005B5F0003913801FC0EB5D8E000133CEE7FF0C9EA0FC0312E7CAC35>I<913807F0 0691383FFE0E9138F80F9E903903E001FE903807800049C7127C131E49143CA2491438A3 13F81630A26D1400A27FEB7F8014F86DB47E15F06D13FC01077F01007F141F02011380EC 003F151F150FA215071218A3150F00381500A2151EA2007C5C007E5C007F5C397B8003E0 39F1F00F8026E07FFEC7FC38C00FF0272F7CAD2B>I<157C4AB4FC913807C380EC0F8715 0FEC1F1FA391383E0E0092C7FCA3147E147CA414FC90383FFFF8A2D900F8C7FCA313015C A413035CA413075CA5130F5CA4131F91C8FCA4133EA3EA383C12FC5BA25B12F0EAE1E0EA 7FC0001FC9FC213D7CAE22>102 D<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01 E0EA07F8EA0E3CEA1C3E123812301270EA607EEAE07C12C013FC485A120012015B12035B A21207EBC04014C0120F13801381381F01801303EB0700EA0F06131EEA07F8EA01F0122E 7EAC18>105 D<15E0EC01F01403A3EC01C091C7FCA9147CEB03FE9038078F80EB0E0713 1C013813C01330EB700F0160138013E013C0EB801F13001500A25CA2143EA2147EA2147C A214FCA25CA21301A25CA21303A25CA2130700385BEAFC0F5C49C7FCEAF83EEAF0F8EA7F F0EA1F801C3B81AC1D>I<131FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA212 0115F89038F003FCEC0F0E0003EB1C1EEC387EEBE07014E03807E1C09038E3803849C7FC 13CEEA0FDC13F8A2EBFF80381F9FE0EB83F0EB01F81300481404150C123EA2007E141C15 18007CEBF038ECF83000FC1470EC78E048EB3FC00070EB0F801F2F7DAD25>I<27078007 F0137E3C1FE01FFC03FF803C18F0781F0783E03B3878E00F1E01263079C001B87F26707F 8013B00060010013F001FE14E000E015C0485A4914800081021F130300015F491400A200 034A13076049133E170F0007027EEC8080188149017C131F1801000F02FCEB3F03053E13 0049495C180E001F0101EC1E0C183C010049EB0FF0000E6D48EB03E0391F7E9D3E>109 D<013F137C9038FFC1FF3A01C1E383803A0380F703C0390700F60F000E13FE4813FC1218 0038EC0700003049C7FCA2EA200100005BA313035CA301075B5D14C000385CD87C0F1306 00FC140E011F130C011B131C39F03BE038D8707113F0393FE0FFC0260F803FC7FC221F7E 9D28>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmsy10 10.95 11 /Fk 11 108 df<007FB812FEBAFCA26C17FE3804799847>0 D<121EEA7F80A2EAFFC0A4 EA7F80A2EA1E000A0A799B19>I<007FB912E0BA12F0A26C18E0CDFCAE007FB912E0BA12 F0A26C18E0CDFCAE007FB912E0BA12F0A26C18E03C287BAA47>17 D<180E183F18FFEF03FEEF0FF8EF3FE0EFFF80933803FE00EE0FF8EE3FE0EEFF80DB03FE C7FCED0FF8ED7FE0913801FF80DA07FEC8FCEC1FF8EC7FC04948C9FCEB07FCEB1FF0EB7F C04848CAFCEA07FCEA1FF0EA7FC048CBFCA2EA7FC0EA1FF0EA07FCEA01FF38007FC0EB1F F0EB07FCEB01FF9038007FC0EC1FF0EC07FE913801FF809138007FE0ED1FF8ED03FE9238 00FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FEEF00FF183F180E1800AE 007FB812FEBAFCA26C17FE384879B947>20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01 FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FE913801FF809138007FE0ED 1FF8ED03FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FEEF00FF A2EF03FEEF0FF8EF3FE0EFFF80933803FE00EE0FF8EE3FE0EEFF80DB03FEC7FCED0FF8ED 7FE0913801FF80DA07FEC8FCEC1FF8EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA 07FCEA1FF0EA7FC048CBFC12FC1270CCFCAE007FB812FEBAFCA26C17FE384879B947>I< D90FF0ED07F0D93FFE151F90B56C147F486E903801F8002703E01FF0EB07E027078007F8 EB0F8048486C6C49C7FC001E6D6C133E001C6D6C5B4891387F80786F6C5A4891381FC1E0 ED0FE10060EDF3C0923807F78000E0EC03FF4893C8FC816F5A828283163F4C7E836C15FF 0060EDF7F0923801E7F8ED03C30070EDC1FC003091380781FE00384AC67E00184A6D7E00 1C023E6D7E6C4A6D7E6C4A6D7E90268003F0EB07FC2703E00FC0903801FFF0C6B5C8FCD9 3FFC153FD90FF015073C297BA747>47 D<047FB612FC0307B8FC031F1780157F4AB9FC91 2903F80FE000011300DA0FC0ED007EDA1F00167C023E17604A011F92C7FC02FC5C495AA2 13034A495A495A5C0106C7FC90C848CAFCA3167E16FEA34B5AA35E150393B612F0A24B5D 614B92C8FC04E0CAFC5E151F5EA24BCBFCA25D157E15FE5DA24A5AA24A5AA24A5AA20003 495A121F486C485A127F486C48CCFCEBE03E387FFC7CEBFFF86C5B6C13C06C5BD801FCCD FC49417FBD41>70 D<153FEC03FFEC0FE0EC3F80EC7E00495A5C495AA2495AB3AA130F5C 131F495A91C7FC13FEEA03F8EA7FE048C8FCEA7FE0EA03F8EA00FE133F806D7E130F8013 07B3AA6D7EA26D7E80EB007EEC3F80EC0FE0EC03FFEC003F205B7AC32D>102 D<12FCEAFFC0EA07F0EA01FCEA007E6D7E131F6D7EA26D7EB3AA801303806D7E1300147F EC1FC0EC07FEEC00FFEC07FEEC1FC0EC7F0014FC1301495A5C13075CB3AA495AA2495A13 3F017EC7FC485AEA07F0EAFFC000FCC8FC205B7AC32D>I<126012F0B3B3B3B3B1126004 5B76C319>106 D<0060131800F0133CB3B3B3B3B000601318165A75C32D>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmsy8 8 4 /Fl 4 55 df0 D<130C131EA50060EB01800078130739FC0C0F C0007FEB3F80393F8C7F003807CCF83801FFE038007F80011EC7FCEB7F803801FFE03807 CCF8383F8C7F397F0C3F8000FCEB0FC039781E078000601301000090C7FCA5130C1A1D7C 9E23>3 D<137813FE1201A3120313FCA3EA07F8A313F0A2EA0FE0A313C0121F1380A3EA 3F00A3123E127E127CA35AA35A0F227EA413>48 D54 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmmi10 10.95 27 /Fm 27 121 df11 D14 D<133F14E0EB07F0EB03FC13016D7EA3147FA26E7EA36E7EA36E7EA36E7EA36E7EA26E7E A36E7EA3157FA36F7E157F15FF4A7F5C913807CFE0EC0F8FEC1F0F91383E07F0147C14FC 49486C7EEB03F0EB07E049486C7EEB1F80EB3F00496D7E13FE4848147F485A485A4848EC 3F80485A123F4848EC1FC048C8FC4816E048150F48ED07F0007015032C407BBE35>21 D<011FB612FE017F15FF48B8FC5A4816FE3B0FC03801C000EA1F00003E1403003C01785B 4813705AECF0075AC712E0010191C7FCA25DEB03C0A313071480A2010F5BA2EB1F0082A2 133EA2137E825B150F0001815B120315075BC648EB038030287DA634>25 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A798919>58 D<121EEA7F8012FF13C0A2 13E0A3127FEA1E601200A413E013C0A312011380120313005A120E5A1218123812300B1C 798919>I61 D<15FF020713E091381F00F80278133E4A7F4948EB0F804948EB07C0 4948EB03E091C7FC496CEB01F002E014F8131F160017FCA25C0107C812FE90C9FCA7EC03 FCEC3FFF9138FE03C1903903F000E149481371D91F80133149C7123B017EEC1BFC5B0001 151F4848140F484815F8A2485A121F17F0485A161F17E0127F5BEE3FC0A200FF168090C8 127F1700A216FEA2484A5A5E007E1403007F4A5A5E6C4A5A6C6C495A4BC7FC6C6C13FE6C 6C485A3903F80FF06CB512C06C6C90C8FCEB0FF82F437CC030>64 D<49B912C0A3D9000190C71201F0003F4B151F190F1A80020316075DA314075D1A00A214 0F4BEB0380A205075B021FED000E4B92C7FC5FA2023F141E5D173EEE01FE4AB55AA3ED80 0102FF6D5A92C71278A34915705C191C05F0133C01034B13384A167894C71270A2010717 F04A5E180161010F16034A4B5AA2180F011F4CC7FC4A5D187E013F16FE4D5A4A140F017F 15FFB95AA260423E7DBD43>69 D<49B6D8C03FB512F81BF01780D900010180C7383FF000 93C85B4B5EA2197F14034B5EA219FF14074B93C7FCA260140F4B5DA21803141F4B5DA218 07143F4B5DA2180F4AB7FC61A20380C7121F14FF92C85BA2183F5B4A5EA2187F13034A5E A218FF13074A93C8FCA25F130F4A5DA21703131F4A5DA2013F1507A24A5D496C4A7EB6D8 E01FB512FCA2614D3E7DBD4C>72 D<49B612C05BA2D90001EB800093C7FC5DA314035DA3 14075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C8FCA35B5CA313035CA313075C A3130F5CA3131F5CA2133FA25CEBFFE0B612E0A32A3E7DBD28>I<49B56C93383FFFF051 13E098B5FCD90001F1E000704B5B03DF933803BF80A2F2077F1403039F040E90C7FC1A1C DB8FE05E02075F030F4C5AA21AE1020FEE01C1020E606F6CEC03811A83021EEE0703021C 040E5BA2F11C07023C16380238606F6C1470F1E00F14780270DB01C05BA2953803801F02 F0ED07004A6C6C5E180E4E133F130102C04B5C601A7F01036D6C5B4A95C8FC4D5A4D485B 130791C749C75A170E047F1401495D010E4B5CA24D1303131E011C4B5C5F013C023F1407 017C5D01FE92C75BD803FF4D7EB500FC013E011FB512F8163C041C5E5C3E7DBD58>77 D<49B712F018FF19C0D9000190C76C7EF00FF84BEC03FC1801020382727E5DA214071A80 5DA2140F4E13005DA2021F5E18034B5D1807023F5E4E5A4B4A5A4E5A027F4B5A06FEC7FC 4BEB03FCEF3FF091B712C005FCC8FC92CBFCA25BA25CA21303A25CA21307A25CA2130FA2 5CA2131FA25CA2133FA25C497EB612E0A3413E7DBD3A>80 DI<49B77E18F818FFD90001D900017F9438003FE04BEC0FF0727E727E14034B 6E7EA30207825DA3020F4B5A5DA24E5A141F4B4A5A614E5A023F4B5A4B4A5A06FEC7FCEF 03FC027FEC0FF04BEBFF8092B500FCC8FC5F9139FF8001FE92C7EA7F80EF1FC084496F7E 4A1407A28413035CA2170F13075C60171F130F5CA3011F033F5B4AEE038018E0013F1707 1A004A021F5B496C160EB600E090380FF01E05075B716C5ACBEAFFE0F03F8041407DBD45 >II97 D99 D101 D<143C14FEA21301A314FCEB00701400AD137E3801FF803803C7C0EA070300 0F13E0120E121C13071238A2EA780F007013C0A2EAF01F14801200133F14005B137EA213 FE5BA212015B0003130E13F0A20007131EEBE01CA2143CEBC0381478147014E013C13803 E3C03801FF00EA007C173E7EBC1F>105 D107 D109 DII115 D<147014FC1301A25CA21303A25CA21307A25CA213 0FA25CA2007FB512F0B6FC15E039001F8000133FA291C7FCA25BA2137EA213FEA25BA212 01A25BA21203A25BA21207EC01C013E01403000F1480A2EBC0071500140E141E5C000713 385C3803E1E03801FF80D8003EC7FC1C3A7EB821>I120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmr8 8 8 /Fn 8 62 df<13031307130E131C1338137013F0EA01E013C01203EA0780A2EA0F00A212 1EA35AA45AA512F8A25AAB7EA21278A57EA47EA37EA2EA0780A2EA03C0120113E0EA00F0 13701338131C130E1307130310437AB11B>40 D<12C07E12707E7E7E120FEA0780120313 C0EA01E0A2EA00F0A21378A3133CA4131EA5131FA2130FAB131FA2131EA5133CA41378A3 13F0A2EA01E0A2EA03C013801207EA0F00120E5A5A5A5A5A10437CB11B>I<130C133C13 7CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23>49 DII<140EA2141E143EA2147E14FEA2EB01BE13 03143E1306130E130C131813381330136013E013C0EA0180120313001206120E120C5A12 3812305A12E0B612FCA2C7EA3E00A9147F90381FFFFCA21E2D7EAC23>I<000CEB018038 0FC01F90B512005C5C14F014C0D80C7EC7FC90C8FCA8EB1FC0EB7FF8380DE07C380F801F 01001380000E130F000CEB07C0C713E0A2140315F0A4127812FCA448EB07E012E0006014 C00070130F6C14806CEB1F006C133E380780F83801FFE038007F801C2D7DAB23>I61 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmr10 10 21 /Fo 21 89 df<121C127FEAFF80A5EA7F00121C0909798817>46 D<1538A3157CA315FEA34A7EA34A6C7EA202077FEC063FA2020E7FEC0C1FA2021C7FEC18 0FA202387FEC3007A202707FEC6003A202C07F1501A2D901807F81A249C77F167FA20106 810107B6FCA24981010CC7121FA2496E7EA3496E7EA3496E7EA213E0707E1201486C81D8 0FFC02071380B56C90B512FEA3373C7DBB3E>65 D<913A01FF800180020FEBE003027F13 F8903A01FF807E07903A03FC000F0FD90FF0EB039F4948EB01DFD93F80EB00FF49C8127F 01FE153F12014848151F4848150FA248481507A2485A1703123F5B007F1601A35B00FF93 C7FCAD127F6DED0180A3123F7F001F160318006C7E5F6C7E17066C6C150E6C6C5D000016 18017F15386D6C5CD91FE05C6D6CEB03C0D903FCEB0F80902701FF803FC7FC9039007FFF FC020F13F002011380313D7BBA3C>67 DIIIIII<013FB512E0A39039001FFC00EC07F8B3B3A3123FEA7F 80EAFFC0A44A5A1380D87F005B0070131F6C5C6C495A6C49C7FC380781FC3801FFF03800 7F80233B7DB82B>IIIIIII82 DI<003FB812E0A3D9C003EB001F273E0001FE130348EE01F000 78160000701770A300601730A400E01738481718A4C71600B3B0913807FF80011FB612E0 A335397DB83C>I86 D<007FB590383FFFFCA3C601F801071380D97FE0D903FCC7FC013F EC01F06D6C5C5F6D6C5C6D6C13034CC8FC6D6C1306160E6D6C5B6DEB8018163891387FC0 306E6C5A16E06E6C5A91380FF18015FB6EB4C9FC5D14036E7EA26E7F6F7EA24B7E15DF91 38019FF09138038FF8150F91380607FC91380E03FE140C4A6C7EEC38000230804A6D7E14 E04A6D7E49486D7E130391C76C7E01066E7E130E010C6E7E011C1401013C8101FE822607 FF80010713E0B500E0013FEBFF80A339397EB83E>88 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmti10 10.95 31 /Fp 31 123 df44 D<171C173C177CA217FCA216011603A21607A24C7EA2161DA216391679167116E1A2ED01 C1A2ED038115071601150EA2031C7FA24B7EA25D15F05D4A5AA24A5AA24AC7FC5C140E5C 