Hidden Markov Models

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Rosalind Briggs
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1 Hdden Mkov Model Ronld J. Wllm CSG220 Spng 2007 Conn evel lde dped fom n Andew Mooe uol on h opc nd few fgue fom Ruell & ovg AIMA e nd Alpydn Inoducon o Mchne Lenng e. A Smple Mkov Chn /2 /3 /3 /3 3 2 /3 0 /2 umbe node epeen pobbly of ng he coepondng e. umbe on c epeen non pobble. /3 A ech me ep,, 2,... new e eleced ndomly ccodng o he dbuon he cuen e. Le be ndom vble fo he e me ep. Le x epeen he cul vlue of he e me. In h exmple, x cn be, 2, o 3. Hdden Mkov Model Slde 2

2 Mkov opey Fo ny, condonlly ndependen of { -, -2, } gven. In ohe wod ( (,ny ele hoy Queon Wh would be he be Bye e ucue o epeen he Jon Dbuon of (, 2,, -,,...? Hdden Mkov Model Slde 3 Mkov opey Fo ny, condonlly ndependen of { -, -2, } gven. In ohe wod ( (,ny ele hoy Queon Wh would be he be Bye e ucue o epeen he Jon Dbuon of (, 2,, -,,...? Anwe Hdden Mkov Model Slde 4 2

3 2 3 Mkov chn Bye ne ( π π 2 π 3 π π oon ( π ( Sme C evey node excep Hdden Mkov Model Slde 5 Mkov Chn Foml Defnon A Mkov chn 3-uple conng of e of poble e {, 2,..., } {π, π 2,.. π } he ng e pobble π ( he e non pobble ( Hdden Mkov Model Slde 6 3

4 Compung uff n Mkov chn Some noon nd umpon Aume me un fom o Recll h he.v. epeenng he e me nd x denoe he cul vlue Ue 2 nd x 2 hohnd fo (,,..., 2 nd (x, x,... x 2, epecvely Ue noon lke (x hohnd fo ( x Hdden Mkov Model Slde 7 Wh (? emp Sep Wok ou how o compue (x fo ny e equence x x x x x ( ( ( ( x x ( x x ( x M Sep 2 Ue h knowledge o ge ( ( ( x equence fo whch x WHY? ( x x ( x x 2 L( x2 x ( x ( x x ( x x L( x x ( x Compuon exponenl n Hdden Mkov Model Slde 8 4

5 5 Hdden Mkov Model Slde 9 Se equence ph ell Exponenlly mny ph, bu ech me ep only goe hough excly one of he e Hdden Mkov Model Slde 0 Wh (? Cleve ppoch Fo ech e, defne Expe nducvely ( ( p π ( ( p ( ( ( p p ( (

6 Wh (? Cleve ppoch Fo ech e, defne ( ( p Expe nducvely ( ( p p ( ( ( p ( π ( e ndex 2 me ep 2... Compuon mple. Ju fll n h ble one column me, fom lef o gh Cell n h ble coepond o node n he ell Hdden Mkov Model Slde Wh (? Cleve ppoch Fo ech e, defne ( ( p Expe nducvely ( ( p p ( ( ( p ( π ( Co of compung p ( fo ll e now O( 2 he f wy w O( h w mple exmple I w men o wm you up o h ck, clled Dynmc ogmmng, becue HMM compuon nvolve mny ck u lke h. Hdden Mkov Model Slde 2 6

7 Inducve ep gphcl epeenon ( p ( p ( p Compe h wh ml depcon of upde we ll ue n HMM Hdden Mkov Model Slde 3 Hdden Se Gven Mkov model of poce, compuon of vou qune of nee (e.g., pobble ghfowd f he e obevble ue echnque lke he one u decbed. Moe elc ume he ue e no obevble only hve obevon h depend on, bu do no fully deemne, he cul e. Exmple Robo loclon e cul locon obevon (noy eno edng Speech ecognon e equence > wod obevon couc gnl In h uon, we y he e hdden Model h ung Hdden Mkov Model (HMM Hdden Mkov Model Slde 4 7

