Three Main Questions on HMMs

Transcription

1 Mache Learg 0-70/ Srg 00 Hdde Marov Model II Erc Xg Lecure Februar 4 00 Readg: Cha. 3 CB Three Ma Quesos o HMMs. Evaluao GIVEN a HMM M ad a sequece FIND Prob M ALGO. Forward. Decodg GIVEN a HMM M ad a sequece FIND he sequece of saes ha mamzes e.g. P M or he mos robable subsequece of saes ALGO. Verb Forward-bacward 3. Learg GIVEN a HMM M wh usecfed raso/emsso robs. ad a sequece FIND arameers θ π a j η ha mamze P θ ALGO. Baum-Welch EM

2 Eamle: FAIR LOADED PF /6 PF /6 P3F /6 P4F /6 P5F /6 P6F / PL /0 PL /0 P3L /0 P4L /0 P5L /0 P6L / α P α a β β P a P P α β P P FAIR LOADED Alha acual Bea acual α PF /6 PF /6 P3F /6 P4F /6 P5F /6 P6F / PL /0 PL /0 P3L /0 P4L /0 P5L /0 P6L / P α a a P + + β + β P P α β P P 4

3 FAIR LOADED Alha logs Bea logs α PF /6 PF /6 P3F /6 P4F /6 P5F /6 P6F / PL /0 PL /0 P3L /0 P4L /0 P5L /0 P6L / P α a a P + + β + β P P α β P P 5 Wha s he robabl of a hdde sae redco? A sgle sae: Wha abou a hdde sae sequece? 6 3

4 Poseror decodg We ca ow calculae P α β P P P The we ca as Wha s he mos lel sae a oso of sequece : * Noe ha hs s a MPA of a sgle hdde sae wha f we wa o a MPA of a whole hdde sae sequece? Poseror Decodg: arg ma P * { : T } L Ths s dffere from MPA of a whole sequece of hdde saes P Ths ca be udersood as b error rae vs. word error rae Eamle: MPA of X? MPA of X Y? Verb decodg GIVEN T we wa o fd T such ha P s mamzed: Le V * argma P argma π P ma... } P { - Probabl of mos lel sequece of saes edg a sae The recurso: N Sae V ma a V Uderflows are a sgfca roblem K K π a La b Lb K These umbers become eremel small uderflow Soluo: Tae he logs of all values: log V log a + V V + ma 8 4

5 The Verb Algorhm dervao Defe he verb robabl: V + + ma {... } P ma {... } P + + ma... } P P { P ma P + ma{... } P ma P + + a V P + + ma a V 9 The Verb Algorhm Iu: T Ialzao: V Ierao: P π V P ma a V Pr arg ma a V Termao: * P ma V T TraceBac: * arg ma V T * * Pr T 0 5

6 Verb Vs. MPA dvdual V log r Seq Veerb MPA N/A Aoher Eamle X V log r Seq Veerb MPA N/A ma 0.6 FAIR 0.4 LOADED 0.6 Same raso robables 0.4 6

7 Comuaoal Comle ad mlemeao deals Wha s he rug me ad sace requred for Forward ad Bacward? α α a β a + + β+ V ma a V Tme: OK N; Sace: OKN. Useful mlemeao echque o avod uderflows Verb: sum of logs Forward/Bacward: rescalg a each oso b mullg b a cosa 3 Three Ma Quesos o HMMs. Evaluao GIVEN a HMM M ad a sequece FIND Prob M ALGO. Forward. Decodg GIVEN a HMM M ad a sequece FIND he sequece of saes ha mamzes e.g. P M or he mos robable subsequece of saes ALGO. Verb Forward-bacward 3. Learg GIVEN a HMM M wh usecfed raso/emsso robs. ad a sequece FIND arameers θ π a j η ha mamze P θ ALGO. Baum-Welch EM 4 7

8 Learg HMM: wo scearos Suervsed learg: esmao whe he rgh aswer s ow Eamles: GIVEN: GIVEN: a geomc rego where we have good eermeal aoaos of he CG slads he caso laer allows us o observe hm oe eveg as he chages dce ad roduces 0000 rolls Usuervsed learg: esmao whe he rgh aswer s uow Eamles: GIVEN: GIVEN: he orcue geome; we do ow how freque are he CG slads here eher do we ow her comoso 0000 rolls of he caso laer bu we do see whe he chages dce QUESTION: Udae he arameers θ of he model o mamze Pθ --- Mamal lelhood ML esmao 5 MLE 6 8

9 Suervsed ML esmao Gve N for whch he rue sae ah N s ow Defe: A j # mes sae raso j occurs # mes sae ems B We ca show ha he mamum lelhood arameers θ are: T j T # j Aj # A ML aj j ' j ' b ML # # T T B B Wha f s couous? We ca rea as N T : : T : N observaos of e.g. a Gaussa ad al learg rules for Gaussa ' ' { } 7 Suervsed ML esmao cd. Iuo: Whe we ow he uderlg saes he bes esmae of θ s he average frequec of rasos & emssos ha occur he rag daa Drawbac: Gve lle daa here ma be overfg: Pθ s mamzed bu θ s ureasoable 0 robables VERY BAD Eamle: Gve 0 caso rolls we observe F F F F F F F F F F The: a FF ; a FL 0 b F b F3.; b F.3; b F4 0; b F5 b F6. 8 9

