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Fh(2)56 b(Arc)n(hitectures)225 1307 y Fm(2.1)48 b(Probabili)o(st)o(ic)13 b(decision)g(trees)225 1392 y Fi(The)i(\\hierarc)o(hical)g(mixture)f(of)g(exp)q(erts")i (\(HME\))g(mo)q(del)d(\(Jordan)i(&)g(Jacobs,)h(1994\))e(is)g(a)225 1438 y(decision)d(tree)g(in)f(whic)o(h)h(the)g(decisions)g(are)g(mo)q (deled)e(probabilistically)m(,)f(as)j(are)g(the)g(outputs.)225 1483 y(The)k(total)g(probabilit)o(y)e(of)i(an)f(output)i(giv)o(en)e(an) h(input)g(is)g(the)g(sum)f(o)o(v)o(er)i(all)d(paths)j(in)e(the)225 1529 y(tree)21 b(from)c(the)j(input)g(to)f(the)h(output.)35 b(The)19 b(HME)h(mo)q(del)e(is)h(sho)o(wn)h(in)f(the)h(graphical)225 1575 y(mo)q(del)12 b(formalism)e(in)k(Figure)g(2.1.)j(Here)e(a)e(no)q (de)i(represen)o(ts)h(a)e(random)e(v)n(ariable,)h(and)g(the)225 1620 y(links)f(represen)o(t)j(probabilistic)d(dep)q(endencies.)20 b(A)13 b(conditional)f(probabilit)o(y)f(distribution)h(is)225 1666 y(asso)q(ciated)18 b(with)e(eac)o(h)i(no)q(de)f(in)f(the)i(graph,) 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b(with)g(the)h(decision)225 1919 y(no)q(des,)d(where)h Ff(\021)499 1904 y Fg(1)518 1919 y Ff(;)7 b(\021)559 1904 y Fg(2)577 1919 y Fi(,)15 b(and)h Ff(\021)709 1904 y Fg(3)743 1919 y Fi(are)g(parameters)g (\(e.g.,)f(Jordan)g(and)h(Jacobs)g(utilized)f(soft-)225 1965 y(max)c(transformations)f(of)i(linear)g(functions)g(of)g Fm(x)g Fi(for)g(these)i(probabilities\).)j(The)12 b(leaf)g(prob-)225 2011 y(abilities)j Ff(P)6 b Fi(\()p Fm(y)q Fk(j)p Fm(x)p Ff(;)h Fm(z)534 1996 y Fg(1)551 2011 y Ff(;)g Fm(z)591 1996 y Fg(2)610 2011 y Ff(;)g Fm(z)650 1996 y Fg(3)668 2011 y Ff(;)g(\022)q Fi(\))16 b(are)g(arbitrary)g(conditional)f (probabilit)o(y)f(mo)q(dels;)i(e.g.,)f(lin-)225 2056 y(ear/Gaussian)f(mo)q(dels)e(for)i(regression)h(problems.)225 2127 y(The)h(k)o(ey)g(calculation)f(in)h(the)g(\014tting)g(of)f(the)h (HME)h(mo)q(del)d(to)i(data)f(is)h(the)h(calculation)e(of)225 2173 y(the)j(p)q(osterior)h(probabilities)e(of)g(the)h(hidden)g (decisions)h(giv)o(en)e(the)h(clamp)q(ed)f(v)n(alues)h(of)f 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begin put /bdef {bind def} bind def /E {exch} bdef /FF {findfont} bdef /MF {makefont} bdef /RO {rotate} bdef /SC {scale} bdef /SF {setfont} bdef /SG {setgray} bdef /TR {translate} bdef /bp {lj lw rgb} bdef /bpbw {lj lw setgray} bdef /c {curveto} bdef /cl {closepath} bdef /fi {eofill} bdef /g {setgray} bdef /gr {grestore} bdef /gs {gsave} bdef /l {lineto} bdef /lj {setlinejoin} bdef /lw {setlinewidth} bdef /m {moveto} bdef /n {newpath} bdef /nx {/x E def} bdef /r {rmoveto} bdef /rl {rlineto} bdef /rgb {setrgbcolor} bdef /s {show} bdef /sd {setdash} bdef /sp {x 0 rmoveto} bdef /ss {currentpoint pop E m} bdef /st {stroke} bdef /BPSIDE 32 def %% pixels per pattern side /PATFREQ 3.0 def %% pattern pixels per mm /dp_mat [PATFREQ 0 0 PATFREQ 0 0] def /dp_pw BPSIDE def %% pattern pixel width /dp_ph BPSIDE def %% pattern pixel height /dp_w dp_pw PATFREQ div def %% pattern mm width /dp_h dp_ph PATFREQ div def %% pattern mm height /savemat matrix def /topmat matrix def /patmat matrix def /ncpoint errordict /nocurrentpoint get def errordict begin /nocurrentpoint { dup /pathbbox load eq {pop 0 0 1 1} {ncpoint} ifelse } bdef end /ar { %% sa ea sx sy rot tx ty matrix currentmatrix 8 1 roll TR RO SC n 0 0 1 5 3 roll arc setmatrix } bdef /arn { %% sa ea