8 HMM An HMM u Mkov chn ugmened wh e of M poble obevon {o, o 2,..., o M } fo ech e, 2,..., dbuon ove poble obevon h mgh be ened n h e We ll le Z be he.v. fo he obevon h occu me (wh epeenng he cul obevon In ddon, we ll ume h he obevon me depend only on he e me, n he ene bou o be decbed Hdden Mkov Model Slde 5 Mkov opey of Obevon Fo ny, Z condonlly ndependen of { -, -2,, Z -, Z -2,..., Z } gven. In ohe wod (Z o (Z o,ny ele hoy Queon Wh would be he be Bye e ucue o epeen he Jon Dbuon of (, Z, 2, Z 2,, -,Z -,, Z,...? Hdden Mkov Model Slde 6 8

9 Mkov opey of Obevon Fo ny, Z condonlly ndependen of { -, -2,, Z -, Z -2,..., Z } gven. In ohe wod (Z o (Z o,ny ele hoy Queon Wh would be he be Bye e ucue o epeen he Jon Dbuon of (, Z, 2, Z 2,, -,Z -,, Z,...? Anwe Z Z 2 Z - Z Hdden Mkov Model Slde 7 HMM Bye e Z Z 2 Z - Z b (o 2 b (o 2 obevon ndex k b (o k M b (o M h he C fo evey Z node e ndex 2 3 b 2 (o b 3 (o b 2 (o 2 b 3 (o 2 b 2 (o k b 3 (o k b 2 (o M b 3 (o M oon b o ( Z o ( k k b (o b (o 2 b (o k b (o M b (o b (o 2 b (o k b (o M Hdden Mkov Model Slde 8 9

10 Ae HMM Ueful? You be!! Robo plnnng & enng unde unceny (e.g. Red Smmon / Sebn hun / Sven Koeng Robo lenng conol (e.g. Yngheng u wok Speech Recognon/Undendng hone Wod, Sgnl phone Humn Genome oec Complced uff you lecue know nohng bou. Conume decon modelng Economc & Fnnce. lu le 5 ohe hng I hven hough of. Hdden Mkov Model Slde 9 Dynmc Bye e An HMM cully pecl ce of moe genel concep Dynmc Bye e (DB Cn decompoe no mulple e vble nd mulple obevon vble ech me lce, wh only dec nfluence epeened explcly ( ode Mkov popey node n ny me lce hve c only fom node n he own o he mmedely pecedng me lce Hghe-ode Mkov model lo ely epeened n h fmewok Hdden Mkov Model Slde 20 0

11 DB Exmple Lne dynmcl yem wh poon eno E.g., ge ckng Hdden Mkov Model Slde 2 Anohe DB Exmple Modelng obo wh poon eno nd bey chge mee Hdden Mkov Model Slde 22

12 Bck o HMM... Summy of ou HMM noon e me (.v. Z obevon me (.v. V 2 (V, V,..., V 2 fo ny me-ndexed.v. V oble e {, 2,..., } oble obevon {o, o 2,..., o M } v cul vlue of.v. V me ep v 2 (v, v,..., v 2 equence of cul vlue of.v. V fom me ep hough 2 Convenen hohnd E.g., (x men ( x Z fnl me ep Hdden Mkov Model Slde 23 HMM Foml Defnon An HMM λ 5-uple conng of e of poble e {, 2,..., } e of M poble obevon {o, o 2,..., o M } {π, π 2,.. π } he ng e pobble π ( he e non pobble 2 ( b (o b (o 2 b (o M b 2 (o b 2 (o 2 b 2 (o M b (o b (o 2 b (o M he obevon pobble b (o k (Z o k Hdden Mkov Model Slde 24 2

13 Hee n HMM /2 uv /3 /3 uw /3 0 Se e {, 2, 3 } Obevon e {u, v, w} 3 /3 π /2 π 2 /2 π 3 0 /3 vw 2 /2 S ndomly n e o 2 Chooe one of he oupu ymbol n ech e ndom. 0 2 /3 3 2 / /3 32 /3 3 /3 b (u /2 b (v /2 b (w 0 b 2 (u 0 b 2 (v /2 b 2 (w /2 b 3 (u /2 b 3 (v 0 b 3 (w /2 Hdden Mkov Model Slde 25 Hee n HMM /2 uv /3 /3 uw /3 0 Se e {, 2, 3 } Obevon e {u, v, w} 3 /3 π /2 π 2 /2 π 3 0 /3 vw 2 /2 S ndomly n e o 2 Chooe one of he oupu ymbol n ech e ndom. Le genee equence of obevon choce beween nd /3 3 2 / /3 32 /3 3 /3 b (u /2 b (v /2 b (w 0 b 2 (u 0 b 2 (v /2 b 2 (w /2 b 3 (u /2 b 3 (v 0 b 3 (w /2 x x 2 x Hdden Mkov Model Slde 26 3

