Learning of Graphical Models Parameter Estimation and Structure Learning

Transcription

1 Learg of Grahal Models Parameer Esmao ad Sruure Learg e Fukumzu he Isue of Sasal Mahemas Comuaoal Mehodology Sasal Iferee II

2 Work wh Grahal Models Deermg sruure Sruure gve by modelg d e.g. Mxure model HMM e Sruure learg sruure Par IV Parameer esmao Parameer gve by some kowledge Parameer esmao wh daa suh as MLE or Bayesa esmao Par IV Iferee Comuao of oseror ad margal robables Already see Par III. a b a \ a arameer 2

3 Parameer Esmao 3

4 Sasal Esmao Esmao from daa Sasal model wh a arameer: : arameer I..d. Daa: D 2 Maxmum lkelhood esmao or arg max L arg max l L Lkelhood fuo l log L log Log lkelhood fuo 4

5 5 Sasal Esmao Bayesa esmao Dsrbuo of he arameer s esmaed Pror robably oseror robably D Bayes rule gves Maxmum a oseror MAP esmao d D D D arg max D MAP

6 Cogey able ML esmao for dsree varables b a {... M} b {... L} D..d. samle a b a b a d e a a : umber of ous a Esmao of robables ML esmaor 6

7 Bayesa Esmao: Dsree Case Bayesa esmao for dsree varables Model: a b a Pror: o b Δ ML Δ ML Δ { R } Lkelhood: D a b Mulomal Bayesa esmao: D D D Δ D D d Δ d hs egral s dfful o omue geeral. 7

8 Drhle Dsrbuo Drhle dsrbuo Desy fuo of -dmesoal Drhle dsrbuo Dr α α Γ α Γ α α α o Δ { R } where α α : arameer α > Γα : Gamma fuo α Γ α e d Γ α α Γ α for α > Γ! for a osve eger. 8

9 Drhle Dsrbuo α 622 α 375 α 234 α 626 Exeao E[ ] α α he mea o s rooroal o he veor α. he mea o s a sable o.e. dffereal ad may be eher maxmum or mmum. 9

10 Drhle Pror Drhle dsrbuo works as a ror o mulomal dsrbuo Poseror s also Drhle -- ougae ror k kk Dr α D Dr ~ α k Dr α d Δ k ~ α α α * α works as a ror ou. MAP esmaor Proof of * MAP ~ α ~ α Dr α D α α Lα α By he ormalzao he rgh had sde mus be Dr α. ~

11 EM Algorhm for Models wh Hdde Varables

12 ML Esmao wh Hdde Varable Sasal model wh hdde varables Suose we a assume hdde uobservable varables addo o observable varables : observable varable : hdde varable : arameer We have daa oly for observable varables: he ML esmao mus be doe wh D 2 log log Bu hs maxmzao s ofe dfful by oleary w.r.. 2

13 3 3 ML Esmao wh Hdde Varable Examle: Gaussa mxure model ML esmao ad are ouled dfful o solve aalyally. x x φ x φ akes values {...}: omoe... Wh hdde varable: Margal of : log max log max φ

14 Esmao wh Comlee Daa Comlee daa Suose... are kow. D { } : omlee daa ML esmao wh D s ofe easer ha esmao wh D. max l D where l D log Comlee log lkelhood 4

15 Esmao wh Comlee Daa Examle: Mxure of Gaussa Redefe he hdde varable by dmesoal bary veor: { aφ x a a } a a... akes values { } lass oe: aφ x a a a 5

16 Esmao wh Comlee Daa ML esmao wh omlee daa: log log { φ } { log logφ } ad are deouled hey a be maxmzed searaely. max max log logφ sub. o Maxmzao s easy. Bu he omlee daa s o avalable rae! 6

17 Exeed Comlee Log Lkelhood Use exeed omlee log lkelhood sead of omlee log lkelhood. Comlee log lkelhood l D log Exeed omlee log lkelhood Suose we have a urre guess Use exeao w.r.. l D log Maxmze of l D 7