sx sy rot tx ty TR RO SC matrix currentmatrix 8 1 roll n 0 0 1 5 3 roll arcn setmatrix } bdef /el { %% sx sy rot tx ty matrix currentmatrix 6 1 roll TR RO SC n 0 0 1 0 360 arc setmatrix cl } bdef /image_raster { %% sw sh sd dw dh xs ys TR SC /sd E def /sh E def /sw E def /imagebuf sw sd mul 7 add 8 idiv string def sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop} image } bdef /imagemask_raster { TR SC /sh E def /sw E def /imagebuf sw 7 add 8 idiv string def sw sh false [sw 0 0 sh 0 0] {currentfile imagebuf readhexstring pop} imagemask } bdef /dither_color_raster { % bool sw sh sd dw dh xs ys TR SC /sd E def /sh E def /sw E def sd 8 eq and { /imagebuf 3 string def /grayval 1 string def sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop pop imagebuf 0 get 0.299 mul imagebuf 1 get 0.587 mul add imagebuf 2 get 0.114 mul add cvi grayval exch 0 exch put grayval } image } { /imagebuf sw 3 mul sd mul 7 add 8 idiv string def sh { currentfile imagebuf readhexstring pop pop } repeat } ifelse } bdef /image_color_raster { % bool sw sh sd dw dh xs ys /colorimage where not { dither_color_raster } { pop TR SC /sd E def /sh E def /sw E def pop /imagebuf sw 3 mul sd mul 7 add 8 idiv string def sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop} false 3 colorimage } ifelse } bdef /patpath { /inv E def topmat setmatrix pathbbox %% get lo - hi indecies /hy E dp_h div floor cvi def /hx E dp_w div floor cvi def /ly E dp_h div floor cvi def /lx E dp_w div floor cvi def lx 1 hx { dp_w mul ly 1 hy { dp_h mul E dup 3 1 roll E patmat currentmatrix pop TR dp_pw dp_ph inv dp_mat dp_proc imagemask patmat setmatrix } for pop } for } bdef /setpattern { /blue E def /green E def /red E def /freq E def /bwidth E def /bpside E def /bstring E def /onbits 0 def /offbits 0 def freq 0 {/y E def /x E def /xindex x 1 add 2 div bpside mul cvi def /yindex y 1 add 2 div bpside mul cvi def bstring yindex bwidth mul xindex 8 idiv add get not 1 7 xindex 8 mod sub bitshift and 0 ne {/onbits onbits 1 add def 1} {/offbits offbits 1 add def 0} ifelse } setscreen {} settransfer systemdict /setcmykcolor known { /fact 1 onbits offbits onbits add div sub def 1 red sub fact mul 1 green sub fact mul 1 blue sub fact mul 0 setcmykcolor } { offbits offbits onbits add div setgray} ifelse } bdef /dmatrix matrix def /dpi 72 0 dmatrix defaultmatrix dtransform dup mul E dup mul add sqrt def /B {gs bp st gr} bdef %% brush: gr lw lj /Bbw {gs bpbw st gr} bdef %% brush: gr lw lj /F {gs rgb eofill gr} bdef %% fill: gr /Fbw {gs setgray eofill gr} bdef %% fill: gr /PB {gs lj lw setpattern st gr} bdef /PF {gs eoclip patpath gr} bdef /BB {gs rgb lj lw strokepath clip patpath gr} bdef /xdef {exch def} bdef /clip_region { /ht xdef /wd xdef /bm xdef /lm xdef newpath lm bm moveto 0 ht rlineto wd 0 rlineto 0 ht neg rlineto closepath clip } bdef /reencode_small_dict 12 dict def /ReencodeSmall { reencode_small_dict begin /new_codes_and_names exch def /new_font_name exch def /base_font_name exch def /base_font_dict base_font_name findfont def /newfont base_font_dict maxlength dict def base_font_dict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName new_font_name put new_codes_and_names aload pop new_codes_and_names length 2 idiv { newfont /Encoding get 3 1 roll put } repeat new_font_name newfont definefont pop end %reencode_small_dict } def /extended_Zapf [ 8#223 /a89 