14 Hee n HMM S ndomly n e o 2 /2 uv /3 /3 uw /3 0 Se e {, 2, 3 } Obevon e {u, v, w} 3 /3 π /2 π 2 /2 π 3 0 /3 vw 2 /2 Chooe one of he oupu ymbol n ech e ndom. Le genee equence of obevon choce beween u nd v 0 2 /3 3 2 / /3 32 /3 3 /3 b (u /2 b (v /2 b (w 0 b 2 (u 0 b 2 (v /2 b 2 (w /2 b 3 (u /2 b 3 (v 0 b 3 (w /2 x x 2 x Hdden Mkov Model Slde 27 Hee n HMM S ndomly n e o 2 /2 uv /3 /3 uw /3 0 Se e {, 2, 3 } Obevon e {u, v, w} 3 /3 π /2 π 2 /2 π 3 0 /3 vw 2 /2 Chooe one of he oupu ymbol n ech e ndom. Le genee equence of obevon Goo 2 wh pobbly /3 o 3 wh pob. 0 2 /3 3 2 / /3 32 /3 3 /3 b (u /2 b (v /2 b (w 0 b 2 (u 0 b 2 (v /2 b 2 (w /2 b 3 (u /2 b 3 (v 0 b 3 (w /2 x x 2 x u Hdden Mkov Model Slde 28 4

15 Hee n HMM S ndomly n e o 2 /2 uv /3 /3 uw /3 0 Se e {, 2, 3 } Obevon e {u, v, w} 3 /3 π /2 π 2 /2 π 3 0 /3 vw 2 /2 Chooe one of he oupu ymbol n ech e ndom. Le genee equence of obevon choce beween u nd w 0 2 /3 3 2 / /3 32 /3 3 /3 b (u /2 b (v /2 b (w 0 b 2 (u 0 b 2 (v /2 b 2 (w /2 b 3 (u /2 b 3 (v 0 b 3 (w /2 x x 2 x u Hdden Mkov Model Slde 29 Hee n HMM S ndomly n e o 2 /2 uv /3 /3 uw /3 0 Se e {, 2, 3 } Obevon e {u, v, w} 3 /3 π /2 π 2 /2 π 3 0 /3 vw 2 /2 Chooe one of he oupu ymbol n ech e ndom. Le genee equence of obevon Ech of he hee nex e eqully lkely 0 2 /3 3 2 / /3 32 /3 3 /3 b (u /2 b (v /2 b (w 0 b 2 (u 0 b 2 (v /2 b 2 (w /2 b 3 (u /2 b 3 (v 0 b 3 (w /2 x x 2 x u u Hdden Mkov Model Slde 30 5

16 Hee n HMM /2 uv /3 /3 uw /3 0 /3 Se e {, 2, 3 } Obevon e {u, v, w} 3 /3 π /2 π 2 /2 π 3 0 vw 2 /2 S ndomly n e o 2 Chooe one of he oupu ymbol n ech e ndom. Le genee equence of obevon choce beween u nd w 0 2 /3 3 2 / /3 32 /3 3 /3 b (u /2 b (v /2 b (w 0 b 2 (u 0 b 2 (v /2 b 2 (w /2 b 3 (u /2 b 3 (v 0 b 3 (w /2 x x 2 x u u Hdden Mkov Model Slde 3 Hee n HMM S ndomly n e o 2 /2 uv /3 /3 uw /3 0 Se e {, 2, 3 } Obevon e {u, v, w} 3 /3 π /2 π 2 /2 π 3 0 /3 vw 2 /2 Chooe one of he oupu ymbol n ech e ndom. Le genee equence of obevon 0 2 /3 3 2 / /3 32 /3 3 /3 b (u /2 b (v /2 b (w 0 b 2 (u 0 b 2 (v /2 b 2 (w /2 b 3 (u /2 b 3 (v 0 b 3 (w /2 x x 2 x u u w Hdden Mkov Model Slde 32 6