021FB6FC4A81A20270C7127FA25C13015C495AA249C8FCA2130E131E131C133C5B01F882 487ED807FEEC01FFB500E0017FEBFF80A25C39417BC044>65 D<9339FF8001C0030F13E0 033F9038F803809239FF807E07913A03FC001F0FDA0FF0EB071FDA1FC0ECBF00DA7F806D B4FC4AC77E495AD903F86E5A495A130F4948157E4948157C495A13FF91C9FC4848167812 035B1207491670120FA2485A95C7FC485AA3127F5BA312FF5BA490CCFCA2170FA2170EA2 171E171C173C173817786C16706D15F04C5A003F5E6D1403001F4B5A6D4AC8FC000F151E 6C6C5C6C6C14F86C6C495A6C6CEB07C090397FC03F8090261FFFFEC9FC010713F0010013 803A4272BF41>67 D71 D<49B6FC5BA2D9000313005D5DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF 92C7FCA35B5CA313035CA313075CA3130F5CA3131F5CA3133F5CA2137FA25C497EB67EA3 283E7BBD23>73 D<49B612C0A25FD9000390C8FC5D5DA314075DA3140F5DA3141F5DA314 3F5DA3147F5DA314FF92C9FCA35B5CA313035C18C0EF01E0010716C05C17031880130F4A 140718005F131F4A141EA2173E013F5D4A14FC1601017F4A5A16074A131F01FFECFFF0B8 FCA25F333E7BBD39>76 D<49B5933807FFFC496062D90003F0FC00505ADBBF805E1A771A EF1407033F923801CFE0A2F1039F020FEE071F020E606F6C140E1A3F021E161C021C0438 5BA2F1707F143C023804E090C7FCF001C0629126780FE0495A02705FF00700F00E0114F0 02E0031C5BA2F03803010116704A6C6C5D18E019070103ED01C00280DA03805BA2943807 000F13070200020E5C5FDB03F8141F495D010E4B5CA24D133F131E011CDAF9C05CEEFB80 197F013C6DB4C7FC013895C8FC5E01784A5C13F8486C4A5CD807FE4C7EB500F04948B512 FE16E01500563E7BBD52>I<92390FF001C0ED7FFE4AB5EA0380913907F80FC791390FC0 03EF91391F8001FF4AC71300027E805C495A4948143EA2495AA2010F153C5CA3011F1538 A38094C7FC80A214FC6DB4FC15F015FE6DEBFFC06D14F06D14FC6D80143F020F7F020180 EC001F150303007F167F163FA2161FA212075A5F120EA2001E153F94C7FCA2163E003E15 7E167C003F15FC4B5A486C5C4B5A6D495AD87DE0EB1F80D8F8F849C8FC017F13FE39F03F FFF8D8E00F13E048C690C9FC32427ABF33>83 D<48B9FCA25A903AFE001FF00101F89138 E0007FD807E0163E49013F141E5B48C75BA2001E147FA2001C4B131C123C003814FFA200 7892C7FC12704A153C00F01738485CC716001403A25DA21407A25DA2140FA25DA2141FA2 5DA2143FA25DA2147FA25DA214FFA292C9FCA25BA25CA21303A25CEB0FFE003FB67E5AA2 383D71BC41>I<001FB500F090B512F0485DA226003FF0C7380FFC004AEC03F04A5D715A 017F1503A24A5DA201FF150795C7FC91C8FCA2485E170E5BA20003161E171C5BA2000716 3C17385BA2000F167817705BA2001F16F05F5BA2003F1501A2495DA2007F1503A2495DA2 160794C8FC48C8FC5E160E161E6C151C163C5E5E5E6C6C13014B5A001F4A5A6C6C011FC9 FC6D133E6C6C13F83903FC07F0C6B512C0013F90CAFCEB07F83C406FBD44>I<147E49B4 7E903907C1C38090391F80EFC090383F00FF017E137F4914804848133F485AA248481400 120F5B001F5C157E485AA215FE007F5C90C7FCA21401485C5AA21403EDF0385AA21407ED E078020F1370127C021F13F0007E013F13E0003E137FECF3E1261F01E313C03A0F8781E3 803A03FF00FF00D800FC133E252977A72E>97 DIIII<167C4BB4FC923807C78092380F83C0ED1F8716 1FED3F3FA2157EA21780EE0E004BC7FCA414015DA414035DA30103B512F8A390260007E0 C7FCA3140F5DA5141F5DA4143F92C8FCA45C147EA414FE5CA413015CA4495AA4495AA449 5A121E127F5C12FF49C9FCA2EAFE1EEAF83C1270EA7878EA3FE0EA0F802A5383BF1C>I< EC03F0EC0FFC91383E0E1C9138FC077E903901F003FE1303903807E001D90FC013FCEB1F 80A2EB3F004914F8137E01FE1303A2484814F0A2150712034914E0A2150F12074914C0A2 151FA216805B153F1203ED7F006D5BA200015B0000495A9038F80F7E90387C1EFEEB1FF8 903807E0FC90C7FC1401A25DA21403A25D001C1307007F5C48130F5D4A5A4AC7FC48137E 00F85B387C03F0381FFFC0D803FEC8FC273B7CA72A>II<1478EB01FCA2 1303A314F8EB00E01400AD137C48B4FC38038F80EA0707000E13C0121E121CEA3C0F1238 A2EA781F00701380A2EAF03F140012005B137E13FE5BA212015BA212035B1438120713E0 000F1378EBC070A214F0EB80E0A2EB81C01383148038078700EA03FEEA00F8163E79BC1C >I108 DII< EC1FC0ECFFF8903803F07C90380FC01FEB1F8090393F000F80017E14C0491307484814E0 485A12075B000F15F0485AA2485AA2ED0FE0127F90C7FCA2151F4815C05AA2ED3F80A2ED 7F00A248147E007C5C007E13015D4A5A003E495A6C495A4A5A260F803EC7FC3807C0FC38 01FFF038003F80242977A72E>I<903903E001F890390FF807FE903A1E7C1E0F80903A1C 3E3C07C0013C137801389038E003E0EB783F017001C013F0ED80019038F07F0001E015F8 147E1603000113FEA2C75AA20101140717F05CA20103140F17E05CA20107EC1FC0A24A14 80163F010F15005E167E5E131F4B5A6E485A4B5A90393FB80F80DA9C1FC7FCEC0FFCEC03 E049C9FCA2137EA213FEA25BA21201A25BA21203A2387FFFE0B5FCA22D3A80A72E>I114 DII<137C48B4141C26038F 80137EEA0707000E7F001E15FE121CD83C0F5C12381501EA781F007001805BA2D8F03F13 03140000005D5B017E1307A201FE5C5B150F1201495CA2151F0003EDC1C0491481A2153F 1683EE0380A2ED7F07000102FF13005C01F8EBDF0F00009038079F0E90397C0F0F1C9039 1FFC07F8903907F001F02A2979A731>I<017CEB01C048B4EB07F038038F80EA0707000E 01C013F8121E001C1403EA3C0F0038EC01F0A2D8781F130000705BA2EAF03F91C712E012 005B017E130116C013FE5B1503000115805BA2ED07001203495B150EA25DA25D15780001 14706D5B0000495A6D485AD97E0FC7FCEB1FFEEB03F0252979A72A>I<137C48B4143826 038F8013FCEA0707000E7F001E1401001C15F8EA3C0F12381503D8781F14F000701380A2 D8F03F1307020013E012005B017E130F16C013FE5B151F1201491480A2153F000315005B A25D157EA315FE5D00011301EBF8030000130790387C1FF8EB3FF9EB07E1EB00035DA214 07000E5CEA3F80007F495AA24A5AD8FF0090C7FC143E007C137E00705B387801F0383803 E0381E0FC06CB4C8FCEA03F8263B79A72C>121 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmbx10 10.95 49 /Fq 49 123 df12 D44 D46 D<140F143F5C495A130F48B5FCB6FCA313F7EAFE071200B3B3A8B712F0A5243C78BB 34>49 D<903803FF80013F13F890B512FE00036E7E4881260FF80F7F261FC0037F4848C6 7F486C6D7E6D6D7E487E6D6D7EA26F1380A46C5A6C5A6C5A0007C7FCC8FC4B1300A25E15 3F5E4B5AA24B5A5E4A5B4A5B4A48C7FC5D4A5AEC1FE04A5A4A5A9139FF000F80EB01FC49 5A4948EB1F00495AEB1F8049C7FC017E5C5B48B7FC485D5A5A5A5A5AB7FC5EA4293C7BBB 34>I<903801FFE0010F13FE013F6D7E90B612E04801817F3A03FC007FF8D807F06D7E82 D80FFC131F6D80121F7FA56C5A5E6C48133FD801F05CC8FC4B5A5E4B5A4A5B020F5B9026 07FFFEC7FC15F815FEEDFFC0D9000113F06E6C7E6F7E6F7E6F7E1780A26F13C0A217E0EA 0FC0487E487E487E487EA317C0A25D491580127F49491300D83FC0495A6C6C495A3A0FFE 01FFF86CB65A6C5DC61580013F49C7FC010313E02B3D7CBB34>II<00071538D80FE0EB01F801FE133F90B6FC5E5E 5E5E93C7FC5D15F85D15C04AC8FC0180C9FCA9ECFFC0018713FC019F13FF90B67E020113 E09039F8007FF0496D7E01C06D7E5B6CC77FC8120F82A31780A21207EA1FC0487E487E12 FF7FA21700A25B4B5A6C5A01805C6CC7123F6D495AD81FE0495A260FFC075B6CB65A6C92 C7FCC614FC013F13F0010790C8FC293D7BBB34>II<16FCA24B7EA24B7EA34B7FA24B7FA34B7FA24B7FA34B7F157C03FC 7FEDF87FA2020180EDF03F0203804B7E02078115C082020F814B7E021F811500824A8102 3E7F027E81027C7FA202FC814A147F49B77EA34982A2D907E0C7001F7F4A80010F835C83 011F8391C87E4983133E83017E83017C81B500FC91B612FCA5463F7CBE4F>65 DI<9226 07FFC0130E92B500FC131E020702FF133E023FEDC07E91B7EAE1FE01039138803FFB4990 39F80003FF4901C01300013F90C8127F4948151FD9FFF8150F48491507485B4A15034817 01485B18004890CAFC197E5A5B193E127FA349170012FFAC127F7F193EA2123FA27F6C18 7E197C6C7F19FC6C6D16F86C6D150119F06C6D15036C6DED07E0D97FFEED0FC06D6CED3F 80010F01C0ECFF006D01F8EB03FE6D9039FF801FFC010091B55A023F15E0020715800200 02FCC7FC030713C03F407ABE4C>IIII<922607FFC0 130E92B500FC131E020702FF133E023FEDC07E91B7EAE1FE01039138803FFB499039F800 03FF4901C01300013F90C8127F4948151FD9FFF8150F48491507485B4A1503481701485B 18004890CAFC197E5A5B193E127FA34994C7FC12FFAB0407B612FC127F7FA3003F92C738 3FFE00A27F7EA26C7FA26C7F6C7FA26C7F6C7FD97FFE157F6D6C7E010F01E014FF6D01F8 13036D9038FF801F010091B512F3023F15C00207ED803E02009138FE000E030701E090C7 FC46407ABE52>III76 DIIII82 D<903A03FFC001C0011FEBF803017FEBFE07 48B6128F4815DF48010013FFD80FF8130F48481303497F4848EB007F127F49143F161F12 FF160FA27F1607A27F7F01FC91C7FCEBFF806C13F8ECFFC06C14FCEDFF806C15E016F86C 816C816C816C16806C6C15C07F010715E0EB007F020714F0EC003F1503030013F8167F16 3F127800F8151FA2160FA27EA217F07E161F6C16E06D143F01E015C001F8EC7F8001FEEB 01FF9026FFE00713004890B55A486C14F8D8F81F5CD8F00314C027E0003FFEC7FC2D407A BE3A>I<003FB912FCA5903BFE003FFE003FD87FF0EE0FFE01C0160349160190C7150019 7E127EA2007C183EA400FC183F48181FA5C81600B3AF010FB712F8A5403D7CBC49>III<007FB6013FB5 12F0A5D8001F01C0D9003FC7FC6D6D147E18FE6D6D5C6D6D495A6D4B5A6F13076D6D5C6E 6C495A4D5A6EEB803F6E01C090C8FC6E147E705A6E13F16EEBF9F86EEBFBF0EEFFE0806F 5B5F816F7F81836F7F81834B7F4B7F5D83DB3F3F7FED7E1F03FE804B6C7F4A486C7F4A48 7E0207814B6C7F4A487E4A4880023F6E7E92C76C7F027E804A8201016F7F4A6E7F495A49 486E7F010F6F7F4A80B600C0017F90B5FCA5483E7DBD4F>88 D<903807FFC0013F13F848 B6FC48812607FE037F260FF8007F6DEB3FF0486C806F7EA36F7EA26C5A6C5AEA01E0C8FC 153F91B5FC130F137F3901FFFE0F4813E0000F1380381FFE00485A5B485A12FF5BA4151F 7F007F143F6D90387BFF806C6C01FB13FE391FFF07F36CEBFFE100031480C6EC003FD91F F890C7FC2F2B7DA933>97 D<13FFB5FCA512077EAFEDFFE0020713FC021FEBFF80027F80 DAFF8113F09139FC003FF802F06D7E4A6D7E4A13074A80701380A218C082A318E0AA18C0 A25E1880A218005E6E5C6E495A6E495A02FCEB7FF0903AFCFF01FFE0496CB55AD9F01F91 C7FCD9E00713FCC7000113C033407DBE3A>II< EE07F8ED07FFA5ED003F161FAFEC7FF0903807FFFE011FEBFF9F017F14DF9039FFF01FFF 48EBC00348EB00014848EB007F485A001F153F5B123FA2127F5BA212FFAA127FA37F123F A26C6C147F120F6D14FF6C6C01037F6C6D48EBFFE06CEBF03F6C6CB512BF6D143F010713 FC010001E0EBE00033407DBE3A>III<903A03FF8007F0013F9038F83FF8499038FCFFFC48B712FE48018313F93A07FC 007FC34848EB3FE1001FEDF1FC4990381FF0F81700003F81A7001F5DA26D133F000F5D6C 6C495A3A03FF83FF8091B5C7FC4814FC01BF5BD80F03138090CAFCA2487EA27F13F06CB6 FC16F016FC6C15FF17806C16C06C16E01207001F16F0393FE000034848EB003F49EC1FF8 00FF150F90C81207A56C6CEC0FF06D141F003F16E001F0147FD81FFC903801FFC02707FF 800F13006C90B55AC615F8013F14E0010101FCC7FC2F3D7DA834>I<13FFB5FCA512077E AFED1FF8EDFFFE02036D7E4A80DA0FE07F91381F007F023C805C4A6D7E5CA25CA35CB3A4 B5D8FE0FB512E0A5333F7CBE3A>II<13FFB5FCA512077EB3B3AF B512FCA5163F7CBE1D>108 D<01FFD91FF8ECFFC0B590B5010713F80203DAC01F13FE4A 6E487FDA0FE09026F07F077F91261F003FEBF8010007013EDAF9F0806C0178ECFBC04A6D B4486C7FA24A92C7FC4A5CA34A5CB3A4B5D8FE07B5D8F03FEBFF80A551297CA858>I<01 FFEB1FF8B5EBFFFE02036D7E4A80DA0FE07F91381F007F0007013C806C5B4A6D7E5CA25C A35CB3A4B5D8FE0FB512E0A533297CA83A>II<01FFEBFFE0B5000713FC021FEBFF80027F80DAFF8113F09139FC007FF80003 01F06D7E4A6D7E4A130F4A6D7E1880A27013C0A38218E0AA4C13C0A318805E18005E6E5C 6E495A6E495A02FCEBFFF0DAFF035B92B55A029F91C7FC028713FC028113C00280C9FCAC B512FEA5333B7DA83A>I<3901FE01FE00FF903807FF804A13E04A13F0EC3F1F91387C3F F8000713F8000313F0EBFFE0A29138C01FF0ED0FE091388007C092C7FCA391C8FCB3A2B6 FCA525297DA82B>114 D<90383FFC1E48B512BE000714FE5A381FF00F383F800148C7FC 007E147EA200FE143EA27E7F6D90C7FC13F8EBFFE06C13FF15C06C14F06C806C806C806C 80C61580131F1300020713C014000078147F00F8143F151F7EA27E16806C143F6D140001 E013FF9038F803FE90B55A15F0D8F87F13C026E00FFEC7FC222B7DA929>III120 D<003FB612F8A4D9F80113F001C014E0495A494813C04A1380007E15005C4A5A 007C5C147F4A5A495B5DC65A495B495BA249EB007C495A5C137F494813FC484913F85C5A 48EBC00114804814034813004848130749131F007FECFFF0B7FCA426287DA72E>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmr10 10.95 77 /Fr 77 125 df<4AB4EB0FE0021F9038E03FFC913A7F00F8FC1ED901FC90383FF03FD907 F090397FE07F80494801FF13FF4948485BD93F805C137F0200ED7F00EF003E01FE6D91C7 FC82ADB97EA3C648C76CC8FCB3AE486C4A7E007FD9FC3FEBFF80A339407FBF35>11 D<4AB4FC021F13C091387F01F0903901FC0078D907F0131C4948133E494813FF49485A13 7F1400A213FE6F5A163893C7FCAA167FB8FCA33900FE00018182B3AC486CECFF80007FD9 FC3F13FEA32F407FBF33>I<4AB47E021F13F791387F00FFEB01F8903807F001EB0FE0EB 1FC0EB3F80137F14008101FE80AEB8FCA3C648C77EB3AE486CECFF80007FD9FC3F13FEA3 2F407FBF33>I<4AB4ECFF80021FD9C00F13E0913B7F01F03F80F8903C01F80078FE003C D907F0D93FF8130E49484948131F49484948EB7F804948484913FF137F02005CA201FE92 C7FC6FED7F0070141C96C7FCAAF13F80BBFCA3C648C76CC7FC197F193FB3AC486C4A6CEB 7FC0007FD9FC3FD9FE1FB5FCA348407FBF4C>I<001E130F397F803FC000FF137F01C013 E0A201E013F0A3007F133F391E600F3000001300A401E01370491360A3000114E04913C0 0003130101001380481303000EEB070048130E0018130C0038131C003013181C1C7DBE2D >34 D38 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A3120113 80120313005A120E5A1218123812300B1C79BE19>I<1430147014E0EB01C0EB03801307 EB0F00131E133E133C5B13F85B12015B1203A2485AA2120F5BA2121F90C7FCA25AA3123E 127EA6127C12FCB2127C127EA6123E123FA37EA27F120FA27F1207A26C7EA212017F1200 7F13787F133E131E7FEB07801303EB01C0EB00E014701430145A77C323>I<12C07E1270 7E7E121E7E6C7E7F12036C7E7F12007F1378137CA27FA2133F7FA21480130FA214C0A313 0714E0A6130314F0B214E01307A614C0130FA31480A2131F1400A25B133EA25BA2137813 F85B12015B485A12075B48C7FC121E121C5A5A5A5A145A7BC323>I<1506150FB3A9007F B912E0BA12F0A26C18E0C8000FC9FCB3A915063C3C7BB447>43 D<121EEA7F8012FF13C0 A213E0A3127FEA1E601200A413E013C0A312011380120313005A120E5A1218123812300B 1C798919>II<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A79 8919>I48 DIII<150E151E153EA2157EA215FE1401A21403EC077E 1406140E141CA214381470A214E0EB01C0A2EB0380EB0700A2130E5BA25B5BA25B5B1201 485A90C7FC5A120E120C121C5AA25A5AB8FCA3C8EAFE00AC4A7E49B6FCA3283E7EBD2D> I<00061403D80780131F01F813FE90B5FC5D5D5D15C092C7FC14FCEB3FE090C9FCACEB01 FE90380FFF8090383E03E090387001F8496C7E49137E497F90C713800006141FC813C0A2 16E0150FA316F0A3120C127F7F12FFA416E090C7121F12FC007015C012780038EC3F8012 3C6CEC7F00001F14FE6C6C485A6C6C485A3903F80FE0C6B55A013F90C7FCEB07F8243F7C BC2D>II<1238123C123F90B612FC A316F85A16F016E00078C712010070EC03C0ED078016005D48141E151C153C5DC8127015 F04A5A5D14034A5A92C7FC5C141EA25CA2147C147814F8A213015C1303A31307A3130F5C A2131FA6133FAA6D5A0107C8FC26407BBD2D>III<121EEA 7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A27 79A619>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3121E127FEAFF80A213C0A4 127F121E1200A412011380A3120313005A1206120E120C121C5A1230A20A3979A619>I< 007FB912E0BA12F0A26C18E0CDFCAE007FB912E0BA12F0A26C18E03C167BA147>61 D63 D<15074B7EA34B7EA34B7EA34B7EA34B7E15E7A2913801C7FC15C3A291380381 FEA34AC67EA3020E6D7EA34A6D7EA34A6D7EA34A6D7EA34A6D7EA349486D7E91B6FCA249 819138800001A249C87EA24982010E157FA2011E82011C153FA2013C820138151FA20178 82170F13FC00034C7ED80FFF4B7EB500F0010FB512F8A33D417DC044>65 DIIIIII< B612F0A3C6EBF000EB3FC0B3B3B2EBFFF0B612F0A31C3E7EBD21>73 D<011FB512FCA3D9000713006E5A1401B3B3A6123FEA7F80EAFFC0A44A5A1380D87F005B 007C130700385C003C495A6C495A6C495A2603E07EC7FC3800FFF8EB3FC026407CBD2F> IIIIIIIIII<003FB91280A3903AF0007FE001018090393FC0003F48C7 ED1FC0007E1707127C00781703A300701701A548EF00E0A5C81600B3B14B7E4B7E0107B6 12FEA33B3D7DBC42>II87 D<486C13C00003130101001380481303000EEB070048130E0018130C0038131C 003013180070133800601330A300E01370481360A400CFEB678039FFC07FE001E013F0A3 007F133FA2003F131F01C013E0390F0007801C1C73BE2D>92 D97 DI<49B4FC010F13E090383F 00F8017C131E4848131F4848137F0007ECFF80485A5B121FA24848EB7F00151C007F91C7 FCA290C9FC5AAB6C7EA3003FEC01C07F001F140316806C6C13076C6C14000003140E6C6C 131E6C6C137890383F01F090380FFFC0D901FEC7FC222A7DA828>II< EB01FE90380FFFC090383F03F09038FC01F848486C7E4848137E48487F000F158049131F 001F15C04848130FA2127F16E090C7FCA25AA290B6FCA290C9FCA67EA27F123F16E06C7E 1501000F15C06C6C13036DEB07806C6C1400C66C131E017E5B90381F80F8903807FFE001 0090C7FC232A7EA828>II<167C903903F801FF903A1FFF078F8090397E0FDE1F9038F803F83803F001A23B07 E000FC0600000F6EC7FC49137E001F147FA8000F147E6D13FE00075C6C6C485AA23901F8 03E03903FE0FC026071FFFC8FCEB03F80006CAFC120EA3120FA27F7F6CB512E015FE6C6E 7E6C15E06C810003813A0FC0001FFC48C7EA01FE003E140048157E825A82A46C5D007C15 3E007E157E6C5D6C6C495A6C6C495AD803F0EB0FC0D800FE017FC7FC90383FFFFC010313 C0293D7EA82D>III<1478EB01FEA2EB03FFA4EB01FEA2EB0078 1400AC147FEB7FFFA313017F147FB3B3A5123E127F38FF807E14FEA214FCEB81F8EA7F01 387C03F0381E07C0380FFF803801FC00185185BD1C>III<2701F801FE14FF00FF902707FFC00313E0913B1E07E00F03 F0913B7803F03C01F80007903BE001F87000FC2603F9C06D487F000101805C01FBD900FF 147F91C75B13FF4992C7FCA2495CB3A6486C496CECFF80B5D8F87FD9FC3F13FEA347287D A74C>I<3901F801FE00FF903807FFC091381E07E091387803F000079038E001F82603F9 C07F0001138001FB6D7E91C7FC13FF5BA25BB3A6486C497EB5D8F87F13FCA32E287DA733 >I<14FF010713E090381F81F890387E007E01F8131F4848EB0F804848EB07C04848EB03 E0000F15F04848EB01F8A2003F15FCA248C812FEA44815FFA96C15FEA36C6CEB01FCA300 1F15F86C6CEB03F0A26C6CEB07E06C6CEB0FC06C6CEB1F80D8007EEB7E0090383F81FC90 380FFFF0010090C7FC282A7EA82D>I<3901FC03FC00FF90381FFF8091387C0FE09039FD E003F03A03FFC001FC6C496C7E91C7127F49EC3F805BEE1FC017E0A2EE0FF0A3EE07F8AA EE0FF0A4EE1FE0A2EE3FC06D1580EE7F007F6E13FE9138C001F89039FDE007F09039FC78 0FC0DA3FFFC7FCEC07F891C9FCAD487EB512F8A32D3A7EA733>I<02FF131C0107EBC03C 90381F80F090397F00387C01FC131CD803F8130E4848EB0FFC150748481303121F485A15 01485AA448C7FCAA6C7EA36C7EA2001F14036C7E15076C6C130F6C7E6C6C133DD8007E13 7990383F81F190380FFFC1903801FE0190C7FCAD4B7E92B512F8A32D3A7DA730>I<3901 F807E000FFEB1FF8EC787CECE1FE3807F9C100031381EA01FB1401EC00FC01FF13304913 00A35BB3A5487EB512FEA31F287EA724>I<90383FC0603901FFF8E03807C03F381F000F 003E1307003C1303127C0078130112F81400A27E7E7E6D1300EA7FF8EBFFC06C13F86C13 FE6C7F6C1480000114C0D8003F13E0010313F0EB001FEC0FF800E01303A214017E1400A2 7E15F07E14016C14E06CEB03C0903880078039F3E01F0038E0FFFC38C01FE01D2A7DA824 >I<131CA6133CA4137CA213FCA2120112031207001FB512C0B6FCA2D801FCC7FCB3A215 E0A912009038FE01C0A2EB7F03013F138090381F8700EB07FEEB01F81B397EB723>IIIIII<001FB61280A2EBE0000180140049485A 001E495A121C4A5A003C495A141F00385C4A5A147F5D4AC7FCC6485AA2495A495A130F5C 495A90393FC00380A2EB7F80EBFF005A5B484813071207491400485A48485BA248485B48 48137F00FF495A90B6FCA221277EA628>III E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%EndSetup %%Page: 1 1 1 0 bop 120 643 a Fr(T)-8 b(o)45 b(app)s(ear:)f(M.)i(I.)e(Jordan,)g (\(Ed.\))p Fq(,)52 b Fp(L)-5 b(e)g(arning)47 b(in)f(Gr)-5 b(aphic)g(al)48 b(Mo)-5 b(dels)p Fq(,)53 b Fr(Klu)m(w)m(er)120 751 y(Academic)31 b(Publishers)p Fq(.)120 967 y(IMPR)m(O)m(VING)k(THE)f (MEAN)h(FIELD)f(APPR)m(O)m(XIMA)-9 b(TION)120 1075 y(VIA)35 b(THE)f(USE)h(OF)f(MIXTURE)h(DISTRIBUTIONS)618 1591 y Fo(TOMMI)28 b(S.)f(JAAKK)n(OLA)618 1708 y Fp(University)33 b(of)f(California)618 1816 y(Santa)i(Cruz,)f(CA)618 1977 y Fo(AND)618 2121 y(MICHAEL)28 b(I.)f(JORD)n(AN)618 2238 y Fp(Massachusetts)34 b(Institute)f(of)g(T)-7 b(e)i(chnolo)g(gy)618 2346 y(Cambridge,)34 b(MA)120 2663 y Fq(Abstract.)f Fr(Mean)g(\014eld)f (metho)s(ds)g(pro)m(vide)g(computationally)g(e\016cien)m(t)h(appro)m (xi-)120 2770 y(mations)28 b(to)i(p)s(osterior)d(probabilit)m(y)g (distributions)d(for)29 b(graphical)e(mo)s(dels.)h(Simple)120 2878 y(mean)43 b(\014eld)f(metho)s(ds)h(mak)m(e)h(a)g(completely)f (factorized)g(appro)m(ximation)g(to)g(the)120 2986 y(p)s(osterior,)g (whic)m(h)f(is)h(unlik)m(ely)e(to)k(b)s(e)e(accurate)i(when)e(the)g(p)s (osterior)g(is)g(m)m(ulti-)120 3094 y(mo)s(dal.)32 b(Indeed,)h(if)f (the)h(p)s(osterior)g(is)f(m)m(ulti-mo)s(dal,)f(only)h(one)i(of)f(the)h (mo)s(des)e(can)120 3202 y(b)s(e)k(captured.)i(T)-8 b(o)37 b(impro)m(v)m(e)g(the)g(mean)h(\014eld)d(appro)m(ximation)h(in)g(suc)m (h)h(cases,)h(w)m(e)120 3310 y(emplo)m(y)f(mixture)e(mo)s(dels)h(as)h (p)s(osterior)f(appro)m(ximations,)g(where)h(eac)m(h)h(mixture)120 3418 y(comp)s(onen)m(t)g(is)e(a)i(factorized)g(distribution.)c(W)-8 b(e)39 b(describ)s(e)c(e\016cien)m(t)j(metho)s(ds)f(for)120 3526 y(optimizing)28 b(the)j(parameters)g(in)e(these)i(mo)s(dels.)120 3967 y Fq(1.)73 b(In)m(tro)s(duction)120 4151 y Fr(Graphical)31 b(mo)s(dels)f(pro)m(vide)h(a)h(con)m(v)m(enien)m(t)h(formalism)d(in)h (whic)m(h)f(to)j(express)e(and)120 4259 y(manipulate)g(conditional)g (indep)s(endence)f(statemen)m(ts.)35 b(Inference)d(algorithms)g(for)120 4367 y(graphical)44 b(mo)s(dels)h(exploit)f(these)i(indep)s(endence)e (statemen)m(ts,)j(using)d(them)i(to)120 4475 y(compute)37 b(conditional)d(probabilities)f(while)i(a)m(v)m(oiding)h(brute)g(force) h(marginaliza-)120 4583 y(tion)25 b(o)m(v)m(er)i(the)f(join)m(t)g (probabilit)m(y)d(table.)j(Man)m(y)g(inference)f(algorithms,)g(in)f (particu-)120 4690 y(lar)k(the)h(algorithms)e(that)i(construct)g(a)g (junction)e(tree,)j(mak)m(e)g(explicit)c(their)i(usage)120 4798 y(of)f(conditional)e(indep)s(endence)f(b)m(y)i(constructing)g(a)h (data)g(structure)g(that)g(captures)p eop %%Page: 2 2 2 1 bop 120 85 a Fr(2)427 b Fo(TOMMI)27 b(S.)h(JAAKK)n(OLA)f(AND)h (MICHAEL)g(I.)g(JORD)n(AN)120 310 y Fr(the)40 b(essen)m(tial)f(Mark)m (o)m(v)i(prop)s(erties)d(underlying)e(the)k(graph.)f(That)g(is,)g(the)h (algo-)120 418 y(rithm)31 b(groups)h(in)m(teracting)h(v)-5 b(ariables)31 b(in)m(to)i(clusters,)f(suc)m(h)h(that)g(the)g(h)m(yp)s (ergraph)120 526 y(of)f(clusters)f(has)g(Mark)m(o)m(v)i(prop)s(erties)d (that)j(allo)m(w)e(simple)e(lo)s(cal)i(algorithms)g(to)h(b)s(e)120 634 y(emplo)m(y)m(ed)43 b(for)g(inference.)g(In)f(the)h(b)s(est)g (case,)h(in)e(whic)m(h)g(the)h(original)e(graph)i(is)120 742 y(sparse)31 b(and)f(without)g(long)h(cycles,)h(the)f(clusters)f (are)i(small)d(and)i(inference)f(is)g(e\016-)120 850 y(cien)m(t.)g(In)f(the)h(w)m(orst)g(case,)h(suc)m(h)f(as)g(the)g(case)h (of)e(a)i(dense)e(graph,)g(the)h(clusters)f(are)120 958 y(large)d(and)f(inference)g(is)f(ine\016cien)m(t)h(\(complexit)m(y)h (scales)g(exp)s(onen)m(tially)e(in)g(the)i(size)120 1066 y(of)f(the)g(largest)g(cluster\).)g(Arc)m(hitectures)g(for)f(whic)m(h)g (the)h(complexit)m(y)f(is)g(prohibitiv)m(e)120 1174 y(include)30 b(the)i(QMR)f(database)i(\(Sh)m(w)m(e,)f(et)h(al.,)f(1991\),)i(the)e (la)m(y)m(ered)g(sigmoid)f(b)s(elief)120 1282 y(net)m(w)m(ork)25 b(\(Neal,)g(1992\),)h(and)e(the)g(factorial)g(hidden)e(Mark)m(o)m(v)k (mo)s(del)d(\(Ghahramani)120 1389 y(&)30 b(Jordan,)g(1996\).)256 1518 y(Mean)i(\014eld)e(theory)h(\(P)m(arisi,)g(1988\))i(pro)m(vides)d (an)h(alternativ)m(e)h(p)s(ersp)s(ectiv)m(e)e(on)120 1626 y(the)h(inference)f(problem.)g(The)h(in)m(tuition)d(b)s(ehind)g (mean)k(\014eld)d(metho)s(ds)h(is)g(that)i(in)120 1734 y(dense)39 b(graphs)g(eac)m(h)i(no)s(de)e(is)g(sub)5 b(ject)40 b(to)g(in\015uences)e(from)h(man)m(y)h(other)g(no)s(des;)120 1842 y(th)m(us,)d(to)i(the)e(exten)m(t)i(that)f(eac)m(h)h(in\015uence)d (is)g(w)m(eak)j(and)e(the)g(total)i(in\015uence)d(is)120 1950 y(roughly)h(additiv)m(e)g(\(on)i(an)f(appropriate)g(scale\),)h (eac)m(h)g(no)s(de)f(should)e(roughly)h(b)s(e)120 2058 y(c)m(haracterized)30 b(b)m(y)f(its)g(mean)g(v)-5 b(alue.)29 b(In)f(particular)g(this)g(c)m(haracterization)i(is)e(v)-5 b(alid)120 2166 y(in)31 b(situations)f(in)h(whic)m(h)f(the)i(la)m(w)g (of)g(large)g(n)m(um)m(b)s(ers)e(can)i(b)s(e)f(applied.)f(The)i(mean) 120 2274 y(v)-5 b(alue)32 b(is)g(unkno)m(wn,)f(but)h(it)g(is)g(related) g(to)i(the)e(mean)h(v)-5 b(alues)32 b(of)h(the)f(other)h(no)s(des.)120 2382 y(Mean)22 b(\014eld)e(theory)i(consists)f(of)h(\014nding)d (consistency)i(relations)g(b)s(et)m(w)m(een)h(the)g(mean)120 2490 y(v)-5 b(alues)31 b(for)g(eac)m(h)h(of)g(the)f(no)s(des)g(and)f (solving)g(the)i(resulting)d(equations)i(\(generally)120 2598 y(b)m(y)k(iteration\).)h(F)-8 b(or)36 b(graphical)e(mo)s(dels)g (these)i(equations)f(generally)f(ha)m(v)m(e)j(simple)120 2706 y(graphical)31 b(in)m(terpretations;)g(for)h(example,)f(P)m (eterson)i(and)e(Anderson)g(\(1987\))j(and)120 2814 y(Saul,)29 b(Jaakk)m(ola,)j(and)e(Jordan)g(\(1996\))j(found,)c(in)g(the)i(cases)g (of)g(Mark)m(o)m(v)h(net)m(w)m(orks)120 2922 y(and)h(Ba)m(y)m(esian)i (net)m(w)m(orks,)g(resp)s(ectiv)m(ely)-8 b(,)34 b(that)g(the)g(mean)g (v)-5 b(alue)33 b(of)h(a)h(giv)m(en)e(no)s(de)120 3029 y(is)c(obtained)h(additiv)m(ely)f(\(on)h(a)h(logit)f(scale\))h(from)e (the)i(mean)f(v)-5 b(alues)30 b(of)g(the)h(no)s(des)120 3137 y(in)e(the)i(Mark)m(o)m(v)h(blank)m(et)e(of)g(the)h(no)s(de.)256 3266 y(Exact)40 b(metho)s(ds)e(and)g(mean)h(\014eld)e(metho)s(ds)h (migh)m(t)g(b)s(e)h(said)e(to)j(b)s(e)e(comple-)120 3374 y(men)m(tary)31 b(in)e(the)i(sense)g(that)g(exact)h(metho)s(ds)e(are)h (b)s(est)f(suited)g(for)g(sparse)g(graphs)120 3482 y(and)35 b(mean)g(\014eld)f(metho)s(ds)h(are)h(b)s(est)f(suited)f(for)h(dense)g (graphs.)g(Both)h(classes)g(of)120 3590 y(metho)s(ds)f(ha)m(v)m(e)i (signi\014can)m(t)d(limitations,)g(ho)m(w)m(ev)m(er,)j(and)e(the)h(gap) f(b)s(et)m(w)m(een)i(their)120 3698 y(resp)s(ectiv)m(e)j(domains)e(of)i (application)e(is)h(large.)h(In)f(particular,)g(the)h(naiv)m(e)g(mean) 120 3806 y(\014eld)29 b(metho)s(ds)i(that)g(w)m(e)g(ha)m(v)m(e)i (referred)d(to)h(ab)s(o)m(v)m(e)h(are)g(based)e(on)h(the)g(appro)m (xima-)120 3914 y(tion)26 b(that)h(eac)m(h)h(no)s(de)e(\015uctuates)h (indep)s(enden)m(tly)d(ab)s(out)i(its)g(mean)h(v)-5 b(alue.)26 b(If)g(there)120 4022 y(are)37 b(a)f(strong)h(in)m(teractions)f(in)f (the)i(net)m(w)m(ork,)g(i.e.)f(if)g(higher-order)e(momen)m(ts)j(are)120 4130 y(imp)s(ortan)m(t,)c(then)g(mean)h(\014eld)e(metho)s(ds)h(will)e (generally)i(fail.)f(One)h(w)m(a)m(y)i(in)d(whic)m(h)120 4238 y(suc)m(h)i(higher-order)e(in)m(teractions)i(ma)m(y)g(manifest)f (themselv)m(es)h(is)f(in)g(the)h(presence)120 4346 y(of)27 b(m)m(ultiple)d(mo)s(des)i(in)f(the)i(distribution;)c(naiv)m(e)k(mean)f (\014eld)f(theory)-8 b(,)28 b(b)m(y)e(assuming)120 4454 y(indep)s(enden)m(t)31 b(\015uctuations,)i(e\013ectiv)m(ely)i(assumes)e (a)h(unimo)s(dal)d(distribution)f(and)120 4562 y(will)e(generally)h (fail)g(if)h(there)g(are)h(m)m(ultiple)d(mo)s(des.)256 4690 y(One)k(approac)m(h)h(to)g(narro)m(wing)e(the)i(gap)f(b)s(et)m(w)m (een)h(exact)h(metho)s(ds)e(and)g(mean)120 4798 y(\014eld)23 b(metho)s(ds)i(is)f(the)h(\\structured)f(mean)h(\014eld")f(metho)s (dology)h(prop)s(osed)e(b)m(y)i(Saul)p eop %%Page: 3 3 3 2 bop 575 85 a Fo(IMPR)n(O)n(VING)27 b(THE)g(MEAN)h(FIELD)g(APPR)n(O) n(XIMA)-7 b(TION)408 b Fr(3)120 310 y(and)30 b(Jordan)f(\(1996\).)k (This)28 b(approac)m(h)i(in)m(v)m(olv)m(es)h(deleting)e(links)f(in)g (the)j(graph)e(and)120 418 y(iden)m(tifying)43 b(a)i(core)h(graphical)e (substructure)f(that)j(can)f(b)s(e)g(treated)h(e\016cien)m(tly)120 526 y(via)35 b(exact)i(metho)s(ds)d(\(e.g.,)k(a)d(forest)h(of)f (trees\).)i(Probabilistic)32 b(dep)s(endencies)h(not)120 634 y(accoun)m(ted)f(for)e(b)m(y)g(the)h(probabilit)m(y)c(distribution) g(on)j(this)f(core)j(substructure)c(are)120 742 y(treated)42 b(via)e(a)h(naiv)m(e)f(mean)h(\014eld)e(appro)m(ximation.)g(F)-8 b(or)42 b(arc)m(hitectures)f(with)e(an)120 850 y(ob)m(vious)j(c)m (hain-lik)m(e)f(or)i(tree-lik)m(e)f(substructure,)g(suc)m(h)g(as)g(the) h(factorial)f(hidden)120 958 y(Mark)m(o)m(v)k(mo)s(del,)d(this)f(metho) s(d)i(is)e(natural)h(and)h(successful.)f(F)-8 b(or)44 b(arc)m(hitectures)120 1066 y(without)31 b(suc)m(h)h(a)g(readily)e (iden)m(ti\014ed)g(substructure,)h(ho)m(w)m(ev)m(er,)j(suc)m(h)d(as)i (the)f(QMR)120 1174 y(database)44 b(and)e(the)h(la)m(y)m(ered)g (sigmoid)f(b)s(elief)f(net)m(w)m(ork,)j(it)e(is)g(not)h(clear)g(ho)m(w) g(to)120 1282 y(dev)m(elop)30 b(a)h(useful)e(structured)g(mean)h (\014eld)f(appro)m(ximation.)256 1396 y(The)34 b(curren)m(t)h(pap)s(er) e(extends)i(the)g(basic)f(mean)h(\014eld)e(metho)s(dology)i(in)e(a)i (dif-)120 1504 y(feren)m(t)30 b(direction.)f(Rather)h(than)f(basing)g (the)h(appro)m(ximation)f(on)g(the)h(assumption)120 1611 y(of)36 b(a)g(unimo)s(dal)c(appro)m(ximating)j(distribution,)1822 1578 y Fn(1)1893 1611 y Fr(w)m(e)h(build)d(in)h(m)m(ultimo)s(dalit)m(y) f(b)m(y)120 1719 y(allo)m(wing)c(the)h(appro)m(ximating)f(distribution) d(to)31 b(tak)m(e)h(the)e(form)f(of)i(a)f(mixture)f(dis-)120 1827 y(tribution.)35 b(The)i(comp)s(onen)m(ts)h(of)f(the)g(mixture)g (are)g(assumed)g(to)h(b)s(e)f(simple)e(fac-)120 1935 y(torized)d(distributions,)c(for)k(reasons)g(of)g(computational)f (e\016ciency)-8 b(.)33 b(Th)m(us,)e(within)120 2043 y(a)g(mo)s(de)g(w)m (e)g(assume)g(indep)s(enden)m(t)e(\015uctuations,)h(and)g(m)m(ultiple)f (mo)s(des)h(are)h(used)120 2151 y(to)g(capture)g(higher-order)e(in)m (teractions.)256 2265 y(In)k(the)g(follo)m(wing)e(sections)i(w)m(e)g (describ)s(e)f(our)g(mixture-based)g(approac)m(h)h(that)120 2373 y(extends)24 b(the)g(basic)g(mean)g(\014eld)e(metho)s(d.)i (Section)g(2)h(describ)s(es)d(the)i(basics)f(of)i(mean)120 2481 y(\014eld)30 b(appro)m(ximation,)h(pro)m(viding)e(enough)i(detail) g(so)g(as)h(to)g(mak)m(e)g(the)g(pap)s(er)e(self-)120 2589 y(con)m(tained.)k(Section)f(3)h(then)f(describ)s(es)e(ho)m(w)i(to) h(extend)g(the)f(mean)h(\014eld)d(appro)m(x-)120 2697 y(imation)h(b)m(y)g(allo)m(wing)f(mixture)h(distributions)d(in)i(the)i (p)s(osterior)e(appro)m(ximation.)120 2805 y(The)g(follo)m(wing)e (sections)i(dev)m(elop)g(the)h(mac)m(hinery)e(whereb)m(y)h(this)f (appro)m(ximation)120 2913 y(can)h(b)s(e)e(carried)h(out)h(in)e (practice.)120 3151 y Fq(2.)73 b(Mean)35 b(\014eld)g(appro)m(ximation) 120 3332 y Fr(Assume)h(w)m(e)g(ha)m(v)m(e)h(a)g(probabilit)m(y)d(mo)s (del)h Fm(P)13 b Fr(\()p Fm(S;)i(S)1909 3299 y Fl(\003)1949 3332 y Fr(\),)36 b(where)g Fm(S)2375 3299 y Fl(\003)2450 3332 y Fr(is)f(the)i(set)f(of)h(in-)120 3440 y(stan)m(tiated)g(or)f (observ)m(ed)g(v)-5 b(ariables)34 b(\(the)j(\\evidence"\))g(and)e(the)h (v)-5 b(ariables)34 b Fm(S)41 b Fr(are)120 3548 y(hidden)35 b(or)j(unobserv)m(ed.)f(W)-8 b(e)40 b(wish)c(to)i(\014nd)e(a)j (tractable)f(appro)m(ximation)f(to)h(the)120 3655 y(p)s(osterior)29 b(probabilit)m(y)f Fm(P)13 b Fr(\()p Fm(S)5 b Fk(j)p Fm(S)1220 3622 y Fl(\003)1260 3655 y Fr(\).)256 3770 y(In)43 b(its)g(simplest)e(form)i(mean)h(\014eld)e(theory)i(assumes)f (that)h(no)s(des)e(\015uctuate)120 3877 y(indep)s(enden)m(tly)g(around) h(their)h(mean)h(v)-5 b(alues.)44 b(W)-8 b(e)46 b(mak)m(e)g(this)e (assumption)f(ex-)120 3985 y(plicit)26 b(b)m(y)h(expressing)f(the)i (appro)m(ximation)f(in)f(terms)h(of)h(a)g(factorized)g(distribution)120 4093 y Fm(Q)192 4108 y Fj(mf)300 4093 y Fr(\()p Fm(S)5 b Fk(j)p Fm(S)482 4060 y Fl(\003)522 4093 y Fr(\):)971 4328 y Fm(Q)1043 4343 y Fj(mf)1150 4328 y Fr(\()p Fm(S)g Fk(j)p Fm(S)1332 4291 y Fl(\003)1372 4328 y Fr(\))26 b(=)1529 4247 y Fi(Y)1570 4430 y Fj(i)1650 4328 y Fm(Q)1722 4342 y Fj(i)1750 4328 y Fr(\()p Fm(S)1841 4342 y Fj(i)1870 4328 y Fk(j)p Fm(S)1956 4291 y Fl(\003)1995 4328 y Fr(\))p Fm(;)902 b Fr(\(1\))224 4601 y Fh(1)258 4632 y Fg(By)24 b(\\unimo)r(dal)h(distributions")g(w)n(e)f(mean)g(distributions)h(that) f(are)h(log-conca)n(v)n(e.)g(Mean)g(\014eld)120 4715 y(distributions)j(that)g(are)h(pro)r(ducts)f(of)h(exp)r(onen)n(tial)f (family)g(distributions)g(are)h(unimo)r(dal)f(in)g(this)120 4798 y(sense.)p eop %%Page: 4 4 4 3 bop 120 85 a Fr(4)427 b Fo(TOMMI)27 b(S.)h(JAAKK)n(OLA)f(AND)h (MICHAEL)g(I.)g(JORD)n(AN)120 310 y Fr(where)f(the)h(parameters)h Fm(Q)1075 324 y Fj(i)1103 310 y Fr(\()p Fm(S)1194 324 y Fj(i)1222 310 y Fk(j)p Fm(S)1308 277 y Fl(\003)1348 310 y Fr(\))f(dep)s(end)e(on)i(eac)m(h)h(other)f(and)f(on)h(the)g (evidence)120 418 y Fm(S)181 385 y Fl(\003)250 418 y Fr(\(i.e.,)j(w)m(e)g(m)m(ust)f(re\014t)g(the)g(mean)g(\014eld)f(appro)m (ximation)g(for)h(eac)m(h)h(con\014guration)120 526 y(of)g(the)f (evidence\).)256 634 y(The)43 b(mean)g(\014eld)f(appro)m(ximation)g (can)i(b)s(e)e(dev)m(elop)s(ed)h(from)f(sev)m(eral)i(p)s(oin)m(ts)120 742 y(of)33 b(view,)f(but)g(a)h(particularly)d(useful)h(p)s(ersp)s (ectiv)m(e)h(for)h(applications)d(to)k(graphical)120 850 y(mo)s(dels)42 b(is)g(the)h(\\v)-5 b(ariational")43 b(p)s(oin)m(t)f(of)h(view.)g(Within)e(this)h(approac)m(h)h(w)m(e)h(use) 120 958 y(the)30 b(KL)f(div)m(ergence)i(as)f(a)g(measure)g(of)g(the)g (go)s(o)s(dness)f(of)h(the)g(appro)m(ximation)f(and)120 1066 y(c)m(ho)s(ose)23 b(v)-5 b(alues)21 b(of)g(the)h(parameters)g Fm(Q)1438 1080 y Fj(i)1466 1066 y Fr(\()p Fm(S)1557 1080 y Fj(i)1585 1066 y Fk(j)p Fm(S)1671 1033 y Fl(\003)1711 1066 y Fr(\))g(that)g(minimize)d(the)j(KL)f(div)m(ergence:)304 1272 y(KL\()f Fm(Q)559 1287 y Fj(mf)667 1272 y Fr(\()p Fm(S)5 b Fk(j)p Fm(S)849 1234 y Fl(\003)889 1272 y Fr(\))21 b Fk(jj)f Fm(P)13 b Fr(\()p Fm(S)5 b Fk(j)p Fm(S)1268 1234 y Fl(\003)1308 1272 y Fr(\))21 b(\))k(=)1520 1191 y Fi(X)1557 1374 y Fj(S)1655 1272 y Fm(Q)1727 1287 y Fj(mf)1835 1272 y Fr(\()p Fm(S)5 b Fk(j)p Fm(S)2017 1234 y Fl(\003)2057 1272 y Fr(\))15 b(log)2250 1209 y Fm(Q)2322 1224 y Fj(mf)2430 1209 y Fr(\()p Fm(S)5 b Fk(j)p Fm(S)2612 1176 y Fl(\003)2652 1209 y Fr(\))p 2250 1250 438 4 v 2305 1334 a Fm(P)13 b Fr(\()p Fm(S)5 b Fk(j)p Fm(S)2558 1308 y Fl(\003)2598 1334 y Fr(\))2697 1272 y Fm(:)235 b Fr(\(2\))120 1504 y(Wh)m(y)24 b(do)f(w)m(e)h(use)f(KL)o(\()p Fm(Q)963 1519 y Fj(mf)1071 1504 y Fk(jj)p Fm(P)13 b Fr(\))24 b(rather)f(than)g(KL\()p Fm(P)13 b Fk(jj)p Fm(Q)2078 1519 y Fj(mf)2186 1504 y Fr(\)?)24 