18 EM Algorhm Ialzao Ialze by some mehod.. Reea he followg ses ul sog rero s sasfed. E-se Comue he exeed omlee log lkelhood M-se Maxmze of l D arg max l D l D Comuaoal dffuly of M-se deeds o a model 8

19 9 9 EM Algorhm for Gaussa Mxure Comlee log lkelhood Exeed omlee log lkelhood E-se { } D log log φ l ] [ E τ φ φ Rao of orbuo of o he -h omoe. { } D log log φ τ l

20 2 2 EM Algorhm for Gaussa Mxure M-se τ τ τ τ τ weghed mea weghed ovarae marx Proof omed. Exerse

21 EM Algorhm for Gaussa Mxure Meag of τ : uobserved τ E SUM

22 Proeres of EM Algorhm EM overges ukly for may roblems. Mooo rease of lkelhood of s guaraeed dsussed laer. EM may be raed by loal oma. he soluo deeds srogly o he al sae. EM algorhm a be aled o ay model wh hdde varables. Mssg value e. 22

23 Demosrao Web se for Gaussa mxure demo: h:// 23

24 heoreal Jusfao of EM 24

25 heoreal Jusfao of EM EM as lkelhood maxmzao he goal s o maxmze he omlee log lkelhood o he exeed omlee log lkelhood. : arbrary.d.f. of may deed o. Defe a auxlary fuo L by L log. heorem E-se: M-se: arg max L arg max L l D ad omue Alerag omzao w.r.. ad. 25

26 26 heoreal Jusfao of EM Prooso L ad lkelhood of Proof For ay ad he log lkelhood of s deomosed as L l log log log log L L l L l I arular for all ad ad he eualy holds f ad oly f.

27 27 27 heoreal Jusfao of EM Prooso 2 L ad exeed omlee lkelhood roof L log l log L log log l log log

28 heoreal Jusfao of EM Proof of heorem E-se: From Prooso l L L deede of maxmze arg max L mmze M-se: From Prooso 2 L M-se s l os. w.r.. max L 28

29 heoreal Jusfao of EM Mooo rease of lkelhood by EM heorem l l for all. Proof l L E-se Pro. L l M-se Pro. 29

30 Remarks o EM Algorhm EM always reases he lkelhood of observable varables bu here are o heoreal guaraees of global maxmzao. I geeral a overge oly o a loal maxmum. here s a suffe odo of overgee by Wu 983. Praally EM overges very ukly. For Gaussa mxure model If he mea ad varae are s arameers he lkelhood fuo a ake a arbrary large value. here s o global maxmum of lkelhood. EM ofe fds a reasoable loal omum by a good hoe of alzao. he resuls deed muh o he alzao. Furher readgs: he EM Algorhm ad Exesos MLahla & rsha 997 Fe Mxure Models MLahla & Peel 2 3

31 EM Algorhm for Hdde Markov Model 3

32 32 32 Maxmum Lkelhood for HMM Paramer model of Gaussa HMM 2 2 A ; y φ Gaussa wh mea ad ovarae A arameer: raso marx al robably y A φ L max log s dfful.

33 33 33 EM for HMM Comlee lkelhood log A φ log A m log2 2 log de 2 2 A log δ δ δ m log2 2 log de 2 2 δ log l

34 34 EM for HMM Exeed omlee lkelhood Suose we already have a esmae : dex for erao 34 log l δ δ δ δ δ δ I reures ξ γ ad a be omued by he forward-bakward algorhm.

35 35 EM for HMM Baum-Welh Algorhm E-se Forward-bakward o omue ad. Exeed omlee log lkelhood M-se l A log ξ γ m log2 2 log de 2 2 γ γ k k A γ ξ ξ ξ γ γ γ γ ξ.f. EM for Gaussa mxure γ

36 Summary: Parameer learg Dsree varables whou hdde varables Maxmum lkelhood esmao s easy by freuees. Bayesa esmao s ofe doe wh Drhle ror. Dsree varables wh hdde varables Maxmum lkelhood esmao a be doe wh EM algorhm. Bayesa aroah omuaoal dffuly. varaoal mehod ad so o. 36