17 Hdden Se S ndomly n e o 2 /2 uv /3 /3 uw /3 0 Se e {, 2, 3 } Obevon e {u, v, w} 3 /3 π /2 π 2 /2 π /3 3 2 / /3 32 /3 3 /3 /3 vw b (u /2 b (v /2 b (w 0 b 2 (u 0 b 2 (v /2 b 2 (w /2 b 3 (u /2 b 3 (v 0 b 3 (w /2 2 /2 Chooe one of he oupu ymbol n ech e ndom. Le genee equence of obevon h wh he obeve h o wok wh x x 2 x 3??? 2 3 u u w Hdden Mkov Model Slde 33 oblem o olve So now we hve n HMM (o, moe genelly, DB h model empol poce of nee Wh e ome of he knd of poblem we d lke o be ble o olve wh h? Hdden Mkov Model Slde 34 7

18 empol Model oblem o Solve Fleng Compue (, λ edcon Compue ( k, λ fo k > Smoohng Compue ( k, λ fo k < Obevon equence lkelhood Compue ( λ Mo pobble ph (e equence Compue x mxmng (x, λ Mxmum lkelhood model Gven e of obevon equence { }, compue λ mxmng ( λ Hdden Mkov Model Slde 35 empol Model oblem o Solve Ued n wde vey of dynmcl yem modelng pplcon fleng pedcon moohng Ued epeclly n HMM pplcon obevon equence lkelhood mo pobble ph mxmum lkelhood model fng Hdden Mkov Model Slde 36 8

19 Fleng (,λ Compue Z Z 2 Z - Z Z obeved nfe (dbuon cuen me Hdden Mkov Model Slde 37 Compue edcon (, fo k k λ > k... Z Z 2 Z - Z Z k obeved nfe (dbuon cuen me Hdden Mkov Model Slde 38 9

20 Compue Smoohng (, fo k k λ < 2... k Z Z 2 Z k Z Z obeved nfe (dbuon cuen me Hdden Mkov Model Slde 39 Obevon Sequence Lkelhood Compue ( λ Z Z 2 Z - Z Z obeved Wh he pobbly of h pcul equence of obevon funcon of he model pmee? Ueful fo uch hng fndng whch of e of HMM model be f n obevon equence, n peech ecognon. Hdden Mkov Model Slde 40 20

21 Mo obble h ( x,λ Compue g mx x Z Z 2 Z - Z Z obeved nfe (only mo pobble o necely he me he equence of ndvdully mo pobble e (obned by moohng Hdden Mkov Model Slde 4 Mxmum Lkelhood Model Aume numbe of e gven Gven e of R obevon equence 2 R Compue 2 R M (, 2, K, (,, K, R R R (,, K, * λ g mx 2 2 R λ 2 R ( λ Hdden Mkov Model Slde 42 2

22 Soluon mehod fo hee poblem Le by condeng he obevon equence lkelhood poblem Gven ( λ, compue Ue ou exmple HMM o llue Hdden Mkov Model Slde 43 ob. of equence of 3 obevon ( ( 3 x ph of lengh 3 ( 3 x ph of lengh 3 x x 3 3 /2 ( x 3 uv /3 /3 /3 uw 3 /3 vw /3 0 2 /2 How do we compue (x 3 fo n by ph x 3? How do we compue ( 3 x 3 fo n by ph x 3? Hdden Mkov Model Slde 44 22

23 ob. of equence of 3 obevon ( ( 3 x ph of lengh 3 ( 3 x ph of lengh 3 x x 3 3 ( x /2 3 uv /3 /3 /3 uw 3 /3 vw /3 0 2 /2 How do we compue (x 3 fo n by ph x 3? How do we compue ( 3 x 3 fo n by ph x 3? (x,x 2,x 3 (x (x 2 x (x 3 x 2 E.g, (, 3, 3 /2 * * /3 /9 Hdden Mkov Model Slde 45 ob. of equence of 3 obevon ( ( 3 x ph of lengh 3 ( 3 x ph of lengh 3 x x 3 3 ( x /2 3 uv /3 /3 /3 uw 3 /3 vw /3 0 2 /2 How do we compue (x 3 fo n by ph x 3? (x,x 2,x 3 (x (x 2 x (x 3 x 2 E.g, (, 3, 3 /2 * * /3 /9 How do we compue ( 3 x 3 fo n by ph x 3? (, 2, 3 x, x 2, x 3 ( x ( 2 x 2 ( 3 x 3 E.g, (uuw, 3, 3 /2 * /2 * /2 /8 Hdden Mkov Model Slde 46 23