b(A)f(pragmatic)h(answ)m(er)120 1612 y(is)37 b(that)h(the)g(use)f(of)h(KL)o(\()p Fm(Q)1096 1627 y Fj(mf)1204 1612 y Fk(jj)p Fm(P)13 b Fr(\))39 b(implies)34 b(that)39 b(a)m(v)m(erages)h(are)e(tak)m(en)g(using)e(the)120 1720 y(tractable)f Fm(Q)580 1735 y Fj(mf)722 1720 y Fr(distribution)c (rather)k(than)f(the)h(in)m(tractable)g Fm(P)47 b Fr(distribution)31 b(\(cf.)120 1828 y(Eq.)i(\(2\)\);)j(only)c(in)g(the)i(former)f(case)h (do)g(w)m(e)g(ha)m(v)m(e)g(an)m(y)g(hop)s(e)f(of)g(\014nding)e(the)j(b) s(est)120 1935 y(appro)m(ximation)22 b(in)f(practice.)i(A)g(more)g (satisfying)e(answ)m(er)h(arises)g(from)h(considering)120 2043 y(the)32 b(lik)m(eliho)s(o)s(d)c(of)k(the)g(evidence,)g Fm(P)13 b Fr(\()p Fm(S)1516 2010 y Fl(\003)1556 2043 y Fr(\).)32 b(Clearly)e(calculation)h(of)h(the)g(lik)m(eliho)s(o)s(d) 120 2151 y(p)s(ermits)d(calculation)h(of)g(the)h(conditional)e (probabilit)m(y)f Fm(P)13 b Fr(\()p Fm(S)5 b Fk(j)p Fm(S)2360 2118 y Fl(\003)2400 2151 y Fr(\).)31 b(Using)f(Jensen's)120 2259 y(inequalit)m(y)h(w)m(e)j(can)f(lo)m(w)m(er)h(b)s(ound)d(the)i (lik)m(eliho)s(o)s(d)d(\(cf.)k(for)f(example)g(Saul,)f(et)i(al.,)120 2367 y(1996\):)793 2532 y(log)16 b Fm(P)d Fr(\()p Fm(S)1092 2494 y Fl(\003)1132 2532 y Fr(\))84 b(=)e(log)1537 2451 y Fi(X)1573 2635 y Fj(S)1672 2532 y Fm(P)13 b Fr(\()p Fm(S;)i(S)1935 2494 y Fl(\003)1975 2532 y Fr(\))1251 2806 y(=)82 b(log)1537 2725 y Fi(X)1573 2909 y Fj(S)1672 2806 y Fm(Q)p Fr(\()p Fm(S)5 b Fk(j)p Fm(S)1926 2768 y Fl(\003)1966 2806 y Fr(\))2011 2744 y Fm(P)13 b Fr(\()p Fm(S;)i(S)2274 2711 y Fl(\003)2314 2744 y Fr(\))p 2011 2785 339 4 v 2016 2868 a Fm(Q)p Fr(\()p Fm(S)5 b Fk(j)p Fm(S)2270 2842 y Fl(\003)2310 2868 y Fr(\))1251 3080 y Fk(\025)1404 2999 y Fi(X)1441 3183 y Fj(S)1539 3080 y Fm(Q)p Fr(\()p Fm(S)g Fk(j)p Fm(S)1793 3043 y Fl(\003)1833 3080 y Fr(\))15 b(log)2026 3019 y Fm(P)e Fr(\()p Fm(S;)i(S)2289 2986 y Fl(\003)2329 3019 y Fr(\))p 2026 3059 V 2031 3142 a Fm(Q)p Fr(\()p Fm(S)5 b Fk(j)p Fm(S)2285 3116 y Fl(\003)2325 3142 y Fr(\))2375 3080 y Fm(;)120 3312 y Fr(for)20 b(arbitrary)f Fm(Q)p Fr(\()p Fm(S)5 b Fk(j)p Fm(S)880 3279 y Fl(\003)920 3312 y Fr(\),)21 b(in)e(particular)f(for)i(the)h(mean)f(\014eld)f (distribution)e Fm(Q)2759 3327 y Fj(mf)2866 3312 y Fr(\()p Fm(S)5 b Fk(j)p Fm(S)3048 3279 y Fl(\003)3088 3312 y Fr(\).)120 3420 y(It)35 b(is)e(easily)h(v)m(eri\014ed)g(that)h(the)f (di\013erence)g(b)s(et)m(w)m(een)i(the)e(left)h(hand)e(side)h(and)f (the)120 3528 y(righ)m(t)42 b(hand)f(side)f(of)j(this)e(inequalit)m(y)f (is)h(the)h(KL)g(div)m(ergence)g(KL\()p Fm(Q)p Fk(jj)p Fm(P)13 b Fr(\);)43 b(th)m(us,)120 3636 y(minimizing)32 b(this)i(div)m(ergence)i(is)e(equiv)-5 b(alen)m(t)35 b(to)h(maximizing)d(a)j(lo)m(w)m(er)g(b)s(ound)d(on)120 3744 y(the)e(lik)m(eliho)s(o)s(d.)256 3852 y(F)-8 b(or)50 b(graphical)d(mo)s(dels,)g(minimization)f(of)j(KL)o(\()p Fm(Q)2124 3867 y Fj(mf)2232 3852 y Fk(jj)p Fm(P)13 b Fr(\))49 b(with)e(resp)s(ect)i(to)120 3960 y Fm(Q)192 3974 y Fj(i)220 3960 y Fr(\()p Fm(S)311 3974 y Fj(i)339 3960 y Fk(j)p Fm(S)425 3927 y Fl(\003)465 3960 y Fr(\))40 b(leads)f(directly)f(to)j(an)f(equation)f(expressing)g Fm(Q)2274 3974 y Fj(i)2302 3960 y Fr(\()p Fm(S)2393 3974 y Fj(i)2421 3960 y Fk(j)p Fm(S)2507 3927 y Fl(\003)2547 3960 y Fr(\))h(in)e(terms)i(of)120 4068 y(the)34 b Fm(Q)352 4082 y Fj(j)389 4068 y Fr(\()p Fm(S)480 4082 y Fj(j)516 4068 y Fk(j)p Fm(S)602 4035 y Fl(\003)642 4068 y Fr(\))g(asso)s(ciated) h(with)e(the)h(no)s(des)g(in)e(the)j(Mark)m(o)m(v)h(blank)m(et)e(of)g (no)s(de)f Fm(i)120 4176 y Fr(\(P)m(eterson)h(&)f(Anderson,)f(1987;)i (Saul,)e(et)i(al.,)f(1996\).)i(These)d(equations)h(are)g(cou-)120 4284 y(pled)c(nonlinear)f(equations)i(and)g(are)h(generally)e(solv)m (ed)i(iterativ)m(ely)-8 b(.)120 4516 y Fq(3.)73 b(Mixture)36 b(appro)m(ximations)120 4690 y Fr(In)41 b(this)g(section)h(w)m(e)g (describ)s(e)e(ho)m(w)i(to)h(extend)f(the)g(mean)f(\014eld)g(framew)m (ork)h(b)m(y)120 4798 y(utilizing)34 b(mixture)i(distributions)e (rather)j(than)g(factorized)h(distributions)33 b(as)k(the)p eop %%Page: 5 5 5 4 bop 575 85 a Fo(IMPR)n(O)n(VING)27 b(THE)g(MEAN)h(FIELD)g(APPR)n(O) n(XIMA)-7 b(TION)408 b Fr(5)120 310 y(appro)m(ximating)44 b(distributions.)d(F)-8 b(or)46 b(notational)e(simplicit)m(y)-8 b(,)43 b(w)m(e)i(will)e(drop)g(the)120 418 y(dep)s(endence)34 b(of)h Fm(Q)p Fr(\()p Fm(S)5 b Fk(j)p Fm(S)974 385 y Fl(\003)1014 418 y Fr(\))35 b(on)g Fm(S)1276 385 y Fl(\003)1350 418 y Fr(in)f(this)g(section)h(and)f(in)g(the)h(remainder)e(of)i(the) 120 526 y(pap)s(er,)d(but)h(it)f(m)m(ust)h(b)s(e)f(b)s(orne)g(in)g (mind)f(that)j(the)f(appro)m(ximation)f(is)g(optimized)120 634 y(separately)40 b(for)g(eac)m(h)h(con\014guration)e(of)i(the)f (evidence)f Fm(S)2201 601 y Fl(\003)2241 634 y Fr(.)h(W)-8 b(e)41 b(also)f(utilize)e(the)120 742 y(notation)26 b Fk(F)9 b Fr(\()p Fm(Q)p Fr(\))27 b(to)g(denote)g(the)f(lo)m(w)m(er)h(b) s(ound)d(on)i(the)g(righ)m(t)g(hand)f(side)g(of)h(Eq.)g(\(3\):)922 973 y Fk(F)9 b Fr(\()p Fm(Q)p Fr(\))26 b Fk(\021)1260 893 y Fi(X)1297 1076 y Fj(S)1395 973 y Fm(Q)p Fr(\()p Fm(S)5 b Fr(\))15 b(log)1756 912 y Fm(P)e Fr(\()p Fm(S;)i(S)2019 879 y Fl(\003)2059 912 y Fr(\))p 1756 952 339 4 v 1824 1036 a Fm(Q)p Fr(\()p Fm(S)5 b Fr(\))2957 973 y(\(3\))120 1232 y(When)23 b Fm(Q)p Fr(\()p Fm(S)5 b Fr(\))23 b(tak)m(es)h(the)f (factorized)h(form)e Fm(Q)1667 1247 y Fj(mf)1775 1232 y Fr(\()p Fm(S)5 b Fr(\))23 b(in)f(Eq.)h(\(1\),)h(w)m(e)f(will)d(refer) j(to)g(the)120 1340 y(appro)m(ximation)29 b(as)h(a)g(\\factorized)h (mean)e(\014eld")g(appro)m(ximation)g(and)g(will)e(denote)120 1448 y(the)k(corresp)s(onding)d(b)s(ound)g(as)j Fk(F)9 b Fr(\()p Fm(Q)1434 1463 y Fj(mf)1542 1448 y Fr(\).)256 1556 y(Our)33 b(prop)s(osal)e(is)i(to)h(utilize)e(a)i(mixture)e (distribution)e(as)j(the)h(appro)m(ximating)120 1664 y(distribution:)935 1843 y Fm(Q)1007 1857 y Fj(mix)1137 1843 y Fr(\()p Fm(S)5 b Fr(\))26 b(=)1390 1762 y Fi(X)1419 1936 y Fj(m)1525 1843 y Fm(\013)1583 1857 y Fj(m)1650 1843 y Fm(Q)1722 1858 y Fj(mf)1830 1843 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))p Fm(;)866 b Fr(\(4\))120 2092 y(where)27 b(eac)m(h)i(of)f(the)g(comp)s(onen)m(t)g(distributions)23 b Fm(Q)1899 2107 y Fj(mf)2007 2092 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))28 b(are)g(factorized)h(distri-)120 2200 y(butions.)h(The)h(mixing)e(prop)s(ortions)g Fm(\013)1506 2214 y Fj(m)1573 2200 y Fr(,)j(whic)m(h)e(are)h(constrained)g(to)h(b)s (e)f(p)s(ositiv)m(e)120 2308 y(and)j(sum)g(to)h(one,)h(are)f (additional)d(parameters)j(to)h(b)s(e)e(\014t)g(via)h(the)g (minimization)120 2415 y(of)c(the)f(KL)g(div)m(ergence.)256 2523 y(In)36 b(the)h(remainder)d(of)j(this)e(section)i(w)m(e)g(fo)s (cus)e(on)i(the)f(relationship)e(b)s(et)m(w)m(een)120 2631 y Fk(F)9 b Fr(\()p Fm(Q)301 2645 y Fj(mix)432 2631 y Fr(\))46 b({)g(the)g(mixture)f(b)s(ound)f({)i(and)f(the)h(factorized) h(mean)e(\014eld)g(b)s(ounds)120 2739 y Fk(F)9 b Fr(\()p Fm(Q)301 2754 y Fj(mf)409 2739 y Fk(j)p Fm(m)p Fr(\))42 b(corresp)s(onding)d(to)k(the)e(mixture)f(comp)s(onen)m(ts.)i(T)-8 b(o)42 b(this)e(end,)h(con-)120 2847 y(sider:)224 3056 y Fk(F)9 b Fr(\()p Fm(Q)405 3070 y Fj(mix)536 3056 y Fr(\))26 b(=)693 2975 y Fi(X)729 3159 y Fj(S)828 3056 y Fm(Q)900 3070 y Fj(mix)1030 3056 y Fr(\()p Fm(S)5 b Fr(\))15 b(log)1319 2995 y Fm(P)e Fr(\()p Fm(S;)i(S)1582 2962 y Fl(\003)1622 2995 y Fr(\))p 1319 3035 V 1321 3118 a Fm(Q)1393 3132 y Fj(mix)1524 3118 y Fr(\()p Fm(S)5 b Fr(\))307 3330 y(=)465 3249 y Fi(X)461 3433 y Fj(m;S)605 3330 y Fm(\013)663 3344 y Fj(m)745 3211 y Fi(\024)788 3330 y Fm(Q)860 3345 y Fj(mf)968 3330 y Fr(\()p Fm(S)g Fk(j)p Fm(m)p Fr(\))15 b(log)1362 3269 y Fm(P)e Fr(\()p Fm(S;)i(S)1625 3236 y Fl(\003)1666 3269 y Fr(\))p 1362 3309 V 1364 3393 a Fm(Q)1436 3407 y Fj(mix)1567 3393 y Fr(\()p Fm(S)5 b Fr(\))1711 3211 y Fi(\025)307 3635 y Fr(=)465 3554 y Fi(X)461 3738 y Fj(m;S)605 3635 y Fm(\013)663 3649 y Fj(m)745 3491 y Fi(")793 3635 y Fm(Q)865 3650 y Fj(mf)973 3635 y Fr(\()p Fm(S)g Fk(j)p Fm(m)p Fr(\))15 b(log)1406 3573 y Fm(P)e Fr(\()p Fm(S;)i(S)1669 3540 y Fl(\003)1709 3573 y Fr(\))p 1367 3614 417 4 v 1367 3697 a Fm(Q)1439 3712 y Fj(mf)1547 3697 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))1814 3635 y(+)20 b Fm(Q)1977 3650 y Fj(mf)2084 3635 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))15 b(log)2479 3573 y Fm(Q)2551 3588 y Fj(mf)2658 3573 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))p 2479 3614 V 2520 3697 a Fm(Q)2592 3711 y Fj(mix)2722 3697 y Fr(\()p Fm(S)g Fr(\))2905 3491 y Fi(#)307 3922 y Fr(=)461 3841 y Fi(X)490 4016 y Fj(m)596 3922 y Fm(\013)654 3936 y Fj(m)721 3922 y Fk(F)k Fr(\()p Fm(Q)902 3937 y Fj(mf)1010 3922 y Fk(j)p Fm(m)p Fr(\))21 b(+)1266 3841 y Fi(X)1262 4025 y Fj(m;S)1405 3922 y Fm(\013)1463 3936 y Fj(m)1530 3922 y Fm(Q)1602 3937 y Fj(mf)1710 3922 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))15 b(log)2104 3860 y Fm(Q)2176 3875 y Fj(mf)2284 3860 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))p 2104 3901 V 2145 3985 a Fm(Q)2217 3999 y Fj(mix)2348 3985 y Fr(\()p Fm(S)g Fr(\))307 4169 y(=)461 4088 y Fi(X)490 4262 y Fj(m)596 4169 y Fm(\013)654 4183 y Fj(m)721 4169 y Fk(F)k Fr(\()p Fm(Q)902 4184 y Fj(mf)1010 4169 y Fk(j)p Fm(m)p Fr(\))21 b(+)f Fm(I)7 b Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))1397 b(\(5\))120 4418 y(where)43 b Fm(I)7 b Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))45 b(is)e(the)h(m)m(utual)g (information)e(b)s(et)m(w)m(een)j(the)f(mixture)e(index)h Fm(m)120 4526 y Fr(and)30 b(the)h(v)-5 b(ariables)30 b Fm(S)36 b Fr(in)30 b(the)h(mixture)e(distribution)1998 4493 y Fn(2)2034 4526 y Fr(.)i(The)g(\014rst)f(term)h(in)f(Eq.)h(\(5\)) 224 4680 y Fh(2)258 4712 y Fg(Recall)i(that)f(the)f(m)n(utual)g (information)i(b)r(et)n(w)n(een)f(an)n(y)f(t)n(w)n(o)i(random)e(v)l (ariables)h Ff(x)g Fg(and)g Ff(y)i Fg(is)120 4795 y(I\(x;y\))24 b(=)421 4735 y Fi(P)508 4822 y Fe(x;y)612 4795 y Ff(P)11 b Fg(\()p Ff(x;)i(y)s Fg(\))f(log)j Ff(P)c Fg(\()p Ff(y)s Fd(j)p Ff(x)p Fg(\))p Ff(=P)g Fg(\()p Ff(y)s Fg(\).)p eop %%Page: 6 6 6 5 bop 120 85 a Fr(6)427 b Fo(TOMMI)27 b(S.)h(JAAKK)n(OLA)f(AND)h (MICHAEL)g(I.)g(JORD)n(AN)120 310 y Fr(is)36 b(a)i(con)m(v)m(ex)h(com)m (bination)d(of)h(factorized)h(mean)f(\014eld)f(b)s(ounds)f(and)i (therefore)g(it)120 418 y(yields)i(no)i(gain)g(o)m(v)m(er)i(factorized) e(mean)h(\014eld)d(as)j(far)f(as)g(the)h(KL)e(div)m(ergence)i(is)120 526 y(concerned.)37 b(It)f(is)g(the)h(second)f(term,)h(i.e.,)g(the)f(m) m(utual)g(information,)f(that)i(c)m(har-)120 634 y(acterizes)32 b(the)f(gain)f(of)h(using)f(the)h(mixture)e(appro)m(ximation.)2287 601 y Fn(3)2357 634 y