Probabilistic Graphical Models

Transcription

1 School of Comuter Scece Probablstc rahcal Models Parameter Est. fully observed Bs Erc Xg X X X X Lecture 7 February 5 04 X X 3 X X X 3 X 3 Readg: KF-cha 7 X 4 X 4 Erc CMU

2 Learg rahcal Models The goal: ve set of deedet samles assgmets of radom varables fd the best the most lely? Bayesa etwor both DA ad CPDs E B E B R A C R A C Structural learg BEACR=TFFTF BEACR=TFTTF.. BEACR=FTTTF E e e e e B b b b b PA EB Parameter learg Erc CMU

3 Learg rahcal Models Scearos: comletely observed Ms drected udrected artally or uobserved Ms drected udrected a oe research toc Estmato rcles: Mamal lelhood estmato MLE Bayesa estmato Mamal codtoal lelhood Mamal "Marg" Mamum etroy We use learg as a ame for the rocess of estmatg the arameters ad some cases the toology of the etwor from data. Erc CMU

4 ML Structural Learg for comletely observed Ms E R B A C Data M M Erc CMU

5 Two Otmal aroaches Otmal here meas the emloyed algorthms guaratee to retur a structure that mamzes the objectves e.g. LogL May heurstcs used to be oular but they rovde o guaratee o attag otmalty terretablty or eve do ot have a elct objectve E.g.: structured EM Module etwor greedy structural search etc. We wll lear two classes of algorthms for guarateed structure learg whch are lely to be the oly ow methods ejoyg such guaratee but they oly aly to certa famles of grahs: Trees: The Chow-Lu algorthm ths lecture Parwse MRFs: covarace selecto eghborhood-selecto later Erc CMU

6 Structural Search How may grahs over odes? O How may trees over odes? O! But t turs out that we ca fd eact soluto of a otmal tree uder MLE! Trc: MLE score decomosable to edge-related elemets Trc: a tree each ode has oly oe aret! Chow-lu algorthm Erc CMU

7 Iformato Theoretc Iterretato of ML M M cout M D D log ˆ log log log log ; l From sum over data ots to sum over cout of varable states Erc CMU

8 Iformato Theoretc Iterretato of ML co'd H M I M M M M M D D ˆ ˆ ˆ log ˆ ˆ ˆ ˆ log ˆ ˆ ˆ ˆ ˆ log ˆ ˆ log ˆ ˆ log ; l Decomosable score ad a fucto of the grah structure Erc CMU

9 Chow-Lu tree learg algorthm Objecto fucto: l ; D log ˆ D M Iˆ M Hˆ C M Iˆ Chow-Lu: For each ar of varable ad j Comute emrcal dstrbuto: Comute mutual formato: ˆ X Iˆ X X X j j cout j M j j log ˆ ˆ j j ˆ ˆ Defe a grah wth ode Edge Ij gets weght Iˆ X X j Erc CMU

10 Chow-Lu algorthm co'd Objecto fucto: Chow-Lu: Otmal tree B Comute mamum weght sag tree Drecto B: c ay ode as root do breadth-frst-search to defe drectos I-equvalece: H M I M D D ˆ ˆ ˆ log ; l I M C ˆ A B C D E C A E B D E C D A B E C I D C I C A I B A I C Erc CMU

11 Structure Learg for geeral grahs Theorem: The roblem of learg a B structure wth at most d arets s P-hard for ay fed d Most structure learg aroaches use heurstcs Elot score decomosto Two heurstcs that elot decomosto dfferet ways reedy search through sace of ode-orders Local search of grah structures Erc CMU

12 Z ML Parameter Est. for comletely observed Ms of gve structure X The data: { z z z z } Erc CMU

13 Parameter Learg Assume s ow ad fed from eert desg from a termedate outcome of teratve structure learg oal: estmate from a dataset of deedet detcally dstrbuted d trag cases D = {... }. I geeral each trag case =... M s a vector of M values oe er ode the model ca be comletely observable.e. every elemet s ow o mssg values o hdde varables or artally observable.e. s.t. s ot observed. I ths lecture we cosder learg arameters for a B wth gve structure ad s comletely observable l ; D log D log log Erc CMU