24 ob. of equence of 3 obevon ( ( 3 x ph of lengh 3 ( 3 x ph of lengh 3 x x 3 3 ( x /2 3 uv /3 /3 /3 uw 3 /3 vw /3 0 2 /2 Bu h um h em n! Exponenl n he lengh of he equence eed o ue dynmc pogmmng ck lke befoe Hdden Mkov Model Slde 47 he pobbly of gven equence of obevon, non-exponenl-co-yle Gven obevon equence (, 2,, Defne he fowd vble ( (, λ fo ( obbly h, n ndom l, we d hve een he f obevon; nd we d hve ended up n he h e ved. Hdden Mkov Model Slde 48 24

25 25 Hdden Mkov Model Slde 49 ( ( ( ( b π ( me ep 2 π oe Fo mplcy, we ll dop explc efeence o condonng on he HMM pmee λ fo mny of he upcomng lde, bu lwy hee mplcly. Be ce Compung he fowd vble Hdden Mkov Model Slde 50 Fowd vble nducve ep ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( b um ove ll poble pevou e

26 26 Hdden Mkov Model Slde 5 Fowd vble nducve ep ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( b pl off l obevon Hdden Mkov Model Slde 52 Fowd vble nducve ep ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( b chn ule

27 27 Hdden Mkov Model Slde 53 Fowd vble nducve ep ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( b le e nd obevon condonlly ndependen of ele obevon gven pevou e Hdden Mkov Model Slde 54 Fowd vble nducve ep ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( b chn ule

28 28 Hdden Mkov Model Slde 55 Fowd vble nducve ep ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( b le obevon condonlly ndependen of ele e gven le e Hdden Mkov Model Slde 56 Fowd vble nducve ep ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( b

29 Fowd vble nducve ep ( ( b ( ( Hdden Mkov Model Slde 57 Obevon Sequence Lkelhood Effcen oluon o he obevon equence lkelhood poblem ung he fowd vble ( λ ( λ ( Hdden Mkov Model Slde 58 29

30 In ou exmple ( ( λ ( b ( π ( b ( ( /2 uv /3 /3 /3 uw 3 /3 vw /3 0 2 /2 Obeved 2 3 u u w ( 2( 0 2( 0 4 ( 2 0 2( 2 0 3( 2 72 ( 3 0 ( 2 3 3( So pobbly of obevng uuw /36 Hdden Mkov Model Slde 59 Fleng Effcen oluon o he fleng poblem ung he fowd vble ( ( ( ( ( Emng cuen e bed on ll obevon up o he cuen me. So n ou exmple, fe obevng uuw, pob. of beng n 0 nd pob. of beng n 2 pob. of beng n 3 /2 Hdden Mkov Model Slde 60 30

31 edcon oe h he (e pedcon poblem cn be vewed pecl ce of he fleng poblem n whch hee e mng obevon. h, yng o compue he pobbly of k gven obevon up hough me ep, wh k >, moun o fleng wh mng obevon me ep, 2,..., k. heefoe, we now focu on he mng obevon poblem. Hdden Mkov Model Slde 6 Mng Obevon Lookng he devon of he nducve ep fo compung he fowd vble, we ee h he l ep nvolve wng ( ( ( ll obevon up hough me b ( ( hu he econd fco gve u pedcon of he e me bed on ll ele obevon, whch we hen mulply by he obevon pobbly me gven he e me. If hee no obevon me, clely he e of obevon mde hough me he me he e of obevon mde hough me. Hdden Mkov Model Slde 62 3

32 Mng Obevon (con. hu we edefne ( ( ll vlble obevon hough me h genele ou ele defnon bu llow fo he pobly h ome obevon e peen nd ohe e mng hen defne b ( ( f hee n obevon me b ohewe I no hd o ee h he coec fowd compuon hould hen poceed ( b ( π b ( ( ( Amoun o popgng e pedcon fowd wheeve hee e no obevon Ineeng pecl ce When hee e no obevon ny me, he vlue e dencl o he p vlue we defned ele fo Mkov chn Hdden Mkov Model Slde 63 Solvng he moohng poblem Defne he bckwd vble ( ( β,λ obbly of obevng,..., gven h yem w n e me ep hee cn be compued effcenly by ng he end (me nd wokng bckwd Be ce β fo ll, ( Vld becue n empy equence of obevon o pobbly Hdden Mkov Model Slde 64 32