Fr(As)h(a)i(non-negativ)m(e)120 742 y(quan)m(tit)m(y)-8 b(,)25 b Fm(I)7 b Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))25 b(increases)f(the)g(lik)m(eliho)s(o)s(d)d (b)s(ound)h(and)i(therefore)h(impro)m(v)m(es)e(the)120 850 y(appro)m(ximation.)28 b(W)-8 b(e)30 b(note,)f(ho)m(w)m(ev)m(er,)i (that)e(since)f Fm(I)7 b Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))26 b Fk(\024)f Fr(log)17 b Fm(M)10 b Fr(,)29 b(where)f Fm(M)39 b Fr(is)120 958 y(the)22 b(n)m(um)m(b)s(er)f(of)h (mixture)f(comp)s(onen)m(ts,)i(the)f(KL)f(div)m(ergence)i(b)s(et)m(w)m (een)g(the)f(mixture)120 1066 y(appro)m(ximation)28 b(and)g(the)h(true) f(p)s(osterior)g(can)h(decrease)h(at)f(most)g(logarithmically)120 1174 y(in)g(the)i(n)m(um)m(b)s(er)e(of)h(mixture)f(comp)s(onen)m(ts.) 120 1406 y Fq(4.)73 b(Computing)34 b(the)h(m)m(utual)f(information)120 1580 y Fr(T)-8 b(o)43 b(b)s(e)e(able)h(to)h(use)e(the)i(mixture)e(b)s (ound)f(of)i(Eq.)g(\(5\))h(w)m(e)g(need)f(to)h(b)s(e)e(able)h(to)120 1688 y(compute)36 b(the)f(m)m(utual)g(information)e(term.)j(The)f (di\016cult)m(y)e(with)h(the)i(expression)120 1796 y(as)23 b(it)e(stands)h(is)f(the)i(need)f(to)h(a)m(v)m(erage)i(the)d(log)h(of)f (the)h(mixture)e(distribution)d(o)m(v)m(er)24 b(all)120 1904 y(con\014gurations)j(of)i(the)f(hidden)e(v)-5 b(ariables.)27 b(Giv)m(en)h(that)h(there)f(are)h(an)f(exp)s(onen)m(tial)120 2012 y(n)m(um)m(b)s(er)k(of)i(suc)m(h)g(con\014gurations,)f(this)g (computation)g(is)g(in)g(general)g(in)m(tractable.)120 2120 y(W)-8 b(e)28 b(will)c(mak)m(e)j(use)f(of)h(a)g(re\014ned)e(form)h (of)h(Jensen's)f(inequalit)m(y)e(to)k(mak)m(e)f(headw)m(a)m(y)-8 b(.)256 2228 y(Let)31 b(us)f(\014rst)g(rewrite)f(the)i(m)m(utual)e (information)g(as)369 2450 y Fm(I)7 b Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))85 b(=)910 2369 y Fi(X)905 2553 y Fj(m;S)1049 2450 y Fm(\013)1107 2464 y Fj(m)1174 2450 y Fm(Q)1246 2465 y Fj(mf)1353 2450 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))15 b(log)1748 2388 y Fm(Q)1820 2403 y Fj(mf)1928 2388 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))p 1748 2429 417 4 v 1789 2512 a Fm(Q)1861 2526 y Fj(mix)1991 2512 y Fr(\()p Fm(S)g Fr(\))2957 2450 y(\(6\))752 2754 y(=)910 2673 y Fi(X)905 2857 y Fj(m;S)1049 2754 y Fm(\013)1107 2768 y Fj(m)1174 2754 y Fm(Q)1246 2769 y Fj(mf)1353 2754 y Fr(\()p Fm(S)g Fk(j)p Fm(m)p Fr(\))1604 2610 y Fi(")1654 2754 y Fk(\000)15 b Fr(log)1923 2693 y Fm(Q)1995 2707 y Fj(mix)2126 2693 y Fr(\()p Fm(S)5 b Fr(\))p 1882 2733 V 1882 2817 a Fm(Q)1954 2832 y Fj(mf)2062 2817 y Fr(\()p Fm(S)g Fk(j)p Fm(m)p Fr(\))2309 2610 y Fi(#)2957 2754 y Fr(\(7\))752 3059 y(=)910 2978 y Fi(X)905 3162 y Fj(m;S)1049 3059 y Fm(\013)1107 3073 y Fj(m)1174 3059 y Fm(Q)1246 3074 y Fj(mf)1353 3059 y Fr(\()p Fm(S)g Fk(j)p Fm(m)p Fr(\))1604 2915 y Fi(")1654 3059 y Fk(\000)15 b Fr(log)1882 2997 y Fm(\013)1940 3011 y Fj(m)2007 2997 y Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))p 1882 3038 432 4 v 1882 3121 a Fm(\013)1940 3135 y Fj(m)2007 3121 y Fm(R)q Fr(\()p Fm(S)g Fk(j)p Fm(m)p Fr(\))2375 2997 y Fm(Q)2447 3011 y Fj(mix)2577 2997 y Fr(\()p Fm(S)g Fr(\))p 2333 3038 417 4 v 2333 3121 a Fm(Q)2405 3136 y Fj(mf)2513 3121 y Fr(\()p Fm(S)g Fk(j)p Fm(m)p Fr(\))2760 2915 y Fi(#)2957 3059 y Fr(\(8\))752 3305 y(=)910 3224 y Fi(X)905 3408 y Fj(m;S)1049 3305 y Fm(\013)1107 3319 y Fj(m)1174 3305 y Fm(Q)1246 3320 y Fj(mf)1353 3305 y Fr(\()p Fm(S)g Fk(j)p Fm(m)p Fr(\))15 b(log)j Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))21 b Fk(\000)2156 3224 y Fi(X)2184 3399 y Fj(m)2291 3305 y Fm(\013)2349 3319 y Fj(m)2430 3305 y Fr(log)c Fm(\013)2621 3319 y Fj(m)905 3610 y Fr(+)995 3529 y Fi(X)991 3713 y Fj(m;S)1135 3610 y Fm(\013)1193 3624 y Fj(m)1260 3610 y Fm(Q)1332 3625 y Fj(mf)1439 3610 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))1690 3466 y Fi(")1740 3610 y Fk(\000)15 b Fr(log)1968 3548 y Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))p 1968 3589 307 4 v 2059 3672 a Fm(\013)2117 3686 y Fj(m)2336 3548 y Fm(Q)2408 3562 y Fj(mix)2538 3548 y Fr(\()p Fm(S)g Fr(\))p 2294 3589 417 4 v 2294 3672 a Fm(Q)2366 3687 y Fj(mf)2474 3672 y Fr(\()p Fm(S)g Fk(j)p Fm(m)p Fr(\))2721 3466 y Fi(#)2957 3610 y Fr(\(9\))120 3894 y(where)20 b(w)m(e)h(ha)m(v)m(e)g (in)m(tro)s(duced)e(additional)f(smo)s(othing)h(distributions)e Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))20 b(whose)120 4002 y(role)j(is)f(to)i(impro)m(v)m(e)f(the)g(con)m(v)m(exit)m(y)i(b)s (ounds)c(that)j(w)m(e)f(describ)s(e)f(b)s(elo)m(w.)2586 3969 y Fn(4)2648 4002 y Fr(With)g(some)120 4109 y(foresigh)m(t)36 b(w)m(e)h(ha)m(v)m(e)h(also)e(in)m(tro)s(duced)e Fm(\013)1557 4123 y Fj(m)1660 4109 y Fr(in)m(to)j(the)f(expression;)f(as)i(will)c(b) s(ecome)224 4263 y Fh(3)258 4295 y Fg(Of)e(course,)h(this)f(is)g(only)f (the)h(p)r(oten)n(tial)g(gain)h(to)f(b)r(e)f(realized)i(from)f(the)f (mixture)f(appro)n(xi-)120 4378 y(mation.)f(W)-6 b(e)29 b(cannot)f(rule)h(out)f(the)h(p)r(ossibilit)n(y)g(that)g(it)f(will)i(b) r(e)f(harder)f(to)h(searc)n(h)g(for)g(the)g(b)r(est)120 4461 y(appro)n(ximating)24 b(distribution)h(in)g(the)g(mixture)e(appro) n(ximation)i(than)f(in)h(the)g(factorized)h(case.)g(In)120 4544 y(particular)34 b(the)f(mixture)e(appro)n(ximation)i(ma)n(y)f(in)n (tro)r(duce)g(additional)i(lo)r(cal)h(minima)d(in)n(to)h(the)120 4627 y(KL)25 b(div)n(ergence.)224 4684 y Fh(4)258 4715 y Fg(In)g(fact,)i(if)g(w)n(e)f(use)g(Jensen's)h(inequalit)n(y)f (directly)f(in)h(Eq.)g(\(6\),)g(w)n(e)h(obtain)f(a)g(v)l(acuous)g(b)r (ound)120 4798 y(of)g(zero,)h(as)f(is)h(easily)f(v)n(eri\014ed.)p eop %%Page: 7 7 7 6 bop 575 85 a Fo(IMPR)n(O)n(VING)27 b(THE)g(MEAN)h(FIELD)g(APPR)n(O) n(XIMA)-7 b(TION)408 b Fr(7)120 310 y(apparen)m(t)29 b(in)e(the)i(follo)m(wing)e(section)h(the)h(forced)f(app)s(earance)h (of)g Fm(\013)2501 324 y Fj(m)2596 310 y Fr(turns)e(out)i(to)120 418 y(b)s(e)h(con)m(v)m(enien)m(t)h(in)e(up)s(dating)g(the)h(mixture)f (co)s(e\016cien)m(ts.)256 540 y(T)-8 b(o)34 b(a)m(v)m(oid)f(the)h(need) e(to)i(a)m(v)m(erage)i(o)m(v)m(er)e(the)f(logarithm)f(of)i(the)f (mixture)f(distri-)120 648 y(bution,)d(w)m(e)i(mak)m(e)g(use)f(of)h (the)g(follo)m(wing)d(con)m(v)m(exit)m(y)k(b)s(ound:)927 890 y Fk(\000)15 b Fr(log)r(\()p Fm(x)p Fr(\))26 b Fk(\025)e(\000)p Fm(\025)15 b(x)21 b Fr(+)e(log)r(\()p Fm(\025)p Fr(\))i(+)f(1)p Fm(;)812 b Fr(\(10\))120 1132 y(replacing)28 b(the)h Fk(\000)15 b Fr(log)r(\()p Fk(\001)p Fr(\))30 b(terms)f(in)f(Eq.)h (\(9\))i(with)d(linear)f(functions,)h(separately)i(for)120 1240 y(eac)m(h)23 b Fm(m)p Fr(.)e(These)h(substitutions)d(imply)g(a)j (lo)m(w)m(er)g(b)s(ound)d(on)j(the)g(m)m(utual)e(information)120 1348 y(giv)m(en)30 b(b)m(y:)366 1590 y Fm(I)7 b Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))84 b Fk(\025)906 1509 y Fi(X)901 1692 y Fj(m;S)1045 1590 y Fm(\013)1103 1604 y Fj(m)1170 1590 y Fm(Q)1242 1605 y Fj(mf)1349 1590 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))15 b(log)j Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))21 b Fk(\000)2152 1509 y Fi(X)2180 1683 y Fj(m)2287 1590 y Fm(\013)2345 1604 y Fj(m)2427 1590 y Fr(log)16 b Fm(\013)2617 1604 y Fj(m)901 1836 y Fk(\000)987 1755 y Fi(X)1016 1930 y Fj(m)1122 1836 y Fm(\025)1175 1850 y Fj(m)1257 1755 y Fi(X)1294 1939 y Fj(S)1392 1836 y Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))p Fm(Q)1770 1850 y Fj(mix)1901 1836 y Fr(\()p Fm(S)g Fr(\))21 b(+)2144 1755 y Fi(X)2172 1930 y Fj(m)2279 1836 y Fm(\013)2337 1850 y Fj(m)2418 1836 y Fr(log)c Fm(\025)2604 1850 y Fj(m)2691 1836 y Fr(+)j(1)84 b(\(11\))748 2049 y Fk(\021)e Fm(I)941 2064 y Fj(\025)987 2049 y Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))p Fm(:)1648 b Fr(\(12\))120 2290 y(The)30 b(tigh)m(tness)g(of)h (the)g(con)m(v)m(exit)m(y)h(b)s(ound)c(is)i(con)m(trolled)g(b)m(y)g (the)h(smo)s(othing)e(func-)120 2398 y(tions)43 b Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\),)45 b(whic)m(h)e(can)i(b)s(e)e (view)m(ed)h(as)g(\\\015attening")h(the)f(logarithm)f(func-)120 2506 y(tion)j(suc)m(h)h(that)g(Jensen's)f(inequalit)m(y)g(is)f(tigh)m (tened.)j(In)e(particular,)f(if)h(w)m(e)h(set)120 2614 y Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))26 b Fk(/)f Fm(Q)620 2629 y Fj(mf)727 2614 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))p Fm(=Q)1080 2628 y Fj(mix)1212 2614 y Fr(\()p Fm(S)g Fr(\))31 b(and)e(maximize)h(o)m(v)m(er)h(the)g(new)e(v)-5 b(ariational)29 b(pa-)120 2722 y(rameters)21 b Fm(\025)538 2736 y Fj(m)625 2722 y Fr(w)m(e)g(reco)m(v)m(er)h(the)f(m)m(utual)f (information)f(exactly)-8 b(.)22 b(Suc)m(h)e(a)h(c)m(hoice)g(w)m(ould) 120 2830 y(not)34 b(reduce)f(the)h(complexit)m(y)f(of)h(our)f (calculations,)g(ho)m(w)m(ev)m(er.)i(T)-8 b(o)34 b(obtain)f(an)g (e\016-)120 2938 y(cien)m(t)25 b(algorithm)f(w)m(e)h(will)d(instead)i (assume)g(that)i(the)f(distributions)20 b Fm(R)26 b Fr(ha)m(v)m(e)g (simple)120 3046 y(factorized)h(forms,)f(and)f(optimize)h(the)h (ensuing)d(lo)m(w)m(er)j(b)s(ound)d(with)h(resp)s(ect)h(to)h(the)120 3154 y(parameters)g(in)e(these)i(distributions.)22 b(It)27 b(will)c(turn)j(out)g(that)h(this)f(assumption)e(will)120 3262 y(p)s(ermit)i(all)g(of)i(the)g(terms)g(in)e(our)h(b)s(ound)f(to)i (b)s(e)f(ev)-5 b(aluated)28 b(e\016cien)m(tly)-8 b(.)28 b(W)-8 b(e)29 b(note)f(\014-)120 3370 y(nally)21 b(that)j(this)e(b)s (ound)e(is)i(nev)m(er)i(v)-5 b(acuous.)23 b(T)-8 b(o)23 b(see)h(wh)m(y)-8 b(,)23 b(simply)d(set)k Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))23 b(equal)120 3478 y(to)30 b(a)g(constan)m(t)g(for)f(all)f Fk(f)p Fm(S;)15 b(m)p Fk(g)31 b Fr(in)d(whic)m(h)f(case)k(max)1954 3493 y Fj(\025)2014 3478 y Fm(I)2054 3493 y Fj(\025)2100 3478 y Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))26 b(=)f(0;)30 b(optimization)120 3586 y(of)h Fm(R)g Fr(can)f(only)g(impro)m(v)m(e)g(the)h(b)s(ound.)120 3832 y Fq(5.)73 b(Finding)36 b(the)e(mixture)g(parameters)120 4021 y Fr(In)29 b(order)f(to)i(\014nd)e(a)i(reasonable)f(mixture)f (appro)m(ximation)g(w)m(e)i(need)f(to)h(b)s(e)f(able)g(to)120 4129 y(optimize)i(the)i(b)s(ound)d(in)h(Eq.)h(\(5\))h(with)e(resp)s (ect)h(to)h(the)f(mixture)f(mo)s(del.)g(Clearly)120 4237 y(a)e(necessary)g(condition)e(for)i(the)g(feasibilit)m(y)d(of)j(this)f (optimization)f(pro)s(cess)h(is)g(that)120 4345 y(the)33 b(comp)s(onen)m(t)h(factorized)f(mean)g(\014eld)f(distributions)d(can) 34 b(b)s(e)e(\014t)h(to)h(the)f(mo)s(del)120 4453 y(tractably)-8 b(.)27 b(That)e(is,)g(w)m(e)h(m)m(ust)f(assume)h(that)g(the)g(follo)m (wing)e(mean)i(\014eld)e(equations:)1089 4681 y Fm(@)5 b Fk(F)k Fr(\()p Fm(Q)1323 4696 y Fj(mf)1432 4681 y Fr(\))p 1089 4722 379 4 v 1116 4805 a Fm(@)c(Q)1241 4819 y Fj(j)1278 4805 y Fr(\()p Fm(S)1369 4819 y Fj(j)1405 4805 y Fr(\))1503 4743 y(=)25 b Fm(constant)962 b Fr(\(13\))p eop %%Page: 8 8 8 7 bop 120 85 a Fr(8)427 b Fo(TOMMI)27 b(S.)h(JAAKK)n(OLA)f(AND)h (MICHAEL)g(I.)g(JORD)n(AN)120 310 y Fr(can)44 b(b)s(e)g(solv)m(ed)g (e\016cien)m(tly)g(\(not)g(necessarily)f(in)g(closed)h(form\))g(for)g (an)m(y)g(of)g(the)120 418 y(marginals)25 b Fm(Q)603 432 y Fj(j)639 418 y Fr(\()p Fm(S)730 432 y Fj(j)767 418 y Fr(\))i(in)e(the)h(factorized)h(mean)f(\014eld)f(distribution)e Fm(Q)2491 433 y Fj(mf)2599 418 y Fr(.)2624 385 y Fn(5)2690 418 y Fr(Examples)120 526 y(of)i(suc)m(h)f(mean)h(\014eld)e(equations)h (are)h(pro)m(vided)e(b)m(y)h(P)m(eterson)i(and)e(Anderson)f(\(1987\)) 120 634 y(and)30 b(Saul,)f(et)i(al.)f(\(1996\).)256 742 y(Let)36 b(us)f(\014rst)f(consider)h(the)g(up)s(dates)g(for)g(the)g (factorized)h(mean)f(\014eld)f(comp)s(o-)120 850 y(nen)m(ts)24 b(of)g(the)f(mixture)g(mo)s(del.)g(W)-8 b(e)25 b(need)e(to)h(tak)m(e)i (deriv)-5 b(ativ)m(es)23 b(of)h(the)g(lo)m(w)m(er)g(b)s(ound:)458 1049 y Fk(F)9 b Fr(\()p Fm(Q)639 1063 y Fj(mix)770 1049 y Fr(\))25 b Fk(\025)926 968 y Fi(X)955 1142 y Fj(m)1061 1049 y Fm(\013)1119 1063 y Fj(m)1186 1049 y Fk(F)9 b Fr(\()p Fm(Q)1367 1064 y Fj(mf)1475 1049 y Fk(j)p Fm(m)p Fr(\))21 b(+)f Fm(I)1767 1064 y Fj(\025)1812 1049 y Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))26 b Fk(\021)f(F)2250 1064 y Fj(\025)2296 1049 y Fr(\()p Fm(Q)2403 1063 y Fj(mix)2534 1049 y Fr(\))342 b(\(14\))120 1313 y(with)32 b(resp)s(ect)i(to)g(the)f (parameters)h(of)g(the)g(comp)s(onen)m(t)f(factorized)h(distributions.) 