14 Revew of desty estmato Ca be vewed as sgle-ode grahcal models M: 3 Istaces of eoetal famly dst. Buldg blocs of geeral M MLE ad Bayesa estmate Erc CMU

15 Beroull dstrbuto: Ber Multomal dstrbuto: Mult Multomal dcator varable:. w.. ad ] [ where [...6] [...6] j j j j j j j X X X X X X X X X X T C A j j T C A j X P j de the dce - face} where { Dscrete Dstrbutos 0 P for for P Erc CMU

16 Multomal dstrbuto: Mult Cout varable: Dscrete Dstrbutos j j K where K K K K!!!!!!!! Erc CMU

17 Eamle: multomal model Data: We observed d de rolls K-sded: D={5 K 3} Reresetato: Ut bass vectors: Model: How to wrte the lelhood of a sgle observato? The lelhood of datasetd={ }: K K P P } th roll the de the de - sde of where { P P... ad {0} where K K ad w.. } {...K X K 3 M: Erc CMU

18 MLE: costraed otmzato wth Lagrage multlers Objectve fucto: l ; D logp D log log We eed to mamze ths subject to the costra Costraed cost fucto wth a Lagrage multler K l log Tae dervatves wrt l 0 MLE or Suffcet statstcs K The couts are suffcet statstcs of data D K MLE Frequecy as samle mea Erc CMU

19 Bayesa estmato: Drchlet dstrbuto: Posteror dstrbuto of : otce the somorhsm of the osteror to the ror such a ror s called a cojugate ror Posteror mea estmato: C P - - P d C d D 3 M: Drchlet arameters ca be uderstood as seudo-couts Erc CMU

20 More o Drchlet Pror: Where s the ormalze costat C come from? Itegrato by arts s the gamma fucto: For regers Margal lelhood: Posteror closed-form: Posteror redctve rate: K K d d C K 0 dt e t t! }... { C C d C P }... { Dr }... { C C d C Erc CMU

21 Sequetal Bayesa udatg Start wth Drchlet ror P Dr : Observe ' samles wth suffcet statstcs '. Posteror becomes: P ' Dr : ' Observe aother " samles wth suffcet statstcs ". Posteror becomes: P ' " Dr : ' " So sequetally absorbg data ay order s equvalet to batch udate. Erc CMU

22 Herarchcal Bayesa Models are the arameters for the lelhood are the arameters for the ror. We ca have hyer-hyer-arameters etc. We sto whe the choce of hyer-arameters maes o dfferece to the margal lelhood; tycally mae hyerarameters costats. Where do we get the ror? Itellget guesses Emrcal Bayes Tye-II mamum lelhood comutg ot estmates of : MLE arg ma Erc CMU

23 Lmtato of Drchlet Pror: Erc CMU

24 - Log Partto Fucto - ormalzato Costat log log e ~ ~ 0 K K K K K e C e L μ Σ The Logstc ormal Pror Pro: co-varace structure Co: o-cojugate we wll dscuss how to solve ths later Erc CMU

25 Logstc ormal Destes Erc CMU

26 Cotuous Dstrbutos Uform Probablty Desty Fucto / b a for a b 0 elsewhere ormal aussa Probablty Desty Fucto f / e The dstrbuto s symmetrc ad s ofte llustrated as a bell-shaed curve. Two arameters mea ad stadard devato determe the locato ad shae of the dstrbuto. The hghest ot o the ormal curve s at the mea whch s also the meda ad mode. The mea ca be ay umercal value: egatve zero or ostve. Multvarate aussa X ; / / e T X X Erc CMU

27 MLE for a multvarate-aussa It ca be show that the MLE for µ ad Σ s where the scatter matr s The suffcet statstcs are ad T. ote that X T X= T may ot be full ra eg. f <D whch case Σ ML s ot vertble S T ML ML MLE MLE T ML ML T T ML ML S K T T T X Erc CMU

28 Bayesa arameter estmato for a aussa There are varous reasos to ursue a Bayesa aroach We would le to udate our estmates sequetally over tme. We may have ror owledge about the eected magtude of the arameters. The MLE for Σ may ot be full ra f we do t have eough data. We wll restrct our atteto to cojugate rors. We wll cosder varous cases order of creasg comlety: Kow σ uow µ Kow µ uow σ Uow µ ad σ Erc CMU

29 Bayesa estmato: uow µ ow σ ormal Pror: Jot robablty: Posteror: 0 / e / P 0 ~ ad / / / / / / ~ where 3 M: Samle mea 0 / e e / / P ~ / ~ e ~ / P Erc CMU

30 Bayesa estmato: uow µ ow σ The osteror mea s a cove combato of the ror ad the MLE wth weghts roortoal to the relatve ose levels. The recso of the osteror /σ s the recso of the ror /σ 0 lus oe cotrbuto of data recso /σ for each observed data ot. Sequetally udatg the mea µ = 0.8 uow σ = 0. ow Effect of sgle data ot Uformatve vague/ flat ror σ ~ / / / / / / Erc CMU

31 Other scearos Kow µ uow λ = /σ The cojugate ror for λ s a amma wth shae a 0 ad rate verse scale b 0 The cojugate ror for σ s Iverse-amma Uow µ ad uow σ The cojugate ror s ormal-iverse-amma Sem cojugate ror Multvarate case: The cojugate ror s ormal-iverse-wshart Erc CMU

32 Estmato of codtoal desty Ca be vewed as two-ode grahcal models Istaces of LIM Q Q Buldg blocs of geeral M MLE ad Bayesa estmate X X See sulemetary sldes Erc CMU

33 MLE for geeral Bs If we assume the arameters for each CPD are globally deedet ad all odes are fully observed the the loglelhood fucto decomoses to a sum of local terms oe er ode: l ; D log D log log X = X 5 =0 X =0 X 5 = Erc CMU

34 Plates A late s a macro that allows subgrahs to be relcated X X X For d echageable data the lelhood s X D We ca rereset ths as a Bayes et wth odes. The rules of lates are smle: reeat every structure a bo a umber of tmes gve by the teger the corer of the bo e.g. udatg the late de varable e.g. as you go. Dulcate every arrow gog to the late ad every arrow leavg the late by coectg the arrows to each coy of the structure. Erc CMU

35 Decomosable lelhood of a B Cosder the dstrbuto defed by the drected acyclc M: Ths s eactly le learg four searate small Bs each of whch cossts of a ode ad ts arets. X X X X X X 3 X X X 3 X 3 X 4 X 4 Erc CMU

36 MLE for Bs wth tabular CPDs Assume each CPD s rereseted as a table multomal where def X j X ote that case of multle arets wll have a comoste state ad the CPD wll be a hgh-dmesoal table The suffcet statstcs are couts of famly cofguratos The log-lelhood s Usg a Lagrage multler j to eforce we get: j j j def j X j l ; D log log j j ML j j j j' j j' j Erc CMU

37 How to defe arameter ror? Earthquae Burglary Factorzato: M X Rado Alarm Call Local Dstrbutos defed by e.g. multomal arameters: j j Assumtos eger & Hecerma 9799: Comlete Model Equvalece lobal Parameter Ideedece Local Parameter Ideedece Lelhood ad Pror Modularty? Erc CMU

38 lobal & Local Parameter Ideedece lobal Parameter Ideedece For every DA model: Earthquae Burglary M m Rado Alarm Local Parameter Ideedece For every ode: q j j P deedet of Call Call AlarmYES P O Call Alarm Erc CMU

39 Parameter Ideedece rahcal Vew lobal Parameter Ideedece X X Local Parameter Ideedece samle X X samle Provded all varables are observed all cases we ca erform Bayesa udate each arameter deedetly!!! Erc CMU

40 Whch PDFs Satsfy Our Assumtos? eger & Hecerma 9799 Dscrete DA Models: Drchlet ror: aussa DA Models: ormal ror: ormal-wshart ror: C P - - Mult ~ j ormal ~ j ' e / /. where W tr e W W / / W w w w w c ormal W W Erc CMU

41 Parameter sharg A X X X 3 X T Cosder a tme-varat statoary st -order Marov model Ital state robablty vector: State trasto robablty matr: The jot: The log-lelhood: Aga we otmze each arameter searately s a multomal frequecy vector ad we've see t before What about A? X : def A j X def X j t X T T X t X t t t T l ; D log log t t A t Erc CMU t

42 Learg a Marov cha trasto matr A s a stochastc matr: j Each row of A s multomal dstrbuto. So MLE of A j s the fracto of trastos from to j Aj A ML j # # j T t j t t T t t Alcato: f the states X t rereset words ths s called a bgram laguage model Sarse data roblem: If j dd ot occur data we wll have A j =0 the ay future sequece wth word ar j wll have zero robablty. A stadard hac: bacoff smoothg or deleted terolato ~ A A t ML Erc CMU

43 Bayesa laguage model lobal ad local arameter deedece A A ' X X X 3 X T A The osteror of A ad A ' s factorzed deste v-structure o X t because X t- acts le a multleer Assg a Drchlet ror to each row of the trasto matr: A Bayes j def # j ' j D A # ML j where # We could cosder more realstc rors e.g. mtures of Drchlets to accout for tyes of words adjectves verbs etc. Erc CMU

44 Eamle: HMM: two scearos Suervsed learg: estmato whe the rght aswer s ow Eamles: IVE: IVE: a geomc rego = where we have good eermetal aotatos of the C slads the caso layer allows us to observe hm oe eveg as he chages dce ad roduces 0000 rolls Usuervsed learg: estmato whe the rght aswer s uow Eamles: IVE: IVE: the orcue geome; we do t ow how frequet are the C slads there ether do we ow ther comosto 0000 rolls of the caso layer but we do t see whe he chages dce QUESTIO: Udate the arameters of the model to mamze P - -- Mamal lelhood ML estmato Erc CMU

45 Recall defto of HMM Trasto robabltes betwee ay two states y y y 3... y T j yt yt a j A A A 3... A T or a a a. yt yt ~ Multomal M I Start robabltes ~ Multomal. y M Emsso robabltes assocated wth each state b b b. t yt ~ Multomal K I or geeral:. y ~ f I t t Erc CMU

46 Suervsed ML estmato ve = for whch the true state ath y = y y s ow Defe: A j B = # tmes state trasto j occurs y = # tmes state y emts We ca show that the mamum lelhood arameters are: a b ML j ML # j # # # T t T t j t t T y t t T What f s cotuous? We ca treat t y t : t : T : as T observatos of e.g. a aussa ad aly learg rules for aussa y y t t y t y t ' A j' j B B A ' j' Erc CMU

47 Suervsed ML estmato ctd. Ituto: Whe we ow the uderlyg states the best estmate of s the average frequecy of trastos & emssos that occur the trag data Drawbac: ve lttle data there may be overfttg: P s mamzed but s ureasoable 0 robabltes VERY BAD Eamle: ve 0 caso rolls we observe = y = 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 =. Erc CMU