33 33 Hdden Mkov Model Slde 65 Bckwd vble nducve ep ( ( ( ( ( ( ( ( ( ( ( ( b β β β Hdden Mkov Model Slde 66 Bckwd vble nducve ep ( β ( ( b β β

34 Solvng he moohng poblem Ue he noon γ ( ( fo he pobbly we wn o compue. hen γ ( c( ( c( ( ( c( ( c ( β ( whee c /( conn of popoonly we cn gnoe long we nomle o ge he cul pob. Hdden Mkov Model Slde 67 Smoohng Effcen oluon o he moohng poblem ung he fowd nd bckwd vble ( ( β ( ( β ( Emng e bed on ll obevon befoe, dung, nd fe h me ep. Fowd-bckwd lgohm Hdden Mkov Model Slde 68 34

35 Solvng he mo pobble ph poblem Wn g mx x One ppoch g mx x x ( x ( g mx g mx ( x ( x ( ( x ( x Ey o compue ech fco fo gven e nd obevon equence, bu numbe of ph exponenl n Ue dynmc pogmmng ned x x Hdden Mkov Model Slde 69 D fo Mo obble h Defne δ ( mx x ( x A ph gvng h mxmum one of lengh - hvng he hghe pobbly of mulneouly occung endng poducng obevon equence Hdden Mkov Model Slde 70 35

36 D fo M (con. We ll how h hee vlue cn be compued by n effcen fowd compuon ml o he compuon of he vlue Bu f, le check h gve u omehng ueful δ mx x ( ( mx x x ( x ( hu vlue of mxmng δ ( denfe e whch epeen he fnl e n ph mxmng x ( Hdden Mkov Model Slde 7 F, be ce δ D fo M (con. ( ( mx one choce ( ( ( b π hen, nce he mx. pob. ph endng me mu go hough ome e me, we cn we δ ( mx x ( x mx mx ( x x ow wok on u h p Cll Δ(, Hdden Mkov Model Slde 72 36

37 37 Hdden Mkov Model Slde 73 D fo M (con. ( ( ( ( ( ( ( ( ( ( ( x b x x x x x, Δ Ung he chn ule nd he Mkov popey, we fnd h he pobbly o be mxmed cn be wen Hdden Mkov Model Slde 74 D fo M (con. Fnlly, hen, we ge h nducve ep Vully dencl o compuon of fowd vble only dffeence h ue mx ned of um Alo need o keep ck of whch e gve mx fo ech e he nex me ep o be ble o deemne cul M, no u pobbly ( ( ( ( [ ] ( ( [ ] ( ( b x b x b x x x δ δ mx mx mx mx mx, mx mx Δ

38 Veb Algohm fo Mo obble h Summy Be ce δ ( b ( π Inducve ep δ b mx δ Compue fo ll e, hen 2, ec. Alo ve ndex gvng mx fo ech e ech me ep (bckwd pone Conuc he M by deemnng e wh lge δ (, hen followng bckwd pone o me ep -, -2, ec. ( ( ( Hdden Mkov Model Slde 75 Veb Algohm Soe wo numbe ech node n h ell, one fo δ nd he ohe bckwd pone o node n he pevou lye gvng he mx fo h node h compued lef o gh. o fnd mo pobble ph, deemne node n he lye wh mx δ vlue, hen follow bckwd pone fom gh o lef. Hdden Mkov Model Slde 76 38

39 Veb lgohm nducve ep ( δ δ ψ ( b ( mx δ ( ( g mx δ ( Hdden Mkov Model Slde 77 ob. of gven non he fnl poblem we wn o dde he HMM nfeence (lenng poblem, gven nng e of obevon equence Mo of he ngeden fo devng mx. lkelhood mehod fo h e n plce Bu hee one moe ub-poblem we ll need o dde Gven n obevon equence, wh he pobbly h he e non o occued me? hu we defne ξ (, ( Hdden Mkov Model Slde 78 39

40 40 Hdden Mkov Model Slde 79 ob. of gven non (con. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( b c b c b c c c c c 2 2, β β ξ c /( nomlng conn we cn gnoe long we mke he um ove ll (, p equl o when compung cul pobble. Hdden Mkov Model Slde 80 ob. of gven non (con. ( ( ( ( ( ( ( ( l k l kl l b k b,, β β ξ

41 Mx. Lkelhood HMM Infeence Gven e e {, 2,..., } nd e of R obevon equence,, K, 2 R deemne pmee e λ (π,{ },{b (o } mxmng * λ g mx 2 R M ( (,, K, R R R (,, K, R λ 2 2 ( 2 R λ Fom now on, we ll mke condonng on λ explc Hdden Mkov Model Slde 8 A che Le f mgne h long wh ech obevon equence (, 2, K, n ocle lo gve u he coepondng e equence ( x x, x x,, 2 K hen we could obn mx. lkelhood eme of ll pmee follow # of equence ng wh ˆ π ol # of equence bˆ ˆ ( o k # of non # of v o e # of v o e whee o # v o e k obeved Hdden Mkov Model Slde 82 4

42 A che (con. Moe fomlly, defne he ndco funcon χ ( ( χ χ ( k f x 0 ohewe f x nd x 0 ohewe f x nd o 0 ohewe k Hdden Mkov Model Slde 83 A che (con. In em of hee ndco funcon, ou ML eme would hen be bˆ ˆ π ˆ ( o k R χ R ( R R R R χ χ χ ( ( ( k ( χ Fo h, we cn ue he l e n ny of he nng equence becue hee no nex e Hdden Mkov Model Slde 84 42

43 he bd new... hee no ocle o ell u he e equence coepondng o ech obevon equence So we don know hee cul ndco funcon vlue So we cn compue hee um Hdden Mkov Model Slde 85 he good new... We cn compue he expeced vlue effcenly ξ γ Alo E χ ( λ ( (, λ E χ ( ( λ (, ( λ E χ (, ( ( k λ ( Z o, λ (, λ ( I ( o k k f ok 0 ohewe γ Uul ndco funcon f ue, 0 f fle Hdden Mkov Model Slde 86 43

44 he good new... We cn compue he expeced vlue effcenly ξ γ Alo E χ ( λ ( (, λ E χ ( ( λ (, ( λ E χ (, ( ( k λ ( Z o, λ (, λ ( I ( o k k f ok 0 ohewe Look lke ob fo EM! γ Uul ndco funcon f ue, 0 f fle Hdden Mkov Model Slde 87 E-ep EM fo HMM (Bum-Welch Ue he cuen eme of model pmee λ o compue ll he γ ( nd ξ, vlue fo ech nng equence. M-ep π R γ b ( k ( R ( R R R ( ( γ I ok R γ ( ξ γ (, ( Hdden Mkov Model Slde 88 44

45 Remk on Bum-Welch Bd new hee my be mny locl mxm Good new he locl mxm e uully deque model of he d Any pobble nled o eo wll emn eo houghou ueful when one wn model wh lmed e non Hdden Mkov Model Slde 89 Summy of oluon mehod Fleng fowd vble ( edcon (modfed fowd vble Smoohng fowd-bckwd lgohm Obevon equence lkelhood fowd vble Mo pobble ph Veb lgohm Mxmum lkelhood model Bum-Welch lgohm Hdden Mkov Model Slde 90 45

46 Some good efeence Sndd HMM efeence L. R. Rbne, "A uol on Hdden Mkov Model nd Seleced Applcon n Speech Recognon," oc. of he IEEE, Vol.77, o.2, pp , 989. Excellen efeence fo Dynmc Bye e unfyng fmewok fo pobblc empol model (ncludng HMM nd Klmn fle Chpe 5 of Afcl Inellgence, A Moden Appoch, 2nd Edon, by Ruell & ovg Hdden Mkov Model Slde 9 Wh You Should Know Wh n HMM Defnon, compuon, nd ue of ( he Veb lgohm Oulne of he EM lgohm fo HMM lenng (Bum-Welch Be comfoble wh he knd of mh needed o deve he HMM lgohm decbed hee Wh DB nd how n HMM pecl ce Appece h DB (nd hu n HMM elly u pecl knd of Bye ne Hdden Mkov Model Slde 92 46

Calculus 241, section 12.2 Limits/Continuity & 12.3 Derivatives/Integrals notes by Tim Pilachowski r r r =, with a domain of real ( )

Calculus 241, section 12.2 Limits/Continuity & 12.3 Derivatives/Integrals notes by Tim Pilachowski r r r =, with a domain of real ( ) Clculu 4, econ Lm/Connuy & Devve/Inel noe y Tm Plchow, wh domn o el Wh we hve o : veco-vlued uncon, ( ) ( ) ( ) j ( ) nume nd ne o veco The uncon, nd A w done wh eul uncon ( x) nd connuy e he componen