8#224 /a90 8#225 /a93 8#226 /a94 8#227 /a91 8#230 /a92 8#231 /a205 8#232 /a85 8#233 /a206 8#234 /a86 8#235 /a87 8#236 /a88 8#237 /a95 8#240 /a96 ] def /extended_Standard [ 29 /thorn 30 /yacute 31 /divide 128 /Acircumflex 129 /Adieresis 130 /Agrave 131 /Aring 132 /Atilde 133 /Ccedilla 134 /Eacute 135 /Ecircumflex 136 /Edieresis 137 /Egrave 138 /Iacute 139 /Icircumflex 140 /Idieresis 141 /Igrave 142 /Ntilde 143 /Oacute 144 /Ocircumflex 145 /Odieresis 146 /Ograve 147 /Otilde 148 /Scaron 149 /Uacute 150 /Ucircumflex 151 /Udieresis 152 /Ugrave 153 /Ydieresis 154 /Zcaron 155 /aacute 156 /acircumflex 157 /adieresis 158 /agrave 159 /aring 160 /atilde 161 /exclamdown 162 /cent 163 /sterling 164 /fraction 165 /yen 166 /florin 167 /section 168 /currency 169 /quotesingle 170 /quotedblleft 171 /guillemotleft 172 /guilsinglleft 173 /guilsinglright 174 /fi 175 /fl 176 /plusminus 177 /endash 178 /dagger 179 /daggerdbl 180 /periodcentered 181 /twosuperior 182 /paragraph 183 /bullet 184 /quotesinglbase 185 /quotedblbase 186 /quotedblright 187 /guillemotright 188 /ellipsis 189 /perthousand 190 /threesuperior 191 /questiondown 192 /mu 193 /grave 194 /acute 195 /circumflex 196 /tilde 197 /macron 198 /breve 199 /dotaccent 200 /dieresis 201 /onesuperior 202 /ring 203 /cedilla 204 /onequarter 205 /hungarumlaut 206 /ogonek 207 /caron 208 /emdash 209 /ccedilla 210 /copyright 211 /eacute 212 /ecircumflex 213 /edieresis 214 /egrave 215 /iacute 216 /icircumflex 217 /idieresis 218 /igrave 219 /logicalnot 220 /minus 221 /ntilde 222 /oacute 223 /ocircumflex 224 /odieresis 225 /AE 226 /onehalf 227 /ordfeminine 228 /ograve 229 /otilde 230 /registered 231 /scaron 232 /Lslash 233 /Oslash 234 /OE 235 /ordmasculine 236 /trademark 237 /uacute 238 /ucircumflex 239 /udieresis 240 /ugrave 241 /ae 242 /ydieresis 243 /zcaron 244 /Aacute 245 /dotlessi 246 /threequarters 247 /Eth 248 /lslash 249 /oslash 250 /oe 251 /germandbls 252 /multiply 253 /Yacute 254 /Thorn 255 /eth ] def /extended_Symbol [ ] def /extend_font { % stack: fontname newfontname exch dup (ZapfDingbats) eq { cvn exch cvn extended_Zapf ReencodeSmall } { dup (Symbol) eq { cvn exch cvn extended_Symbol ReencodeSmall } { cvn exch cvn extended_Standard ReencodeSmall } ifelse } ifelse } bind def /extend_font_name { % stack: font_name_string dup length 1 add string /extended_font_name exch def extended_font_name 0 (_) putinterval extended_font_name 1 3 -1 roll putinterval extended_font_name } bind def /gf { /f exch def f cvn where { f exch begin cvn load exec setfont end } { f 0 f length 8 sub getinterval dup /localfont exch extend_font_name def localfont extend_font localfont findfont /xsz f f length 4 sub 4 getinterval cvi def /ysz f f length 8 sub 4 getinterval cvi def [ xsz 0 0 ysz neg 0 0 ] makefont dup f cvn exch def setfont } ifelse } bind def /ul { % space drop thickness gs currentpoint currentlinewidth currentpoint n m 6 -3 roll lw 0 exch r 0 rl st lw m gr } bind def /nxtab { currentpoint pop 1000.0 mul cvi tab mod tab exch sub 1000.0 div 0 rmoveto } bind def /nx { /x exch def } bind def 0. nx gsave 2.83465 -2.83465 scale 0 -279.4 translate topmat currentmatrix pop n 97.167 55 m 97.167 55 115.81 68.444 116.72 90.75 c 117.64 113.06 96.861 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/nocurrentpoint { dup /pathbbox load eq {pop 0 0 1 1} {ncpoint} ifelse } bdef end /ar { %% sa ea sx sy rot tx ty matrix currentmatrix 8 1 roll TR RO SC n 0 0 1 5 3 roll arc setmatrix } bdef /arn { %% sa ea sx sy rot tx ty TR RO SC matrix currentmatrix 8 1 roll n 0 0 1 5 3 roll arcn setmatrix } bdef /el { %% sx sy rot tx ty matrix currentmatrix 6 1 roll TR RO SC n 0 0 1 0 360 arc setmatrix cl } bdef /image_raster { %% sw sh sd dw dh xs ys TR SC /sd E def /sh E def /sw E def /imagebuf sw sd mul 7 add 8 idiv string def sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop} image } bdef /imagemask_raster { TR SC /sh E def /sw E def /imagebuf sw 7 add 8 idiv string def sw sh false [sw 0 0 sh 0 0] {currentfile imagebuf readhexstring pop} imagemask } bdef /dither_color_raster { % bool sw sh sd dw dh xs ys TR SC /sd E def /sh E def /sw E def sd 8 eq and { /imagebuf 3 string def /grayval 1 string def sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop pop imagebuf 0 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/onbits 0 def /offbits 0 def freq 0 {/y E def /x E def /xindex x 1 add 2 div bpside mul cvi def /yindex y 1 add 2 div bpside mul cvi def bstring yindex bwidth mul xindex 8 idiv add get not 1 7 xindex 8 mod sub bitshift and 0 ne {/onbits onbits 1 add def 1} {/offbits offbits 1 add def 0} ifelse } setscreen {} settransfer systemdict /setcmykcolor known { /fact 1 onbits offbits onbits add div sub def 1 red sub fact mul 1 green sub fact mul 1 blue sub fact mul 0 setcmykcolor } { offbits offbits onbits add div setgray} ifelse } bdef /dmatrix matrix def /dpi 72 0 dmatrix defaultmatrix dtransform dup mul E dup mul add sqrt def /B {gs bp st gr} bdef %% brush: gr lw lj /Bbw {gs bpbw st gr} bdef %% brush: gr lw lj /F {gs rgb eofill gr} bdef %% fill: gr /Fbw {gs setgray eofill gr} bdef %% fill: gr /PB {gs lj lw setpattern st gr} bdef /PF {gs eoclip patpath gr} bdef /BB {gs rgb lj lw strokepath clip patpath gr} bdef /xdef {exch def} bdef /clip_region { /ht xdef /wd xdef /bm xdef /lm xdef newpath lm bm moveto 0 ht rlineto wd 0 rlineto 0 ht neg rlineto closepath clip } bdef /reencode_small_dict 12 dict def /ReencodeSmall { reencode_small_dict begin /new_codes_and_names exch def /new_font_name exch def /base_font_name exch def /base_font_dict base_font_name findfont def /newfont base_font_dict maxlength dict def base_font_dict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName new_font_name put new_codes_and_names aload pop new_codes_and_names length 2 idiv { newfont /Encoding get 3 1 roll put } repeat new_font_name newfont definefont pop end %reencode_small_dict } def /extended_Zapf [ 8#223 /a89 8#224 /a90 8#225 /a93 8#226 /a94 8#227 /a91 8#230 /a92 8#231 /a205 8#232 /a85 8#233 /a206 8#234 /a86 8#235 /a87 8#236 /a88 8#237 /a95 8#240 /a96 ] def /extended_Standard [ 29 /thorn 30 /yacute 31 /divide 128 /Acircumflex 129 /Adieresis 130 /Agrave 131 /Aring 132 /Atilde 133 /Ccedilla 134 /Eacute 135 /Ecircumflex 136 /Edieresis 137 /Egrave 138 /Iacute 139 /Icircumflex 140 /Idieresis 141 /Igrave 142 /Ntilde 143 /Oacute 144 /Ocircumflex 145 /Odieresis 146 /Ograve 147 /Otilde 148 /Scaron 149 /Uacute 150 /Ucircumflex 151 /Udieresis 152 /Ugrave 153 /Ydieresis 154 /Zcaron 155 /aacute 156 /acircumflex 157 /adieresis 158 /agrave 159 /aring 160 /atilde 161 /exclamdown 162 /cent 163 /sterling 164 /fraction 165 /yen 166 /florin 167 /section 168 /currency 169 /quotesingle 170 /quotedblleft 171 /guillemotleft 172 /guilsinglleft 173 /guilsinglright 174 /fi 175 /fl 176 /plusminus 177 /endash 178 /dagger 179 /daggerdbl 180 /periodcentered 181 /twosuperior 182 /paragraph 183 /bullet 184 /quotesinglbase 185 /quotedblbase 186 /quotedblright 187 /guillemotright 188 /ellipsis 189 /perthousand 190 /threesuperior 191 /questiondown 192 /mu 193 /grave 194 /acute 195 /circumflex 196 /tilde 197 /macron 198 /breve 199 /dotaccent 200 /dieresis 201 /onesuperior 202 /ring 203 /cedilla 204 /onequarter 205 /hungarumlaut 206 /ogonek 207 /caron 208 /emdash 209 /ccedilla 210 /copyright 211 /eacute 212 /ecircumflex 213 /edieresis 214 /egrave 215 /iacute 216 /icircumflex 217 /idieresis 218 /igrave 219 /logicalnot 220 /minus 221 /ntilde 222 /oacute 223 /ocircumflex 224 /odieresis 225 /AE 226 /onehalf 227 /ordfeminine 228 /ograve 229 /otilde 230 /registered 231 /scaron 232 /Lslash 233 /Oslash 234 /OE 235 /ordmasculine 236 /trademark 237 /uacute 238 /ucircumflex 239 /udieresis 240 /ugrave 241 /ae 242 /ydieresis 243 /zcaron 244 /Aacute 245 /dotlessi 246 /threequarters 247 /Eth 248 /lslash 249 /oslash 250 /oe 251 /germandbls 252 /multiply 253 /Yacute 254 /Thorn 255 /eth ] def /extended_Symbol [ ] def /extend_font { % stack: fontname newfontname exch dup (ZapfDingbats) eq { cvn exch cvn extended_Zapf ReencodeSmall } { dup (Symbol) eq { cvn exch cvn extended_Symbol ReencodeSmall } { cvn exch cvn extended_Standard ReencodeSmall } ifelse } ifelse } bind def 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/FontName new_font_name put new_codes_and_names aload pop new_codes_and_names length 2 idiv { newfont /Encoding get 3 1 roll put } repeat new_font_name newfont definefont pop end %reencode_small_dict } def /extended_Zapf [ 8#223 /a89 8#224 /a90 8#225 /a93 8#226 /a94 8#227 /a91 8#230 /a92 8#231 /a205 8#232 /a85 8#233 /a206 8#234 /a86 8#235 /a87 8#236 /a88 8#237 /a95 8#240 /a96 ] def /extended_Standard [ 29 /thorn 30 /yacute 31 /divide 128 /Acircumflex 129 /Adieresis 130 /Agrave 131 /Aring 132 /Atilde 133 /Ccedilla 134 /Eacute 135 /Ecircumflex 136 /Edieresis 137 /Egrave 138 /Iacute 139 /Icircumflex 140 /Idieresis 141 /Igrave 142 /Ntilde 143 /Oacute 144 /Ocircumflex 145 /Odieresis 146 /Ograve 147 /Otilde 148 /Scaron 149 /Uacute 150 /Ucircumflex 151 /Udieresis 152 /Ugrave 153 /Ydieresis 154 /Zcaron 155 /aacute 156 /acircumflex 157 /adieresis 158 /agrave 159 /aring 160 /atilde 161 /exclamdown 162 /cent 163 /sterling 164 /fraction 165 /yen 166 /florin 167 /section 168 /currency 169 /quotesingle 170 /quotedblleft 171 /guillemotleft 172 /guilsinglleft 173 /guilsinglright 174 /fi 175 /fl 176 /plusminus 177 /endash 178 /dagger 179 /daggerdbl 180 /periodcentered 181 /twosuperior 182 /paragraph 183 /bullet 184 /quotesinglbase 185 /quotedblbase 186 /quotedblright 187 /guillemotright 188 /ellipsis 189 /perthousand 190 /threesuperior 191 /questiondown 192 /mu 193 /grave 194 /acute 195 /circumflex 196 /tilde 197 /macron 198 /breve 199 /dotaccent 200 /dieresis 201 /onesuperior 202 /ring 203 /cedilla 204 /onequarter 205 /hungarumlaut 206 /ogonek 207 /caron 208 /emdash 209 /ccedilla 210 /copyright 211 /eacute 212 /ecircumflex 213 /edieresis 214 /egrave 215 /iacute 216 /icircumflex 217 /idieresis 218 /igrave 219 /logicalnot 220 /minus 221 /ntilde 222 /oacute 223 /ocircumflex 224 /odieresis 225 /AE 226 /onehalf 227 /ordfeminine 228 /ograve 229 /otilde 230 /registered 231 /scaron 