48 Pseudocouts Soluto for small trag sets: Add seudocouts A j B = # tmes state trasto j occurs y + R j = # tmes state y emts + S R j S j are seudocouts reresetg our ror belef Total seudocouts: R = j R j S = S --- "stregth" of ror belef --- total umber of magary staces the ror Larger total seudocouts strog ror belef Small total seudocouts: just to avod 0 robabltes --- smoothg Ths s equvalet to Bayesa est. uder a uform ror wth "arameter stregth" equals to the seudocouts Erc CMU

49 Summary: Learg M For fully observed B the log-lelhood fucto decomoses to a sum of local terms oe er ode; thus learg s also factored Structural learg Chow lu eghborhood selecto Learg sgle-ode M desty estmato: eoetal famly dst. Tycal dscrete dstrbuto Tycal cotuous dstrbuto Cojugate rors Learg two-ode B: LIM Codtoal Desty Est. Classfcato Learg B wth more odes Local oeratos Erc CMU

50 Sulemetal revew: Erc CMU

51 Two ode fully observed Bs Codtoal mtures Lear/Logstc Regresso Classfcato eeratve ad dscrmatve aroaches Q X Q X Erc CMU

52 Classfcato: oal: Wsh to lear f: X Y eeratve: Modelg the jot dstrbuto of all data Dscrmatve: Modelg oly ots at the boudary Erc CMU

53 Codtoal aussa The data: y y y y 3 3 Both odes are observed: Y s a class dcator vector M: Y X y mult y : X s a codtoal aussa varable wth a class-secfc mea y e - / y - y : Erc CMU y

54 Data log-lelhood MLE log log ; y y y D θ l ; ma y D rg a MLE MLE θ l the fracto of samles of class m MLE MLE y y y D ; ma arg θ l the average of samles of class m MLE of codtoal aussa M: Y X Erc CMU

55 Pror: Posteror mea Bayesa est. Bsyesa estmato of codtoal aussa M: Y X : Dr P : ormal P K Bayes ML Bayes ad / / / / / / ML Bayes Erc CMU

56 Classfcato aussa Dscrmatve Aalyss: The jot robablty of a datum ad t label s: ve a datum we redct ts label usg the codtoal robablty of the label gve the datum: Ths s basc ferece troduce evdece ad the ormalze / - - e y y y ' ' / ' / - - e - - e y Y X Erc CMU

57 Trasductve classfcato ve X what s ts corresodg Y whe we ow the aswer for a set of trag data? Frequetst redcto: we ft ad from data frst ad the Bayesa: we comute the osteror dst. of the arameters frst M: Y X K Y X ' ' ' y y Erc CMU

58 Lear Regresso The data: y y y y Both odes are observed: X s a ut vector Y s a resose vector we frst cosder y as a geerc cotuous resose vector the we cosder the secal case of classfcato where y s a dscrete dcator 3 3 Y X A regresso scheme ca be used to model y drectly rather tha y Erc CMU

59 A dscrmatve robablstc model Let us assume that the target varable ad the uts are related by the equato: where ε s a error term of umodeled effects or radom ose ow assume that ε follows a aussa 0σ the we have: By deedece assumto: T y e ; T y y T y y L e ; Erc CMU

60 Lear regresso Hece the log-lelhood s: T l log y Do you recogze the last term? Yes t s: T J y It s same as the MSE! Erc CMU

61 A reca: LMS udate rule Pros: o-le low er-ste cost Cos: coordate maybe slow-covergg Steeest descet Pros: fast-covergg easy to mlemet Cos: a batch ormal equatos t t * t t y T t Pros: a sgle-shot algorthm! Easest to mlemet. Cos: eed to comute seudo-verse X T X - eesve umercal ssues e.g. matr s sgular.. T t y T T X X X y Erc CMU

62 Bayesa lear regresso Erc CMU

63 Smle Ms are the buldg blocs of comle Bs Desty estmato Parametrc ad oarametrc methods Regresso Lear codtoal mture oarametrc Classfcato eeratve ad dscrmatve aroach X X Q X X Y Q X Erc CMU