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/hyphen/registered/macron/degree/plusminus/twosuperior/threesuperior /acute/mu/paragraph/periodcentered/cedilla/onesuperior/ordmasculine /guillemotright/onequarter/onehalf/threequarters/questiondown /Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla /Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex /Idieresis/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis /multiply/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute /Thorn/germandbls/agrave/aacute/acircumflex/atilde/adieresis /aring/ae/ccedilla/egrave/eacute/ecircumflex/edieresis/igrave /iacute/icircumflex/idieresis/eth/ntilde/ograve/oacute/ocircumflex /otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex/udieresis /yacute/thorn/ydieresis ] def /Helvetica reencodeISO def /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { dup -1 eq { /patternGrayLevel idef /patternString idef } { /patternGrayLevel idef } ifelse false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l w add b translate w neg h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 1 printSize sub descender sub def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def /SSten { fgred fggreen fgblue setrgbcolor dup true exch 1 0 0 -1 0 6 -1 roll matrix astore } def /FSten { dup 3 -1 roll dup 4 1 roll exch newpath 0 0 moveto dup 0 exch lineto exch dup 3 1 roll exch lineto 0 lineto closepath bgred bggreen bgblue setrgbcolor eofill SSten } def /Rast { exch dup 3 1 roll 1 0 0 -1 0 6 -1 roll matrix astore } def %I Idraw 10 Grid 8 8 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.796717 0 0 0.796717 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 -0 -0 1 163 157 ] concat %I 123 415 5 5 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 -0 -0 1 162 180 ] concat %I 123 415 5 5 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 -0 -0 1 160 225 ] concat %I 123 415 5 5 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 -0 -0 1 164 133 ] concat %I 123 415 5 5 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 -0 -0 1 240 175 ] concat %I 123 415 5 5 Elli End Begin %I Line %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 -0 -0 1 175 232 ] concat %I 116 317 117 317 Line %I 1 End Begin %I Line %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 -0 -0 1 175 232 ] concat %I 118 316 189 357 Line %I 1 End Begin %I Line %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 -0 -0 1 175 232 ] concat %I 114 340 188 359 Line %I 1 End Begin %I Line %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 -0 -0 1 175 232 ] concat %I 117 362 188 360 Line %I 1 End Begin %I Line %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 -0 -0 1 175 232 ] concat %I 112 408 191 359 Line %I 1 End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-12-*-*-*-*-*-*-* Helvetica 12 SetF %I t [ 1 0 0 1 381 600 ] concat %I [ (k) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-12-*-*-*-*-*-*-* Helvetica 12 SetF %I t [ 1 0 0 1 285 511 ] concat %I [ (t) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-12-*-*-*-*-*-*-* Helvetica 12 SetF %I t [ 1 0 0 1 362 515 ] concat %I [ (t+1) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-12-*-*-*-*-*-*-* Helvetica 12 SetF %I t [ 1 0 0 1 307 495 ] concat %I [ (TIME) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-12-*-*-*-*-*-*-* Helvetica 12 SetF %I t [ -4.37114e-08 1 -1 -4.37114e-08 218.5 571.5 ] concat %I [ (STATES) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-12-*-*-*-*-*-*-* Helvetica 12 SetF %I t [ 1 0 0 1 269 550 ] concat %I [ (1) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-12-*-*-*-*-*-*-* Helvetica 12 SetF %I t [ 1 0 0 1 268 576 ] concat %I [ (2) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-12-*-*-*-*-*-*-* Helvetica 12 SetF %I t [ 1 0 0 1 270 600 ] concat %I [ (3) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-12-*-*-*-*-*-*-* Helvetica 12 SetF %I t [ 1 0 0 1 264 643 ] concat %I [ (N) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-12-*-*-*-*-*-*-* Helvetica 12 SetF %I t [ 1 0 0 1 324 637 ] concat %I [ (aNk) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-12-*-*-*-*-*-*-* Helvetica 12 SetF %I t [ 1 0 0 1 321 561 ] concat %I [ (a1K) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-12-*-*-*-*-*-*-* Helvetica 12 SetF %I t [ 1 0 0 1 308 607 ] concat %I [ (a3k) ] Text End End %I eop showpage end 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140 a Fk(3.)25 b(Consider)h(the)h(observ)-5 b(ation)27 b(sequence)541 308 y Fi(~)523 327 y Fh(O)21 b Fi(=)f(\()p Fh(H)r(;)12 b(T)6 b(;)12 b(T)6 b(;)12 b(H)r(;)g(T)6 b(;)12 b(H)r(;)g(H)r(;)g(T)6 b(;)12 b(T)6 b(;)12 b(H)5 b Fi(\))p Fh(:)182 513 y Fk(Ho)n(w)17 b(w)n(ould)h(y)n(our)f(answ)n (ers)g(to)h(parts)g(1)h(and)f(2)h(c)n(hange?)93 692 y(4.)25 b(If)j(the)f(state)g(transition)g(probabilities)g(w)n(ere:)687 983 y Fh(A)741 954 y Fe(0)776 983 y Fi(=)848 768 y Fd(2)848 841 y(6)848 866 y(6)848 891 y(6)848 915 y(6)848 940 y(6)848 965 y(6)848 990 y(6)848 1015 y(6)848 1040 y(6)848 1065 y(6)848 1091 y(4)913 856 y Fi(0)p Fh(:)p Fi(9)57 b(0)p Fh(:)p Fi(45)41 b(0)p Fh(:)p Fi(45)896 993 y(0)p Fh(:)p Fi(05)58 b(0)p Fh(:)p Fi(1)f(0)p Fh(:)p Fi(45)896 1130 y(0)p Fh(:)p Fi(05)41 b(0)p Fh(:)p Fi(45)58 b(0)p Fh(:)p Fi(1)1356 768 y Fd(3)1356 841 y(7)1356 866 y(7)1356 891 y(7)1356 915 y(7)1356 940 y(7)1356 965 y(7)1356 990 y(7)1356 1015 y(7)1356 1040 y(7)1356 1065 y(7)1356 1091 y(5)1395 983 y Fh(;)182 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{1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -24.0 507.0 translate 1 -1 scale .9 .9 scale % to make patterns same scale as in xfig % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def 1.1111 1.1111 scale %restore scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def 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F2psBegin10setmiterlimitn0792m00l6120l612792lcpclip0.060000.06000sc7.500slwF2psEnd rs end %%EndDocument endTexFig eop %%Page: 21 21 21 20 bop 1374 -123 a Fq(HMM)13 b(T)m(utorial/)1629 -124 y Fp(c)1621 -123 y Fo(\015)d Fq(T)m(apas)h(Kan)o(ungo{21)-36 -88 y Fn(')p -36 2675 4 2683 v -36 2760 a(&)2033 -88 y($)p 2033 2675 V 2033 2760 a(\045)p 49 2760 1903 4 v 49 -88 V 448 162 a Fl(Bac)m(kw)m(ard)38 b(Algorithm)p 60 299 1860 6 v 122 539 a Fj(\017)24 b Fk(De\014ne)j(bac)n(kw)n(ard)e (v)-5 b(ariable)28 b Fh(\014)1160 550 y Ff(t)1179 539 y Fi(\()p Fh(i)p Fi(\))g Fk(as:)505 726 y Fh(\014)546 737 y Ff(t)565 726 y Fi(\()p Fh(i)p Fi(\))20 b(=)g Fh(P)10 b Fi(\()p Fh(o)848 737 y Ff(t)p Fg(+1)930 726 y Fh(;)i(o)997 737 y Ff(t)p Fg(+2)1079 726 y Fh(;)g(:)g(:)g(:)f(;)h(o)1273 737 y Ff(T)1310 726 y Fj(j)p Fh(q)1362 737 y Ff(t)1402 726 y Fi(=)19 b Fh(i;)12 b(\025)p Fi(\))122 934 y Fj(\017)24 b Fh(\014)223 945 y Ff(t)242 934 y Fi(\()p Fh(i)p Fi(\))39 b Fk(is)g(the)f(probabilit)n(y)g(of)h(observing)f(the)g(partial)182 1071 y(sequence)24 b Fi(\()p Fh(o)586 1082 y Ff(t)p Fg(+1)669 1071 y Fh(;)12 b(o)736 1082 y Ff(t)p Fg(+2)829 1071 y Fh(:)g(:)g(:)g(;)g(o)992 1082 y Ff(T)1029 1071 y Fi(\))27 b Fk(suc)n(h)e(that)i(the)f(state)g Fh(q)1818 1082 y Ff(t)1865 1071 y Fk(is)182 1208 y Fh(i)p Fk(.)122 1386 y Fj(\017)e Fk(Induction:)201 1565 y(1.)g(Initialization:)37 b Fh(\014)850 1576 y Ff(T)887 1565 y Fi(\()p Fh(i)p Fi(\))20 b(=)g(1)201 1723 y Fk(2.)k(Induction:)646 1909 y Fh(\014)687 1920 y Ff(t)706 1909 y Fi(\()p Fh(i)p Fi(\))42 b(=)937 1850 y Ff(N)929 1862 y Fd(X)916 1964 y Ff(j)s Fg(=1)1014 1909 y Fh(a)1052 1920 y Ff(ij)1093 1909 y Fh(b)1124 1920 y Ff(j)1149 1909 y Fi(\()p Fh(o)1209 1920 y Ff(t)p Fg(+1)1291 1909 y Fi(\))p Fh(\014)1357 1920 y Ff(t)p Fg(+1)1439 1909 y Fi(\()p Fh(j)t Fi(\))p Fh(;)1081 2059 y Fi(1)20 b Fj(\024)g Fh(i)f Fj(\024)h Fh(N)t(;)1081 2208 y(t)g Fi(=)g Fh(T)25 b Fj(\000)16 b Fi(1)p Fh(;)c(:)g(:)g(:)g(;)g Fi(1)p eop %%Page: 22 22 22 21 bop 1374 -123 a Fq(HMM)13 b(T)m(utorial/)1629 -124 y Fp(c)1621 -123 y Fo(\015)d Fq(T)m(apas)h(Kan)o(ungo{22)-36 -88 y Fn(')p -36 2675 4 2683 v -36 2760 a(&)2033 -88 y($)p 2033 2675 V 2033 2760 a(\045)p 49 2760 1903 4 v 49 -88 V 413 162 a Fl(Solution)42 b(to)e(Problem)i(2)p 60 299 1860 6 v 122 539 a Fj(\017)24 b Fk(Cho)r(ose)j(the)g(most)g(lik) n(ely)f(path)122 718 y Fj(\017)e Fk(Find)38 b(the)g(path)g Fi(\()p Fh(q)800 729 y Fg(1)827 718 y Fh(;)12 b(q)891 729 y Ff(t)911 718 y Fh(;)g(:)g(:)g(:)f(;)h(q)1102 729 y Ff(T)1140 718 y Fi(\))38 b Fk(that)h(maximizes)d(the)182 855 y(lik)n(eliho)r(o)r(d:)734 992 y Fh(P)10 b Fi(\()p Fh(q)847 1003 y Fg(1)873 992 y Fh(;)i(q)937 1003 y Fg(2)964 992 y Fh(;)g(:)g(:)g(:)f(;)h(q)1155 1003 y Ff(T)1193 992 y Fj(j)p Fh(O)r(;)g(\025)p Fi(\))122 1177 y Fj(\017)24 b Fk(Solution)j(b)n(y)g(Dynamic)f(Programming)122 1355 y Fj(\017)e Fk(De\014ne:)282 1542 y Fh(\016)314 1553 y Ff(t)334 1542 y Fi(\()p Fh(i)p Fi(\))c(=)89 b(max)501 1570 y Ff(q)523 1580 y Fc(1)545 1570 y Ff(;q)581 1580 y Fc(2)603 1570 y Ff(;:::)o(;q)694 1580 y Fb(t)p Fa(\000)p Fc(1)775 1542 y Fh(P)10 b Fi(\()p Fh(q)888 1553 y Fg(1)915 1542 y Fh(;)i(q)979 1553 y Fg(2)1005 1542 y Fh(;)g(:)g(:)g(:)g(;)g(q) 1197 1553 y Ff(t)1236 1542 y Fi(=)20 b Fh(i;)12 b(o)1400 1553 y Fg(1)1425 1542 y Fh(;)g(o)1492 1553 y Fg(2)1519 1542 y Fh(;)g(:)g(:)g(:)f(;)h(o)1713 1553 y Ff(t)1732 1542 y Fj(j)p Fh(\025)p Fi(\))122 1750 y Fj(\017)24 b Fh(\016)214 1760 y Ff(t)234 1750 y Fi(\()p Fh(i)p Fi(\))k Fk(is)e(the)i(highest)e(prob.)37 b(path)27 b(ending)g(in)g(state)g Fh(i)122 1928 y Fj(\017)d Fk(By)j(induction)f(w)n(e)h(ha)n(v)n(e:)586 2115 y Fh(\016)618 2126 y Ff(t)p Fg(+1)700 2115 y Fi(\()p Fh(j)t Fi(\))20 b(=)f(max)929 2154 y Ff(i)999 2115 y Fi([)p Fh(\016)1049 2126 y Ff(t)1068 2115 y Fi(\()p Fh(i)p Fi(\))p Fh(a)1181 2126 y Ff(ij)1223 2115 y Fi(])d Fj(\001)g Fh(b)1324 2126 y Ff(j)1349 2115 y Fi(\()p Fh(o)1409 2126 y Ff(t)p Fg(+1)1491 2115 y Fi(\))p eop %%Page: 23 23 23 22 bop 1374 -123 a Fq(HMM)13 b(T)m(utorial/)1629 -124 y Fp(c)1621 -123 y Fo(\015)d Fq(T)m(apas)h(Kan)o(ungo{23)-36 -88 y Fn(')p -36 2675 4 2683 v -36 2760 a(&)2033 -88 y($)p 2033 2675 V 2033 2760 a(\045)p 49 2760 1903 4 v 49 -88 V 517 162 a Fl(Viterbi)42 b(Algorithm)p 60 299 1860 6 v 168 644 a 25954845 18945146 0 0 38811238 28220375 startTexFig 168 644 a %%BeginDocument: forward.eps %Magnification: 1.00 /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def F2psDictbeginF2psDict begin F2psDictbeginF2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -24.0 507.0 translate 1 -1 scale .9 .9 scale % to make patterns same scale as in xfig % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def 1.1111 1.1111 scale %restore scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def F2psBegin10setmiterlimitn0792m00l6120l612792lcpclip0.060000.06000sc7.500slwF2psBegin 10 setmiterlimit n 0 792 m 0 0 l 612 0 l 612 792 l cp clip 0.06000 0.06000 sc 7.500 slw % Polyline n 1500 5700 m 9900 5700 l gs col-1 s gr % Polyline n 1500 5100 m 9900 5100 l gs col-1 s gr % Polyline n 1500 4500 m 9900 4500 l gs col-1 s gr % Polyline n 1500 3900 m 9900 3900 l gs col-1 s gr % Polyline n 1500 3300 m 9900 3300 l gs col-1 s gr % Polyline n 1500 2700 m 9900 2700 l gs col-1 s gr % Polyline n 1500 2100 m 9900 2100 l gs col-1 s gr % Polyline n 1500 1500 m 9900 1500 l gs col-1 s gr % Polyline n 1500 1500 m 1500 5700 l gs col-1 s gr % Polyline n 2700 1500 m 2700 5700 l gs col-1 s gr % Polyline n 3900 1500 m 3900 5700 l gs col-1 s gr % Polyline n 5100 1500 m 5100 5700 l gs col-1 s gr % Polyline n 6300 1500 m 6300 5700 l gs col-1 s gr % Polyline n 7500 1500 m 7500 5700 l gs col-1 s gr % Polyline n 8700 1500 m 8700 5700 l gs col-1 s gr % Polyline n 9900 1500 m 9900 5700 l gs col-1 s gr 30.000 slw % Polyline n 5100 1500 m 6300 2700 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 340.00 100.00] PATmp PATsp ef gr PATusp gs col-1 s gr % Polyline n 5100 2100 m 6300 2700 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 340.00 140.00] PATmp PATsp ef gr PATusp gs col-1 s gr % Polyline n 5100 2700 m 6300 2700 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 340.00 180.00] PATmp PATsp ef gr PATusp gs col-1 s gr % Polyline n 5100 3300 m 6300 2700 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 340.00 180.00] PATmp PATsp ef gr PATusp gs col-1 s gr % Polyline n 5100 3900 m 6300 2700 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 340.00 180.00] PATmp PATsp ef gr PATusp gs col-1 s gr % Polyline n 5100 4500 m 6300 2700 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 340.00 180.00] PATmp PATsp ef gr PATusp gs col-1 s gr % Polyline n 5100 5100 m 6300 2700 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 340.00 180.00] PATmp PATsp ef gr PATusp gs col-1 s gr % Polyline n 5100 5700 m 6300 2700 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 340.00 180.00] PATmp PATsp ef gr PATusp gs col-1 s gr /Times-Roman ff 360.00 scf sf 1500 6225 m gs 1 -1 sc (1) col-1 sh gr /Times-Roman ff 360.00 scf sf 2700 6225 m gs 1 -1 sc (2) col-1 sh gr /Times-Roman ff 360.00 scf sf 6300 6225 m gs 1 -1 sc (t+1) col-1 sh gr /Times-Roman ff 360.00 scf sf 5100 6225 m gs 1 -1 sc (t) col-1 sh gr /Times-Roman ff 360.00 scf sf 600 4200 m gs 1 -1 sc 90.0 rot (STATES) col-1 sh gr /Times-Roman ff 360.00 scf sf 900 5700 m gs 1 -1 sc (1) col-1 sh gr /Times-Roman ff 360.00 scf sf 900 5100 m gs 1 -1 sc (2) col-1 sh gr /Times-Roman ff 360.00 scf sf 900 4500 m gs 1 -1 sc (3) col-1 sh gr /Times-Roman ff 360.00 scf sf 900 3900 m gs 1 -1 sc (4) col-1 sh gr /Times-Roman ff 360.00 scf sf 900 1500 m gs 1 -1 sc (N) col-1 sh gr /Times-Roman ff 360.00 scf sf 3900 6225 m gs 1 -1 sc (t-1) col-1 sh gr /Times-Roman ff 360.00 scf sf 9825 6225 m gs 1 -1 sc (T) col-1 sh gr /Times-Roman ff 360.00 scf sf 8550 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y Fh(O)r(:)201 1664 y Fk(3.)24 b(If)k Fi(log)13 b Fh(P)d Fi(\()p Fh(O)r Fj(j)p Fh(\025)p Fi(\))16 b Fj(\000)g Fi(log)c Fh(P)e Fi(\()p Fh(O)r Fj(j)p Fh(\025)1079 1675 y Fg(0)1106 1664 y Fi(\))20 b Fh(<)g(D)r(E)t(LT)10 b(A)27 b Fk(stop.)201 1822 y(4.)d(Else)j(set)g Fh(\025)636 1833 y Fg(0)682 1822 y Fj( )20 b Fh(\025)27 b Fk(and)h(goto)g(step)f (2.)p eop %%Page: 26 26 26 25 bop 1374 -123 a Fq(HMM)13 b(T)m(utorial/)1629 -124 y Fp(c)1621 -123 y Fo(\015)d Fq(T)m(apas)h(Kan)o(ungo{26)-36 -88 y Fn(')p -36 2675 4 2683 v -36 2760 a(&)2033 -88 y($)p 2033 2675 V 2033 2760 a(\045)p 49 2760 1903 4 v 49 -88 V 271 162 a Fl(Baum-W)-10 b(elc)m(h:)53 b(Prelimi)q(nari)q(es)p 60 299 1860 6 v 122 539 a Fj(\017)24 b Fk(De\014ne)30 b Fh(\030)s Fi(\()p Fh(i;)12 b(j)t Fi(\))31 b Fk(as)f(the)g(probabilit) n(y)f(of)i(b)r(eing)f(in)g(state)182 676 y Fh(i)d Fk(at)h(time)f Fh(t)g Fk(and)h(in)f(state)g Fh(j)k Fk(at)d(time)f Fh(t)16 b Fi(+)f(1)p Fh(:)425 863 y(\030)s Fi(\()p Fh(i;)d(j)t Fi(\))42 b(=)741 815 y 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26 bop 1374 -123 a Fq(HMM)13 b(T)m(utorial/)1629 -124 y Fp(c)1621 -123 y Fo(\015)d Fq(T)m(apas)h(Kan)o(ungo{27)-36 -88 y Fn(')p -36 2675 4 2683 v -36 2760 a(&)2033 -88 y($)p 2033 2675 V 2033 2760 a(\045)p 49 2760 1903 4 v 49 -88 V 259 162 a Fl(Baum-W)-10 b(elc)m(h:)53 b(Up)s(date)41 b(Rules)p 60 299 1860 6 v 122 539 a Fj(\017)29 b Fi(\026)-38 b Fh(\031)223 550 y Ff(i)268 539 y Fk(=)27 b(exp)r(ected)f(frequency)f (in)h(state)h Fh(i)f Fk(at)i(time)e Fi(\()p Fh(t)20 b Fi(=)g(1\))182 676 y Fk(=)28 b Fh(\015)311 687 y Fg(1)337 676 y Fi(\()p Fh(i)p Fi(\))p Fh(:)122 855 y Fj(\017)e Fi(\026)-35 b Fh(a)220 866 y Ff(ij)281 855 y Fi(=)25 b Fk(\(exp)r(ected)f(n)n(um)n(b)r(er)g(of)h(transition)g(from)g(state)f Fh(i)182 992 y Fk(to)c(state)f Fh(j)t Fk(\)/)h(\(exp)r(ected)e(n)n (ubmer)h(of)h(transitions)f(from)182 1129 y(state)27 b Fh(i)p Fk(\):)839 1266 y Fi(\026)-36 b Fh(a)874 1277 y Ff(ij)936 1266 y Fi(=)1013 1179 y Fd(P)1068 1217 y Fh(\030)1099 1228 y Ff(t)1119 1217 y Fi(\()p Fh(i;)12 b(j)t 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