232 /Lslash 233 /Oslash 234 /OE 235 /ordmasculine 236 /trademark 237 /uacute 238 /ucircumflex 239 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b(idea)h(is)f(to)g(asso)q(ciate)h(with)f(the)h(in)o (tractable)f(distribution)g Ff(P)21 b Fi(a)15 b(simpli\014ed)e (distribu-)225 2176 y(tion)i Ff(Q)h Fi(that)g(retains)h(certain)f(of)g (the)g(terms)g(in)g Ff(P)21 b Fi(and)16 b(neglects)h(others,)g (replacing)f(them)225 2222 y(with)f(parameters)h Ff(\026)563 2228 y Fc(i)592 2222 y Fi(that)f(w)o(e)h(will)e(refer)i(to)g(as)f(\\v)n (ariational)e(parameters.")23 b(Graphically)225 2267 y(the)18 b(metho)q(d)e(can)h(b)q(e)h(view)o(ed)f(as)g(deleting)g(arcs)h (from)e(the)h(original)e(graph)i(un)o(til)g(a)f(forest)225 2313 y(of)i(tractable)g(substructures)k(is)c(obtained.)31 b(Arcs)19 b(that)f(remain)f(in)h(the)h(simpli\014ed)d(graph)225 2359 y(corresp)q(ond)k(to)e(terms)g(that)g(are)h(retained)g(in)e Ff(Q)p Fi(;)j(arcs)f(that)f(are)h(deleted)g(corresp)q(ond)h(to)225 2404 y(v)n(ariational)12 b(parameters.)225 2475 y(T)m(o)i(obtain)g(the) i(b)q(est)g(p)q(ossible)f(appro)o(ximation)d(of)j Ff(P)20 b Fi(w)o(e)15 b(minimi)o(ze)e(the)j(Kullbac)o(k-Liebler)p eop %%Page: 5 5 5 4 bop 225 42 a Fi(div)o(ergence)18 b Ff(K)s(L)p Fi(\()p Ff(Q)p Fk(jj)p Ff(P)6 b Fi(\))15 b(with)i(resp)q(ect)h(to)f(the)g (parameters)g Ff(\026)1247 48 y Fc(i)1261 42 y Fi(.)26 b(The)17 b(result)g(is)f(a)h(coupled)225 87 y(set)f(of)e(equations)g (that)h(are)g(solv)o(ed)g(iterativ)o(ely)m(.)k(These)d(equations)f(mak) o(e)e(reference)k(to)d(the)225 133 y(v)n(alues)i(of)g(exp)q(ectations)h (of)f(no)q(des)h(in)e(the)i(tractable)g(substructures;)j(th)o(us)c(the) h(\(e\016cien)o(t\))225 178 y(algorithms)8 b(that)j(pro)o(vide)f(suc)o (h)i(exp)q(ectations)f(are)g(run)g(as)g(subroutines.)18 b(Based)11 b(on)g(the)g(p)q(os-)225 224 y(terior)g(exp)q(ectations)g (computed)g(under)g Ff(Q)p Fi(,)f(the)h(parameters)g(de\014ning)f Ff(P)16 b Fi(are)11 b(adjusted.)18 b(The)225 270 y(algorithm)11 b(as)i(a)g(whole)g(is)g(guaran)o(teed)h(to)f(increase)h(a)f(lo)o(w)o (er)g(b)q(ound)g(on)g(the)h(log)e(lik)o(eliho)q(o)q(d.)225 371 y Fm(3.2)48 b(A)16 b(forest)e(of)h(c)o(hains)225 456 y Fi(The)10 b(HMDT)g(can)g(b)q(e)h(view)o(ed)f(as)g(a)f(coupled)i (set)g(of)e(c)o(hains,)h(with)g(couplings)f(induced)i(directly)225 502 y(via)f(the)h(decision)g(tree)h(structure)h(and)e(indirectly)g(via) e(the)j(common)c(coupling)i(to)g(the)i(output)225 547 y(v)o(ector.)20 b(If)13 b(these)j(couplings)e(are)g(remo)o(v)o(ed)f(in) h(the)h(v)n(ariational)d(appro)o(ximation,)e(w)o(e)15 b(obtain)225 593 y(a)20 b Ff(Q)h Fi(distribution)f(whose)h(graph)g(is)f (a)h(forest)g(of)f(c)o(hains.)39 b(There)21 b(are)h(sev)o(eral)f(w)o(a) o(ys)f(to)225 639 y(parameterize)10 b(this)g(graph;)h(in)e(the)i (curren)o(t)g(pap)q(er)g(w)o(e)f(in)o(v)o(estigate)g(a)f (parameterization)g(with)225 684 y(time-v)n(arying)k(transition)i (matrices)h(and)f(time-v)n(arying)e(\014elds.)25 b(Th)o(us)16 b(the)g Ff(Q)f Fi(distribution)225 730 y(is)f(giv)o(en)f(b)o(y)307 840 y Ff(Q)p Fi(\()p Fk(f)p Fm(z)398 823 y Fg(1)398 850 y Fc(t)416 840 y Ff(;)7 b Fm(z)456 823 y Fg(2)456 850 y Fc(t)475 840 y Ff(;)g Fm(z)515 823 y Fg(3)515 850 y Fc(t)533 840 y Fk(g)k(j)g(f)p Fm(y)634 846 y Fc(t)649 840 y Fk(g)p Ff(;)c Fk(f)p Fm(x)735 846 y Fc(t)749 840 y Fk(g)p Fi(\))41 b(=)924 812 y(1)p 906 830 57 2 v 906 868 a Ff(Z)934 874 y Fc(Q)989 788 y(T)975 800 y Fb(Y)974 888 y Fc(t)p Fg(=2)1036 840 y Fi(~)-21 b Ff(a)1058 823 y Fg(1)1058 850 y Fc(t)1076 840 y Fi(\()p Fm(z)1113 823 y Fg(1)1113 850 y Fc(t)1132 840 y Fk(j)p Fm(z)1165 823 y Fg(1)1165 850 y Fc(t)p Fl(\000)p Fg(1)1222 840 y Fi(\))q(~)f Ff(a)1260 823 y Fg(2)1260 850 y Fc(t)1279 840 y Fi(\()p Fm(z)1316 823 y Fg(2)1316 850 y Fc(t)1335 840 y Fk(j)p Fm(z)1368 823 y Fg(2)1368 850 y Fc(t)p Fl(\000)p Fg(1)1425 840 y Fi(\))q(~)g Ff(a)1463 823 y Fg(3)1463 850 y Fc(t)1481 840 y Fi(\()p Fm(z)1518 823 y Fg(3)1518 850 y Fc(t)1537 840 y Fk(j)p Fm(z)1570 823 y Fg(3)1570 850 y Fc(t)p Fl(\000)p Fg(1)1627 840 y Fi(\))916 933 y Fc(T)901 945 y Fb(Y)901 1033 y Fc(t)p Fg(=1)965 985 y Fi(~)e Ff(q)982 968 y Fg(1)981 995 y Fc(t)1001 985 y Fi(\()p Fm(z)1038 968 y Fg(1)1038 995 y Fc(t)1057 985 y Fi(\))s(~)g Ff(q)1093 968 y Fg(2)1092 995 y Fc(t)1112 985 y Fi(\()p Fm(z)1149 968 y Fg(2)1149 995 y Fc(t)1168 985 y Fi(\))s(~)g Ff(q)1204 968 y Fg(3)1203 995 y Fc(t)1222 985 y Fi(\()p Fm(z)1259 968 y Fg(3)1259 995 y Fc(t)1278 985 y Fi(\))225 1098 y(where)16 b(~)-22 b Ff(a)367 1082 y Fc(i)367 1108 y(t)382 1098 y Fi(\()p Fm(z)419 1082 y Fc(i)419 1108 y(t)434 1098 y Fk(j)p Fm(z)467 1082 y Fc(i)467 1108 y(t)p Fl(\000)p Fg(1)523 1098 y Fi(\))14 b(and)j(~)-24 b Ff(q)654 1082 y Fc(i)653 1108 y(t)668 1098 y Fi(\()p Fm(z)705 1082 y Fc(i)705 1108 y(t)720 1098 y Fi(\))14 b(are)g(p)q(oten)o(tials)g(that)g(pro)o(vide)f (the)i(v)n(ariational)c(parameter-)225 1143 y(ization.)225 1245 y Fm(3.3)48 b(A)16 b(forest)e(of)h(decision)f(trees)225 1330 y Fi(Alternativ)o(ely)19 b(w)o(e)h(can)g(drop)g(the)g(horizon)o (tal)f(couplings)h(in)f(the)h(HMDT)g(and)f(obtain)g(a)225 1375 y(v)n(ariational)c(appro)o(ximation)f(in)i(whic)o(h)h(the)h 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Fg(1)1456 1541 y(1)1475 1531 y Ff(;)7 b Fm(z)1515 1513 y Fg(2)1515 1541 y(1)1533 1531 y Fi(\))225 1638 y(Note)15 b(that)g(a)g(decision)g(tree)h (is)f(a)f(fully)g(coupled)h(graphical)f(mo)q(del;)f(th)o(us)j(w)o(e)f (can)g(view)f(the)225 1683 y(partially)c(factorized)i(appro)o(ximation) d(in)j(this)f(case)i(as)f(a)f(completely)g(factorized)h(mean)f(\014eld) 225 1729 y(appro)o(ximation)g(on)k(\\sup)q(er-no)q(des")h(whose)f (con\014gurations)f(include)h(all)e(p)q(ossible)i(con\014gu-)225 1775 y(rations)f(of)f(the)h(decision)h(tree.)225 1876 y Fm(3.4)48 b(A)16 b(Viterbi-l)o(i)o(k)o(e)d(appro)o(ximation)225 1961 y Fi(In)k(hidden)g(Mark)o(o)o(v)f(mo)q(deling)e(it)j(is)g(often)f (found)h(that)g(a)f(particular)h(sequence)i(of)d(states)225 2007 y(has)k(signi\014can)o(tly)f(higher)g(probabilit)o(y)g(than)g(an)o (y)h(other)g(sequence.)38 b(In)20 b(suc)o(h)g(cases)h(the)225 2052 y(Viterbi)14 b(algorithm,)d(whic)o(h)j(calculates)h(only)e(the)h (most)f(probable)h(path,)g(pro)o(vides)g(a)g(useful)225 2098 y(computational)d(alternativ)o(e.)225 2169 y(W)m(e)e(can)g(dev)o (elop)h(a)f(Viterbi-lik)o(e)f(algorithm)f(b)o(y)i(utilizing)f(an)h (appro)o(ximation)d Ff(Q)j Fi(that)h(assigns)225 2214 y(probabilit)o(y)i(one)i(to)g(a)g(single)f(path)h Fk(f)832 2213 y Fi(\026)832 2214 y Fm(z)853 2199 y Fg(1)853 2224 y Fc(t)872 2214 y Ff(;)890 2213 y Fi(\026)891 2214 y Fm(z)912 2199 y Fg(2)912 2224 y Fc(t)930 2214 y Ff(;)949 2213 y Fi(\026)949 2214 y Fm(z)970 2199 y Fg(3)970 2224 y Fc(t)988 2214 y Fk(g)p Fi(:)460 2310 y Ff(Q)p Fi(\()p Fk(f)p Fm(z)551 2293 y Fg(1)551 2321 y Fc(t)570 2310 y Ff(;)7 b Fm(z)610 2293 y Fg(2)610 2321 y Fc(t)628 2310 y Ff(;)g Fm(z)668 2293 y Fg(3)668 2321 y Fc(t)686 2310 y Fk(g)12 b(j)f(f)p Fm(y)788 2316 y Fc(t)802 2310 y Fk(g)p Ff(;)c Fk(f)p Fm(x)888 2316 y Fc(t)902 2310 y Fk(g)p Fi(\))41 b(=)1054 2252 y Fb(\032)1106 2288 y Fi(1)g(if)27 b Fm(z)1241 2272 y Fc(i)1241 2298 y(t)1268 2288 y Fi(=)1312 2287 y(\026)1311 2288 y Fm(z)1332 2272 y Fc(i)1332 2298 y(t)1347 2288 y Ff(;)34 b Fk(8)p Ff(t;)7 b(i)1106 2333 y Fi(0)41 b(otherwise)1672 2310 y(\(1\))225 2429 y(Note)15 b(that)f(the)g(en)o(trop)o(y)h Ff(Q)7 b Fi(ln)f Ff(Q)14 b Fi(is)g(zero,)h(moreo)o(v)o(er)e(the)h(ev)n(aluation)f(of)h(the)g (energy)h Ff(Q)7 b Fi(ln)g Ff(P)225 2475 y Fi(reduces)18 b(to)e(substituting)661 2474 y(\026)661 2475 y Fm(z)682 2460 y Fc(i)682 2485 y(t)713 2475 y Fi(for)f Fm(z)799 2460 y Fc(i)799 2485 y(t)830 2475 y Fi(in)h Ff(P)6 b Fi(.)23 b(Th)o(us)16 b(the)h(v)n(ariational)d(is)h(particularly)g (simple)p eop %%Page: 6 6 6 5 bop 375 0 a 18945146 8525330 3289088 18089984 35719495 35193241 startTexFig 375 0 a %%BeginDocument: ps/toyfig.ps /MathWorks 150 dict begin /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef /portraitMode 0 def /landscapeMode 1 def /dpi2point 0 def /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath fill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 387 917 5916 3121 MR c np 76 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef /Helvetica /ISOLatin1Encoding 144 FMSR 1 j c1 0 0 6912 5184 PR 6 w DO 4 w SO 6 w c0 898 3630 mt 3157 3630 L 898 1370 mt 3157 1370 L 3157 3630 mt 3157 1370 L 898 3630 mt 898 1370 L 3157 1370 mt 3157 1370 L 898 1370 mt 898 1370 L 898 3630 mt 3157 3630 L 898 3630 mt 898 1370 L 898 3630 mt 898 3630 L 898 3630 mt 898 3607 L 898 1370 mt 898 1393 L 858 3799 mt (0) s 1463 3630 mt 1463 3607 L 1463 1370 mt 1463 1393 L 1383 3799 mt (50) s 2028 3630 mt 2028 3607 L 2028 1370 mt 2028 1393 L 1908 3799 mt (100) s 2592 3630 mt 2592 3607 L 2592 1370 mt 2592 1393 L 2472 3799 mt (150) s 3157 3630 mt 3157 3607 L 3157 1370 mt 3157 1393 L 3037 3799 mt (200) s 898 3630 mt 921 3630 L 3157 3630 mt 3134 3630 L 619 3683 mt (-10) s 898 3065 mt 921 3065 L 3157 3065 mt 3134 3065 L 699 3118 mt (-5) s 898 2500 mt 921 2500 L 3157 2500 mt 3134 2500 L 783 2553 mt (0) s 898 1935 mt 921 1935 L 3157 1935 mt 3134 1935 L 783 1988 mt (5) s 898 1370 mt 921 1370 L 3157 1370 mt 3134 1370 L 703 1423 mt (10) s 898 1370 mt 3157 1370 L 898 3630 mt 3157 3630 L 898 3630 mt 898 1370 L 3157 3630 mt 3157 1370 L 3157 3630 mt 3157 3630 L 898 3630 mt 898 3630 L gs 898 1370 2260 2261 MR c np 11 3 12 567 11 -31 11 0 11 -1118 12 43 11 -78 11 -518 12 546 11 -424 11 411 12 48 11 -572 11 -20 11 116 12 -28 11 36 11 -27 12 -113 11 19 11 -20 11 12 12 5 11 -35 11 509 12 -8 11 7 11 -490 12 485 11 -358 11 -111 11 104 12 451 11 -76 11 -10 12 25 11 69 11 -436 12 -104 11 99 11 -23 11 21 12 -128 11 49 11 526 12 -570 11 -12 11 1 11 26 12 -33 11 6 11 -6 12 105 11 1130 11 -72 12 -39 11 163 11 -149 11 -10 12 33 11 -14 11 1 12 11 11 17 11 -1158 11 559 12 -16 11 -433 11 444 12 -99 11 -446 11 -4 12 -15 11 5 11 -1 11 25 12 96 11 -2 11 -109 12 135 11 -151 11 40 11 14 12 523 11 -559 11 94 12 -108 11 139 11 -140 12 441 11 -14 11 158 11 -40 12 52 11 952 11 50 12 -436 11 537 11 34 2039 1623 100 MP stroke 11 -182 12 -1540 11 533 11 -547 12 25 11 105 11 -31 12 -126 11 184 11 -160 11 131 12 -150 11 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