120 1421 y(The)20 b(imp)s(ortan)m(t)f(prop)s(ert)m(y)h(of)g Fm(I)1211 1436 y Fj(\025)1257 1421 y Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))21 b(is)e(that)i(it)f(is)f(linear)g(in)g(an)m(y)h(of)h (the)f(marginals)120 1529 y Fm(Q)192 1543 y Fj(j)228 1529 y Fr(\()p Fm(S)319 1543 y Fj(j)356 1529 y Fk(j)p Fm(k)s Fr(\),)32 b(where)e Fm(k)k Fr(indicates)29 b(the)i(mixture)f (comp)s(onen)m(t)h(\(see)g(App)s(endix)d(A\).)k(As)120 1637 y(a)f(result,)e(w)m(e)i(ha)m(v)m(e:)1083 1822 y Fm(@)5 b(I)1176 1837 y Fj(\025)1222 1822 y Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))p 1078 1862 401 4 v 1078 1946 a Fm(@)g(Q)1203 1960 y Fj(j)1240 1946 y Fr(\()p Fm(S)1331 1960 y Fj(j)1368 1946 y Fk(j)p Fm(k)s Fr(\))1514 1883 y(=)25 b Fm(constant)951 b Fr(\(15\))120 2139 y(where)30 b(the)g(constan)m(t)h(is)f(indep)s(enden)m(t)e(of)i(the)g(marginal)f Fm(Q)2222 2153 y Fj(j)2258 2139 y Fr(\()p Fm(S)2349 2153 y Fj(j)2386 2139 y Fk(j)p Fm(k)s Fr(\))i(\(and)f(is)f(gener-)120 2247 y(ally)34 b(dep)s(enden)m(t)g(on)g(the)h(other)g(marginals)f(in)f (the)i(particular)f(comp)s(onen)m(t)h(mean)120 2355 y(\014eld)29 b(distribution\).)e(It)k(follo)m(ws)e(from)h(our)g(assumption)f(that)i (the)f(equations:)580 2546 y Fm(@)5 b Fk(F)698 2561 y Fj(\025)744 2546 y Fr(\()p Fm(Q)851 2560 y Fj(mix)982 2546 y Fr(\))p 580 2586 437 4 v 599 2670 a Fm(@)g(Q)724 2684 y Fj(j)760 2670 y Fr(\()p Fm(S)851 2684 y Fj(j)888 2670 y Fk(j)p Fm(k)s Fr(\))1052 2608 y(=)25 b Fm(\013)1206 2623 y Fj(k)1259 2545 y Fm(@)5 b Fk(F)1377 2560 y Fj(\025)1423 2545 y Fr(\()p Fm(Q)1530 2560 y Fj(mf)1638 2545 y Fk(j)p Fm(k)s Fr(\))p 1259 2586 490 4 v 1304 2670 a Fm(@)g(Q)1429 2684 y Fj(j)1466 2670 y Fr(\()p Fm(S)1557 2684 y Fj(j)1593 2670 y Fk(j)p Fm(k)s Fr(\))1779 2608 y(+)1885 2546 y Fm(@)g(I)1978 2561 y Fj(\025)2023 2546 y Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))p 1880 2586 401 4 v 1880 2670 a Fm(@)g(Q)2005 2684 y Fj(j)2041 2670 y Fr(\()p Fm(S)2132 2684 y Fj(j)2169 2670 y Fk(j)p Fm(k)s Fr(\))2315 2608 y(=)25 b(0)455 b(\(16\))120 2864 y(can)38 b(b)s(e)f(solv)m(ed)g (e\016cien)m(tly)g(for)h(an)m(y)g(sp)s(eci\014c)e(marginal)g Fm(Q)2228 2878 y Fj(j)2264 2864 y Fr(\()p Fm(S)2355 2878 y Fj(j)2392 2864 y Fk(j)p Fm(k)s Fr(\).)j(W)-8 b(e)38 b(can)g(th)m(us)120 2972 y(\014nd)31 b(the)h(b)s(est)g(\014tting)g (mixture)f(comp)s(onen)m(ts)h(b)m(y)h(iterativ)m(ely)f(optimizing)e (eac)m(h)k(of)120 3080 y(the)d(marginals)d(in)i(the)g(mo)s(del.)256 3188 y(Let)d(us)e(no)m(w)g(turn)g(to)h(the)g(mixture)f(co)s(e\016cien)m (ts)h Fm(\013)2017 3203 y Fj(k)2060 3188 y Fr(.)g(W)-8 b(e)27 b(note)f(\014rst)f(that,)h(apart)120 3296 y(from)k(the)h(en)m (trop)m(y)g(term)f Fk(\000)1126 3231 y Fi(P)1214 3318 y Fj(k)1272 3296 y Fm(\013)1330 3311 y Fj(k)1388 3296 y Fr(log)16 b Fm(\013)1578 3311 y Fj(k)1621 3296 y Fr(,)31 b(these)g(co)s(e\016cien)m(ts)g(app)s(ear)f(linearly)e(in)120 3403 y Fm(I)160 3418 y Fj(\025)205 3403 y Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\).)36 b(The)d(same)i(is)e(therefore)i(true)f (for)g Fk(F)1824 3418 y Fj(\025)1870 3403 y Fr(\()p Fm(Q)1977 3417 y Fj(mix)2107 3403 y Fr(\))h(\(see)g(Eq.)f(\(14\)\),)j(and)c(w)m (e)120 3511 y(can)e(write:)625 3710 y Fk(F)690 3725 y Fj(\025)736 3710 y Fr(\()p Fm(Q)843 3724 y Fj(mix)973 3710 y Fr(\))83 b(=)1245 3629 y Fi(X)1286 3814 y Fj(k)1380 3710 y Fm(\013)1438 3725 y Fj(k)1481 3710 y Fr(\()p Fk(\000)p Fm(E)5 b Fr(\()p Fm(k)s Fr(\)\))22 b Fk(\000)1927 3629 y Fi(X)1968 3814 y Fj(k)2062 3710 y Fm(\013)2120 3725 y Fj(k)2178 3710 y Fr(log)16 b Fm(\013)2368 3725 y Fj(k)2431 3710 y Fr(+)k(1)344 b(\(17\))120 3990 y(where)30 b Fk(\000)p Fm(E)5 b Fr(\()p Fm(k)s Fr(\))31 b(is)f(the)g(collection)g(of)h(terms)f (linear)f(in)g Fm(\013)2103 4005 y Fj(k)2146 3990 y Fr(:)584 4201 y Fk(\000)p Fm(E)5 b Fr(\()p Fm(k)s Fr(\))84 b Fk(\021)f(F)9 b Fr(\()p Fm(Q)1266 4216 y Fj(mf)1374 4201 y Fk(j)p Fm(k)s Fr(\))21 b(+)1596 4120 y Fi(X)1633 4304 y Fj(S)1731 4201 y Fm(Q)1803 4216 y Fj(mf)1911 4201 y Fr(\()p Fm(S)5 b Fk(j)p Fm(k)s Fr(\))15 b(log)j Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(k)s Fr(\))1085 4434 y(+)1171 4353 y Fi(X)1200 4528 y Fj(m)1306 4434 y Fm(\025)1359 4448 y Fj(m)1441 4353 y Fi(X)1477 4537 y Fj(S)1576 4434 y Fm(R)q Fr(\()p Fm(S)g Fk(j)p Fm(m)p Fr(\))p Fm(Q)1954 4449 y Fj(mf)2062 4434 y Fr(\()p Fm(S)g Fk(j)p Fm(k)s Fr(\))21 b(+)f(log)d Fm(\025)2566 4449 y Fj(k)2911 4434 y Fr(\(18\))224 4684 y Fh(5)258 4715 y Fg(In)27 b(the)g(ordinary)h(mean)e(\014eld)i (equations)f(the)h(constan)n(t)f(is)h(zero,)h(but)d(for)j(man)n(y)d (probabilit)n(y)120 4798 y(mo)r(dels)g(of)g(in)n(terest)g(this)g(sligh) n(t)g(di\013erence)g(p)r(oses)g(no)g(additional)h(di\016culties.)p eop %%Page: 9 9 9 8 bop 575 85 a Fo(IMPR)n(O)n(VING)27 b(THE)g(MEAN)h(FIELD)g(APPR)n(O) n(XIMA)-7 b(TION)408 b Fr(9)120 310 y(No)m(w)23 b(Eq.)g(\(17\))h(has)e (the)h(form)f(of)h(a)g(free)g(energy)-8 b(,)24 b(and)e(the)g(mixture)g (co)s(e\016cien)m(ts)h(that)120 418 y(maximize)34 b(the)h(lo)m(w)m(er)g (b)s(ound)d Fk(F)1283 433 y Fj(\025)1329 418 y Fr(\()p Fm(Q)1436 432 y Fj(mix)1567 418 y Fr(\))j(therefore)g(come)h(from)e (the)h(Boltzmann)120 526 y(distribution:)1160 765 y Fm(\013)1218 780 y Fj(k)1286 765 y Fr(=)1487 704 y Fm(e)1529 671 y Fl(\000)p Fj(E)t Fn(\()p Fj(k)r Fn(\))p 1392 744 441 4 v 1392 766 a Fi(P)1479 853 y Fj(k)1518 834 y Fc(0)1559 830 y Fm(e)1601 804 y Fl(\000)p Fj(E)t Fn(\()p Fj(k)1778 785 y Fc(0)1800 804 y Fn(\))1842 765 y Fm(:)1044 b Fr(\(19\))120 1016 y(\(This)27 b(fact)i(is)e(easily)h(v)m(eri\014ed)f(using)g (Lagrange)i(m)m(ultipliers)c(to)k(optimize)f(Eq.)g(\(17\))120 1124 y(with)h(resp)s(ect)h(to)i Fm(\013)808 1138 y Fj(m)874 1124 y Fr(\).)256 1232 y(Finally)-8 b(,)39 b(w)m(e)i(m)m(ust)f (optimize)f(the)i(b)s(ound)d(with)h(resp)s(ect)h(to)h(the)f(parameters) 120 1340 y Fm(\025)173 1354 y Fj(m)272 1340 y Fr(and)32 b(the)g(parameters)h(of)f(the)h(smo)s(othing)e(distribution.)e(W)-8 b(e)33 b(discuss)e(the)h(lat-)120 1448 y(ter)h(up)s(dates)f(in)f(App)s (endix)f(A.)j(Di\013eren)m(tiating)f(the)h(b)s(ound)e(with)g(resp)s (ect)i(to)g(the)120 1556 y(parameters)e Fm(\025)644 1570 y Fj(m)710 1556 y Fr(,)g(w)m(e)g(readily)d(obtain:)816 1769 y Fm(\025)869 1783 y Fj(m)961 1769 y Fr(=)1559 1708 y Fm(\013)1617 1722 y Fj(m)p 1067 1748 1109 4 v 1067 1767 a Fi(P)1155 1854 y Fj(k)1212 1831 y Fm(\013)1270 1846 y Fj(k)1328 1767 y Fi(P)1416 1854 y Fj(S)1482 1831 y Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))p Fm(Q)1860 1846 y Fj(mf)1968 1831 y Fr(\()p Fm(S)g Fk(j)p Fm(k)s Fr(\))2185 1769 y Fm(;)701 b Fr(\(20\))120 2017 y(where)29 b(a)i(simpli\014ed)26 b(form)j(for)h(the)g(expression)e(in)h(the)h (denominator)f(is)g(presen)m(ted)120 2125 y(in)g(the)i(App)s(endix.)120 2358 y Fq(6.)73 b(Discussion)120 2532 y Fr(W)-8 b(e)24 b(ha)m(v)m(e)g(presen)m(ted)f(a)g(general)g(metho)s(dology)g(for)f (using)f(mixture)h(distributions)d(in)120 2640 y(appro)m(ximate)32 b(probabilistic)d(calculations.)i(W)-8 b(e)33 b(ha)m(v)m(e)g(sho)m(wn)e (that)i(the)f(b)s(ound)e(on)120 2748 y(the)d(lik)m(eliho)s(o)s(d)d (resulting)i(from)g(mixture-based)g(appro)m(ximations)g(is)h(comp)s (osed)g(of)120 2856 y(t)m(w)m(o)d(terms,)f(one)g(a)g(con)m(v)m(ex)h (com)m(bination)e(of)h(simple)e(factorized)i(mean)g(\014eld)e(b)s (ounds)120 2964 y(and)31 b(the)h(other)g(the)g(m)m(utual)f(information) f(b)s(et)m(w)m(een)i(the)g(mixture)f(lab)s(els)e(and)i(the)120 3072 y(hidden)c(v)-5 b(ariables)28 b Fm(S)5 b Fr(.)30 b(It)f(is)g(the)g(latter)h(term)f(whic)m(h)f(represen)m(ts)i(the)f (impro)m(v)m(emen)m(t)120 3179 y(a)m(v)-5 b(ailable)30 b(from)g(the)g(use)g(of)h(mixture)e(mo)s(dels.)256 3287 y(The)37 b(basic)g(mixture)g(appro)m(ximation)f(utilizes)g(parameters)i Fm(\013)2499 3301 y Fj(m)2603 3287 y Fr(\(the)h(mixing)120 3395 y(prop)s(ortions\),)25 b Fm(Q)737 3409 y Fj(i)765 3395 y Fr(\()p Fm(S)856 3409 y Fj(i)885 3395 y Fk(j)p Fm(m)p Fr(\))i(\(the)g(parameters)g(of)g(the)g(factorized)h(comp)s (onen)m(t)f(distri-)120 3503 y(butions\).)k(These)h(parameters)g(are)g (\014t)g(via)g(Eq.)g(\(19\))h(and)f(Eq.)g(\(16\),)i(resp)s(ectiv)m(ely) -8 b(.)120 3611 y(W)g(e)45 b(also)e(in)m(tro)s(duced)f(v)-5 b(ariational)42 b(parameters)i Fm(\025)1955 3625 y Fj(m)2065 3611 y Fr(and)f Fm(R)2324 3625 y Fj(i)2352 3611 y Fr(\()p Fm(S)2443 3625 y Fj(i)2472 3611 y Fk(j)p Fm(m)p Fr(\))g(to)i(pro)m (vide)120 3719 y(tigh)m(t)29 b(appro)m(ximations)e(to)h(the)h(m)m (utual)e(information.)g(Up)s(dates)h(for)g(these)h(param-)120 3827 y(eters)i(w)m(ere)g(presen)m(ted)f(in)f(Eq.)i(\(20\))g(and)f(Eq.)h (\(29\).)256 3935 y(Bishop,)c(et)h(al.)f(\(1998\))i(ha)m(v)m(e)g (presen)m(ted)e(empirical)e(results)h(using)g(the)h(mixture)120 4043 y(appro)m(ximation.)j(In)f(exp)s(erimen)m(ts)h(with)f(random)h (sigmoid)f(b)s(elief)f(net)m(w)m(orks,)k(they)120 4151 y(sho)m(w)m(ed)38 b(that)g(the)f(mixture)g(appro)m(ximation)f(pro)m (vides)g(a)i(b)s(etter)g(appro)m(ximation)120 4259 y(than)c(simple)d (factorized)k(mean)e(\014eld)g(b)s(ounds,)f(and)h(moreo)m(v)m(er)i (that)f(the)g(appro)m(x-)120 4367 y(imation)42 b(impro)m(v)m(es)i (monotonically)e(with)g(the)i(n)m(um)m(b)s(er)e(of)i(comp)s(onen)m(ts)f (in)f(the)120 4475 y(mixture.)256 4583 y(An)f(in)m(teresting)f (direction)g(for)h(further)f(researc)m(h)h(w)m(ould)f(in)m(v)m(olv)m(e) h(the)h(study)120 4690 y(of)34 b(h)m(ybrids)d(that)j(com)m(bine)f(the)h (use)f(of)h(the)f(mixture)g(appro)m(ximation)f(with)g(exact)120 4798 y(metho)s(ds.)23 b(Muc)m(h)h(as)h(in)d(the)i(case)h(of)f(the)g (structured)f(mean)h(\014eld)f(appro)m(ximation)g(of)p eop %%Page: 10 10 10 9 bop 120 85 a Fr(10)382 b Fo(TOMMI)27 b(S.)h(JAAKK)n(OLA)f(AND)h (MICHAEL)g(I.)g(JORD)n(AN)120 310 y Fr(Saul)22 b(and)h(Jordan)g (\(1996\),)j(it)d(should)e(b)s(e)i(p)s(ossible)e(to)j(utilize)e(appro)m (ximations)g(that)120 418 y(iden)m(tify)31 b(core)i(substructures)d (that)j(are)g(treated)g(exactly)-8 b(,)34 b(while)c(the)j(in)m (teractions)120 526 y(b)s(et)m(w)m(een)42 b(these)g(substructures)e (are)i(treated)g(via)f(the)h(mixture)e(approac)m(h.)i(This)120 634 y(w)m(ould)30 b(pro)m(vide)g(a)i(\015exible)e(mo)s(del)g(for)h (dealing)f(with)g(a)i(v)-5 b(ariet)m(y)31 b(of)h(forms)e(of)i(high-)120 742 y(order)e(in)m(teractions,)g(while)f(remaining)f(tractable)j (computationally)-8 b(.)120 982 y Fq(References)120 1131 y Fg(Bishop,)37 b(C.,)h(La)n(wrence,)f(N.,)f(Jaakk)n(ola,)i(T.)f(S.,)f (&)g(Jordan,)h(M.)g(I.)g(Appro)n(ximating)d(p)r(osterior)247 1214 y(distributions)20 b(in)g(b)r(elief)h(net)n(w)n(orks)f(using)g (mixtures.)f(In)g(M.)i(I.)f(Jordan,)g(M.)h(J.)f(Kearns,)h(&)e(S.)h(A.) 247 1297 y(Solla,)i Fb(A)l(dvanc)l(es)i(in)e(Neur)l(al)i(Information)e (Pr)l(o)l(c)l(essing)j(Systems)f(10)p Fg(,)d(MIT)f(Press,)i(Cam)n (bridge)247 1380 y(MA)k(\(1998\).)120 1463 y(Ghahramani,)j(Z.)g(&)g (Jordan,)h(M.)g(I.)f(F)-6 b(actorial)31 b(Hidden)d(Mark)n(o)n(v)h(mo)r (dels.)g(In)f(D.)h(S.)g(T)-6 b(ouretzky)g(,)247 1546 y(M.)27 b(C.)g(Mozer,)g(&)f(M.)g(E.)h(Hasselmo)f(\(Eds.\),)h Fb(A)l(dvanc)l(es)i(in)f(Neur)l(al)g(Information)g(Pr)l(o)l(c)l(essing) 247 1629 y(Systems)h(8)p Fg(,)d(MIT)h(Press,)g(Cam)n(bridge)e(MA)h (\(1996\).)120 1712 y(Neal,)40 b(R.)g(Connectionist)g(learning)h(of)f (b)r(elief)h(net)n(w)n(orks,)f Fb(A)n(rti\014cial)h(Intel)t(ligenc)l(e) p Fg(,)f Fb(56)p Fg(:)g(71-113)247 1795 y(\(1992\).)120 1878 y(P)n(arisi,)27 b(G.)g Fb(Statistic)l(al)h(Field)g(The)l(ory)p Fg(.)f(Addison-W)-6 b(esley:)25 b(Redw)n(o)r(o)r(d)h(Cit)n(y)g (\(1988\).)120 1961 y(P)n(eterson,)k(C.)g(&)f(Anderson,)g(J.)g(R.)g(A)g (mean)f(\014eld)h(theory)g(learning)h(algorithm)f(for)h(neural)f(net-) 247 2044 y(w)n(orks.)e Fb(Complex)h(Systems)f Fa(1)p Fg(:995{1019)i(\(1987\).)120 2127 y(Saul,)19 b(L.)g(K.,)g(Jaakk)n(ola,) h(T.)f(S.,)g(&)f(Jordan,)i(M.)f(I.)g(Mean)g(\014eld)f(theory)g(for)i (sigmoid)f(b)r(elief)g(net)n(w)n(orks.)247 2210 y Fb(Journal)29 b(of)e(A)n(rti\014cial)g(Intel)t(ligenc)l(e)h(R)l(ese)l(ar)l(ch)p Fg(,)g Fb(4)p Fg(:61{76)g(\(1996\).)120 2293 y(Saul,)h(L.)g(K.)f(&)h (Jordan,)g(M.)g(I.)g(Exploiting)g(tractable)g(substructures)g(in)f(in)n (tractable)h(net)n(w)n(orks.)247 2376 y(In)35 b(D.)g(S.)h(T)-6 b(ouretzky)g(,)34 b(M.)j(C.)f(Mozer,)h(&)e(M.)h(E.)g(Hasselmo)g (\(Eds.\),)g Fb(A)l(dvanc)l(es)i(in)e(Neur)l(al)247 2459 y(Information)28 b(Pr)l(o)l(c)l(essing)h(Systems)g(8)p Fg(,)d(MIT)h(Press,)g(Cam)n(bridge)e(MA)h(\(1996\).)120 2543 y(Sh)n(w)n(e,)21 b(M.)h(A.,)f(Middleton,)h(B.,)h(Hec)n(k)n(erman,) d(D.)h(E.,)h(Henrion,)f(M.,)h(Horvitz,)g(E.)g(J.,)g(Lehmann,)e(H.)247 2626 y(P)-6 b(.,)21 b(&)e(Co)r(op)r(er,)j(G.)e(F.)h(Probabilistic)g (diagnosis)h(using)e(a)g(reform)n(ulation)h(of)g(the)e(INTERNIST-)247 2709 y(1/QMR)26 b(kno)n(wledge)h(base.)f Fb(Meth.)i(Inform.)e(Me)l(d.)g (30)p Fg(:)g(241-255)i(\(1991\).)120 3036 y Fq(A.)73 b(Optimization)34 b(of)h(the)f(smo)s(othing)h(distribution)120 3217 y Fr(Let)c(us)f(\014rst)f(adopt)i(the)f(follo)m(wing)f(notation:) 693 3439 y Fm(\031)745 3453 y Fj(R;Q)877 3439 y Fr(\()p Fm(m;)15 b(m)1112 3401 y Fl(0)1136 3439 y Fr(\))84 b Fk(\021)1408 3358 y Fi(X)1445 3542 y Fj(S)1543 3439 y Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))p Fm(Q)1921 3454 y Fj(mf)2029 3439 y Fr(\()p Fm(S)g Fk(j)p Fm(m)2230 3401 y Fl(0)2254 3439 y Fr(\))622 b(\(21\))1255 3755 y(=)1408 3674 y Fi(Y)1449 3857 y Fj(i)1529 3586 y Fi(2)1529 3736 y(4)1585 3674 y(X)1610 3858 y Fj(S)1653 3868 y Fe(i)1720 3755 y Fm(R)1789 3769 y Fj(i)1817 3755 y Fr(\()p Fm(S)1908 3769 y Fj(i)1936 3755 y Fk(j)p Fm(m)p Fr(\))p Fm(Q)2148 3769 y Fj(i)2177 3755 y Fr(\()p Fm(S)2268 3769 y Fj(i)2296 3755 y Fk(j)p Fm(m)2401 3718 y Fl(0)2425 3755 y Fr(\))2460 3586 y Fi(3)2460 3736 y(5)2911 3983 y Fr(\(22\))650 4128 y Fm(H)7 b Fr(\()p Fm(Q)25 b Fk(k)h Fm(R)g Fk(j)p Fm(m)p Fr(\))84 b Fk(\021)1408 4048 y Fi(X)1445 4231 y Fj(S)1543 4128 y Fm(Q)1615 4143 y Fj(mf)1723 4128 y Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))15 b(log)i Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))498 b(\(23\))1255 4362 y(=)1408 4281 y Fi(X)1456 4463 y Fj(i)1543 4281 y Fi(X)1569 4465 y Fj(S)1612 4475 y Fe(i)1678 4362 y Fm(Q)1750 4376 y Fj(i)1778 4362 y Fr(\()p Fm(S)1869 4376 y Fj(i)1898 4362 y Fk(j)p Fm(m)p Fr(\))15 b(log)i Fm(R)2255 4376 y Fj(i)2283 4362 y Fr(\()p Fm(S)2374 4376 y Fj(i)2402 4362 y Fk(j)p Fm(m)p Fr(\))2911 4585 y(\(24\))938 4730 y Fm(H)7 b Fr(\()p Fm(m)p Fr(\))84 b Fk(\021)e(\000)1494 4649 y Fi(X)1523 4823 y Fj(m)1629 4730 y Fm(\013)1687 4744 y Fj(m)1769 4730 y Fr(log)17 b Fm(\013)1960 4744 y Fj(m)2911 4730 y Fr(\(25\))p eop %%Page: 11 11 11 10 bop 575 85 a Fo(IMPR)n(O)n(VING)27 b(THE)g(MEAN)h(FIELD)g(APPR)n (O)n(XIMA)-7 b(TION)363 b Fr(11)120 310 y(where)26 b(w)m(e)i(ha)m(v)m (e)g(ev)-5 b(aluated)27 b(the)g(sums)f(o)m(v)m(er)i Fm(S)k Fr(b)m(y)26 b(making)h(use)f(of)h(the)g(assumption)120 418 y(that)k Fm(R)q Fr(\()p Fm(S)5 b Fk(j)p Fm(m)p Fr(\))30 b(factorizes.)h(W)-8 b(e)32 b(denote)e(the)g(marginals)f(of)h(these)h (factorized)f(forms)120 526 y(b)m(y)g Fm(R)315 540 y Fj(i)343 526 y Fr(\()p Fm(S)434 540 y Fj(i)463 526 y Fk(j)p Fm(m)p Fr(\).)h(Eq.)f(\(11\))i(no)m(w)e(reads)h(as:)246 738 y Fm(I)286 753 y Fj(\025)331 738 y Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))84 b(=)820 657 y Fi(X)849 831 y Fj(m)955 738 y Fm(\013)1013 752 y Fj(m)1080 738 y Fm(H)7 b Fr(\()p Fm(Q)25 b Fk(k)h Fm(R)g Fk(j)p Fm(m)p Fr(\))20 b(+)g Fm(H)7 b Fr(\()p Fm(m)p Fr(\))820 1020 y Fk(\000)906 939 y Fi(X)934 1113 y Fj(m)1041 1020 y Fm(\025)1094 1034 y Fj(m)1176 876 y Fi(")1224 939 y(X)1242 1124 y Fj(m)1304 1105 y Fc(0)1359 1020 y Fm(Q)p Fr(\()p Fm(m)1546 982 y Fl(0)1569 1020 y Fr(\))21 b Fm(\031)1677 1034 y Fj(R;Q)1810 1020 y Fr(\()p Fm(m;)15 b(m)2045 982 y Fl(0)2068 1020 y Fr(\))2103 876 y Fi(#)2172 1020 y Fr(+)2263 939 y Fi(X)2292 1113 y Fj(m)2398 1020 y Fm(\013)2456 1034 y Fj(m)2538 1020 y Fr(log)i Fm(\025)2724 1034 y Fj(m)2810 1020 y Fr(+)j(1)-35 b(\(26\))120 1300 y(T)-8 b(o)34 b(optimize)e(the)i(b)s (ound)d Fm(I)1111 1315 y Fj(\025)1156 1300 y Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))34 b(with)e(resp)s(ect)i(to)g Fm(R)q Fr(,)f(w)m(e)h(need)f(the)g(follo)m(wing)120 1408 y(deriv)-5 b(ativ)m(es:)304 1576 y Fm(@)26 b(H)7 b Fr(\()p Fm(Q)25 b Fk(k)h Fm(R)g Fk(j)p Fm(m)p Fr(\))p 304 1616 596 4 v 393 1700 a Fm(@)f(R)535 1714 y Fj(j)572 1700 y Fr(\()p Fm(S)663 1714 y Fj(j)700 1700 y Fk(j)p Fm(k)s Fr(\))992 1637 y(=)83 b Fm(\016)1186 1652 y Fj(k)r(;m)1341 1576 y Fm(Q)1413 1590 y Fj(j)1450 1576 y Fr(\()p Fm(S)1541 1590 y Fj(j)1577 1576 y Fk(j)p Fm(k)s Fr(\))p 1341 1616 347 4 v 1342 1700 a Fm(R)1411 1714 y Fj(j)1448 1700 y Fr(\()p Fm(S)1539 1714 y Fj(j)1576 1700 y Fk(j)p Fm(k)s Fr(\))2911 1637 y(\(27\))347 1870 y Fm(@)25 b(\031)472 1884 y Fj(R;Q)605 1870 y Fr(\()p Fm(m;)15 b(m)840 1837 y Fl(0)864 1870 y Fr(\))p 347 1911 553 4 v 414 1994 a Fm(@)26 b(R)557 2008 y Fj(j)593 1994 y Fr(\()p Fm(S)684 2008 y Fj(j)721 1994 y Fk(j)p Fm(k)s Fr(\))992 1932 y(=)83 b Fm(\016)1186 1947 y Fj(k)r(;m)1331 1932 y Fm(Q)1403 1946 y Fj(j)1440 1932 y Fr(\()p Fm(S)1531 1946 y Fj(j)1567 1932 y Fk(j)p Fm(m)1672 1894 y Fl(0)1696 1932 y Fr(\))1749 1851 y Fi(Y)1746 2035 y Fj(i)p Fl(6)p Fn(=)p Fj(j)1872 1763 y Fi(2)1872 1912 y(4)1928 1851 y(X)1953 2035 y Fj(S)1996 2045 y Fe(i)2063 1932 y Fm(R)2132 1946 y Fj(i)2160 1932 y Fr(\()p Fm(S)2251 1946 y Fj(i)2279 1932 y Fk(j)p Fm(m)p Fr(\))p Fm(Q)2491 1946 y Fj(i)2520 1932 y Fr(\()p Fm(S)2611 1946 y Fj(i)2639 1932 y Fk(j)p Fm(m)2744 1894 y Fl(0)2767 1932 y Fr(\))2802 1763 y Fi(3)2802 1912 y(5)2873 1932 y Fm(:)13 b Fr(\(28\))120 2246 y(Denoting)28 b(the)h(pro)s(duct)e(in)g (Eq.)h(\(28\))i(b)m(y)e Fm(\031)1640 2202 y Fj(j)1637 2273 y(R;Q)1769 2246 y Fr(\()p Fm(m;)15 b(m)2004 2213 y Fl(0)2028 2246 y Fr(\),)29 b(w)m(e)f(can)h(no)m(w)f(c)m(haracterize) 120 2354 y(the)j(optimal)e Fm(R)i Fr(via)f(the)g(consistency)h (equations:)146 2557 y Fm(@)26 b(I)260 2572 y Fj(\025)305 2557 y Fr(\()p Fm(m)p Fr(;)15 b Fm(S)5 b Fr(\))p 143 2597 418 4 v 143 2681 a Fm(@)25 b(R)285 2695 y Fj(j)322 2681 y Fr(\()p Fm(S)413 2695 y Fj(j)449 2681 y Fk(j)p Fm(k)s Fr(\))595 2619 y(=)g Fm(\013)749 2634 y Fj(k)802 2557 y Fm(Q)874 2571 y Fj(j)910 2557 y Fr(\()p Fm(S)1001 2571 y Fj(j)1038 2557 y Fk(j)p Fm(k)s Fr(\))p 802 2597 347 4 v 803 2681 a Fm(R)872 2695 y Fj(j)909 2681 y Fr(\()p Fm(S)1000 2695 y Fj(j)1037 2681 y Fk(j)p Fm(k)s Fr(\))1179 2619 y Fk(\000)20 b Fm(\025)1323 2634 y Fj(k)1381 2475 y Fi(")1429 2538 y(X)1447 2723 y Fj(m)1509 2704 y Fc(0)1564 2619 y Fm(Q)p Fr(\()p Fm(m)1751 2581 y Fl(0)1775 2619 y Fr(\))g Fm(\031)1885 2574 y Fj(j)1882 2646 y(R;Q)2015 2619 y Fr(\()p Fm(k)s(;)15 b(m)2220 2581 y Fl(0)2244 2619 y Fr(\))p Fm(Q)2351 2633 y Fj(j)2388 2619 y Fr(\()p Fm(S)2479 2633 y Fj(j)2515 2619 y Fk(j)p Fm(m)2620 2581 y Fl(0)2644 2619 y Fr(\))2679 2475 y Fi(#)2753 2619 y Fr(=)25 b(0)17 b(\(29\))120 2894 y(Note)42 b(that)f(the)g(second)g (term)f(do)s(es)g(not)h(dep)s(end)e(on)i(the)f(smo)s(othing)g(marginal) 120 3002 y Fm(R)189 3016 y Fj(j)226 3002 y Fr(\()p Fm(S)317 3016 y Fj(j)353 3002 y Fk(j)p Fm(k)s Fr(\))k(and)f(therefore)h(these)g (consistency)f(equations)g(are)h(easily)f(solv)m(ed)g(for)120 3110 y(an)m(y)37 b(sp)s(eci\014c)f(marginal.)g(The)h(b)s(est)f (\014tting)g Fm(R)i Fr(is)e(found)g(b)m(y)h(iterativ)m(ely)f(optimiz-) 120 3218 y(ing)g(eac)m(h)i(of)f(its)g(marginals.)e(W)-8 b(e)38 b(note)g(\014nally)d(that)i(the)h(iterativ)m(e)f(solutions)e (are)120 3325 y(not)h(necessarily)f(normalized.)g(This)g(is)g(of)h(no)g (consequence,)h(ho)m(w)m(ev)m(er,)h(since)d(the)120 3433 y(information)25 b(b)s(ound)g(of)i(Eq.)f(\(11\)|once)j(maximized)d(o)m (v)m(er)i(the)f Fm(\025)f Fr(parameters|is)120 3541 y(in)m(v)-5 b(arian)m(t)30 b(to)h(the)f(normalization)f(of)i Fm(R)q Fr(.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF