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1 /3/0 COMP4 Lecure 8 Hidden Mrkov Models Hidden Mrkov Models Imgine he s nd s re hidden, so he d roduced is sequence of nd D generion is esy, D decoding is miguous,? Ses emi feures or, u heir originis no known Mrkov Chins: reminder A Mrkov chin is generivemodel of sequences Hidden Mrkov Model (HMM An HMM is model of sequence of feures or feure vecors x x x 3... x - x x s s s s 3... s - s... s where s is n ojec from he se sce S I hs he roery h (s s -, s -,..., s (s s - he roiliy of sequence is ( s ( s ( s + s 3 genered ccording o emission roiliies (x s he underlying se sequence is from Mrkov chin model s s s s 3 Ls sls ( s ( s ( s + s u he se sequence is hiddenfrom us 4
2 /3/0 HMM Exmles Feures re x {, } nd ses re s {,,, } Emission roiliies ( x s ( x s 0 x s x s ec ( 0 ( rnsiion roiliies s s 0. ( ( s s ec 5 his gives similr sequences HMM Exmles Emission roiliies ( x s ( x s 0 ( x s 0. 5 ( x s 0 ( x s ( x s 0. 5 Se cn emi feure or wih equl roiliy 6 So he difference.0 HMM for seech yes no 0.0 Emission roiliies: (x s SIL, (x s yes, (x s no x is he MFCC feure vecor for segmen of he seech signl We cn fi norml densiies o he feure disriuion for ech se (slighly more flexile disriuions re used in rcice his model ws used o cro he seech h you use in L 7 8
3 /3/0 Join roiliy of ses nd feures Emission roiliies: (x s (x s 0 (x s (x s (x s 0 (x s (x s -- x x x x x x 9 Join roiliy of ses nd feures Esy mulily he emission nd rnsiion roiliies ( x,, s, s, s ( s ( x s ( s s ( x s L ( s s ( x s ( s ( s ( s+ s ( x s Bu: he se h s, s,, s is unknown i s hidden 0 HMM inference We don know he hidden ses, so he join roiliy of ses nd feures isn useful hing o comue We will consider wo more useful sks: Clssificion: Modelling differen clsses of d, e.g. yes nd no : Finding he mos likely ses given feure vecor Comuing hese is hrder nd requires he use of clever lgorihms Clssificion Build model for ech clss of d, e.g. C yes, C no Comue (x, x,, x C i for ech clss C i Aly Byes rule ( ( x, C ( C C x, ( x, Ci ( Ci i Aly clssificion rule, e.g. selec he mos likely clss yes 0.0 3
4 /3/0 Clssificion Build model for ech clss of d, e.g. C yes, C no Comue (x, x,, x C i for ech clss C i Aly Byes rule ( ( x, C ( C C x, ( x, Ci ( Ci i Aly clssificion rule, e.g. selec he mos likely clss yes 0.0 Clssificion Need wy o comue (x, x,, x for ech model Requires sum over ll ossile hs hrough he model ( x, L ( x,, s, s, s s S s S s S Wors cse: S erms in his sum 3 4 Clssificion Need n efficienwy o comue (x, x,, x for ech model Use similr recursion relion s for Mrkov chin cse ( x, s ( x s ( s ( x, x, s ( x s ( s s ( x, x, s for ( x, ( x,, s ( s s S s S Quesion in Exmles shee 8 is similr ide: his is clled he Forwrd Algorihm Clssificion cn del wih limied numer of models Less useful for hrses or senences An lernive roch is o decodehe d: s, s, s ( s x, x, x * s rg mx : find he mos likely h hrough he hidden ses 5 6 4
5 /3/0 D decoding is miguous:? Mos likely h is Less likely h is s, s, s ( s x, x, x * s rg mx Requires serchfor h mximising he quniy on he righ here cn e s mny s S ossile hs exhusive serch isn ossile he VieriAlgorihm uses recursion o find he oiml h efficienly his is n exmle of n oimisionrolem. 7 8 rining Is esy if d is lelled, i.e. he se h of he rining d is known Lelling is ime consuming, difficul nd error-rone he Bum-Welch lgorihm llows rining wih unlelled d Hels if we know somehing ou he d, e.g. reding from scri. his mssively reduces he serch sce. Seech Recogniion Now we hve mos of he ingrediens for seech recogniion 9 0 5
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