10 Pseudocous Soluo for small rag ses: Add seudocous A j # mes sae raso j occurs + R j B # mes sae ems + S R j S j are seudocous rereseg our ror belef Toal seudocous: R Σ j R j S Σ S --- "sregh" of ror belef --- oal umber of magar saces he ror Larger oal seudocous srog ror belef Small oal seudocous: jus o avod 0 robables --- smoohg 9 Usuervsed ML esmao 0 0

11 Usuervsed ML esmao Gve N for whch he rue sae ah N s uow EXPECTATION MAXIMIZATION 0. Sarg wh our bes guess of a model M arameers θ:. Esmae A j B he rag daa How? Udae θ accordg o A j B B j j A g j Now a "suervsed learg" roblem. Reea & ul covergece Ths s called he Baum-Welch Algorhm We ca ge o a rovabl more or equall lel arameer se θ each erao The Baum Welch algorhm The comlee log lelhood T T l l θ l The eeced comlee log lelhood EM The E se c log log ; θ l + + T T j j c b a log log log ; θ π l The M se "smbolcall" decal o MLE γ j j j ξ T T j ML j a γ ξ T T ML b γ γ N ML γ π

12 The Baum-Welch algorhm -- commes Tme Comle: # eraos OK N Guaraeed o crease he log lelhood of he model No guaraeed o fd globall bes arameers Coverges o local omum deedg o al codos Too ma arameers / oo large model: Over-fg 3 Summar: he HMM algorhms Quesos: Evaluao: Wha s he robabl of he observed sequece? Forward Decodg: Wha s he robabl ha he sae of he 3rd roll s loaded gve he observed sequece? Forward- Bacward Decodg: Wha s he mos lel de sequece? Verb Learg: Uder wha arameerzao are he observed sequeces mos robable? Baum-Welch EM 4

13 Alcaos of HMMs Some earl alcaos of HMMs face bu we ever saw hem seech recogo modellg o chaels I he md-lae 980s HMMs eered geecs ad molecular bolog ad he are ow frml ereched. Some curre alcaos of HMMs o bolog mag chromosomes algg bologcal sequeces redcg sequece srucure ferrg evoluoar relaoshs fdg gees DNA sequece 5 Tcal srucure of a gee 6 3

14 GENSCAN Burge & Karl 5'UTR Forward + srad Reverse - srad E0 E E GAGAACGTGTGAGAGAGAGGCAAGCCGAAAAATCAGCCGC CGAAGGATACACTATCGTCGTCCTTGTCCGACGAACCGGT GGTCATGCAAACAACGCACAGAACAAATTAATTTTCAAAT TGTTCAATAAATGTCCCACTTGCTTCTGTTGTTCCCCCCT TTCCGCTAGCGGAATTTTTTATATTCTTTTGGGGGCGCTC TTTCGTTGACTTTTCGAGCACTTTTTCGATTTTCGCGCGC TGTCGAACGGCAGCGTATTTATTTACAATTTTTTTTGTTA GCGGCCGCCGTTGTTTGTTGCAGATACACAGCGCACACAT I 0 I I ATAAGCTTGCACACTGATGCACACACACCGACACGTTGTC ACCGAAATGAACGGGACGGCCATATGACTGGCTGGCGCTC GGTATGTGGGTGCAAGCGAGATACCGCGATCAAGACTCGA ACGAGACGGGTCAGCGAGTGATACCGATTCTCTCTCTTTT E E GCGATTGGGAATAATGCCCGACTTTTTACACTACATGCGT TGGATCTGGTTATTTAATTATGCCATTTTTCTCAGTATAT CGGCAATTGGTTGCATTAATTTTGCCGCAAAGTAAGGAAC θ ACAAACCGATAGTTAAGATCCAACGTCCCTGCTGCGCCTC θ GCGTGCACAATTTGCGCCAATTTCCCCCCTTTTCCAGTTT TTTTCAACCCAGCACCGCTCGTCTCTTCCTCTTCTTAACG E s 3'UTR θ TTAGCATTCGTACGAGGAACAGTGCTGTCATTGTGGCCGC 3 TGTGTAGCTAAAAAGCGTAATTATTCATTATCTAGCTATC θ4 TTTTCGGATATTATTGTCATTTGCCTTTAATCTTGTGTAT TTATATGGATGAAACGTGCTATAATAACAATGCAGAATGA ol-a AGAACTGAAGAGTTTCAAAACCTAAAAATAATTGGAATAT G G G C CC GG romoer ergec rego Forward + srad Reverse - srad AAAGTTTGGTTTTACAATTTGATAAAACTCTATTGTAAGT GGAGCGTAACATAGGGTAGAAAACAGTGCAAATCAAAGTA CCTAAATGGAATACAAATTTTAGTTGTACAATTGAGTAAA ATGAGCAAAGCGCCTATTTTGGATAATATTTGCTGTTTAC AAGGGGAACATATTCATAATTTTCAGGTTTAGGTTACGCA TATGTAGGCGTAAAGAAATAGCTATATTTGTAGAAGTGCA TATGCACTTTATAAAAAATTATCCTACATTAACGTATTTT ATTTGCTTTAAAACCTATCTGAGATATTCCAATAAGGTAA GTGCAGTAATACAATGTAAATAATTGCAAATAATGTTGTA ACTAAATACGTAAACAATAATGTAGAAGTCCGGCTGAAAG CCCCAGCAGCTATAGCCGATATCTATATGATTTAAACTCT TGTCTGCAACGTTCTAATAAATAAATAAAATGCAAAATAT AACCTATTGAGACAATACATTTATTTTATTTTTTTATATC ATCAATCATCTACTGATTTCTTTCGGTGTATCGCCTAATC CATCTGTGAAATAGAAATGGCGCCACCTAGGTTAAGAAAA GATAAACAGTTGCCTTTAGTTGCATGACTTCCCGCTGGAT 7 Shorcomgs of Hdde Marov Model Y Y Y X X X HMM models caure deedeces bewee each sae ad ol s corresodg observao NLP eamle: I a seece segmeao as each segmeal sae ma deed o jus o a sgle word ad he adjace segmeal sages bu also o he o-local feaures of he whole le such as le legh deao amou of whe sace ec. Msmach bewee learg objecve fuco ad redco objecve fuco HMM lears a jo dsrbuo of saes ad observaos PY X bu a redco as we eed he codoal robabl PYX 8 4

15 Recall Geerave vs. Dscrmave Classfers Goal: Wsh o lear f: X Y e.g. PYX Geerave classfers e.g. Naïve Baes: Assume some fucoal form for PXY PY Ths s a geerave model of he daa! Esmae arameers of PXY PY drecl from rag daa Use Baes rule o calculae PYX Y X Dscrmave classfers e.g. logsc regresso Drecl assume some fucoal form for PYX Ths s a dscrmave model of he daa! Esmae arameers of PYX drecl from rag daa Y X 9 Srucured Codoal Models Y Y Y : Codoal robabl Plabel sequece observao sequece raher ha jo robabl P Secf he robabl of ossble label sequeces gve a observao sequece Allow arbrar o-deede feaures o he observao sequece X The robabl of a raso bewee labels ma deed o as ad fuure observaos Rela srog deedece assumos geerave models 30 5

16 Codoal Dsrbuo If he grah G V E of Y s a ree he codoal dsrbuo over he label sequece Y gve X b he Hammersle Clfford heorem of radom felds s: θ e λ f e e + µ g v v e E v V s a daa sequece s a label sequece v s a vere from vere se V se of label radom varables e s a edge from edge se E over V Y Y Y 5 X X f ad g are gve ad fed. g s a Boolea vere feaure; f s a Boolea edge feaure s he umber of feaures θ λ are arameers o be esmaed λ L λ ; µ µ L µ ; λ adµ e s he se of comoes of defed b edge e v s he se of comoes of defed b vere v 3 Codoal Radom Felds Y Y Y : CRF s a arall dreced model Dscrmave model Usage of global ormalzer Z Models he deedece bewee each sae ad he ere observao sequece 3 6

17 Codoal Radom Felds A 3 A A T A T Geeral aramerc form: Y Y Y : 33 Codoal Radom Felds θ e θc fc c Z θ c Allow arbrar deedeces o u Clque deedeces o labels Use aromae ferece for geeral grahs 34 7

18 CRFs: Iferece Gve CRF arameers λ ad µ fd he * ha mamzes P Ca gore Z because s o a fuco of Ru he ma-roduc algorhm o he juco-ree of CRF: Y Y Y : Same as Verb decodg used HMMs! Y Y Y Y Y - Y - Y Y 3 3. Y - Y Y - Y - 35 CRF learg Gve { d d } dn fd λ* µ* such ha Comug he grade w.r. λ: Grade of he log-aro fuco a eoeal faml s he eecao of he suffce sascs. 36 8

19 CRFs: some emrcal resuls Comarso of error raes o shec daa MEMM error MEMM error CRF error HMM error CRF error HMM error Daa s creasgl hgher order he dreco of arrow CRFs acheve he lowes error rae for hgher order daa 37 CRFs: some emrcal resuls Pars of Seech aggg Usg same se of feaures: HMM >< CRF > MEMM Usg addoal overlag feaures: CRF + > MEMM + >> HMM 38 9

20 Summar Codoal Radom Felds s a dscrmave Srucured Iu Ouu model! Y HMM s a geerave srucured I/O model X Comlemear sregh ad weaess:.. 3. Y X 39 0