Advanced Machine Learning

Transcription

1 dvaced ache Learg Learg rahcal odels Learg full observed ad arall observed BN rc g Lecure 4 ugus Readg: rc g rc U Iferece ad Learg BN descrbes a uque robabl dsrbuo P Tcal asks: Task : How do we aswer queres abou P? We use ferece as a ae for he rocess of coug aswers o such queres So far we have leared several algorhs for eac ad aro. ferece Task : How do we esae a lausble odel fro daa D?. We use learg as a ae for he rocess of obag o esae of.. Bu for Baesa he seek D whch s acuall a ferece roble.. Whe o all varables are observable eve coug o esae of eed o do ferece o ue he ssg daa. rc g rc U

2 Learg rahcal odels The goal: ve se of deede sales assges of rado varables fd he bes he os lkel? grahcal odel boh he grah ad he PDs B B R R BRTFFTF BRTFTTF.. BRFTTTF rc g rc U e e e e B b b b b P B Learg rahcal odels Scearos: coleel observed s dreced udreced arall observed s dreced udreced a oe research oc sao rcles: aal lkelhood esao L Baesa esao aal codoal lkelhood aal "arg" We use learg as a ae for he rocess of esag he araeers ad soe cases he oo of he ework fro daa. rc g rc U

3 Score-based aroach Daa Possble srucures B Lear araeers Score sruc/ara R au lkelhood 0 5 K K K K R B Baesa odoal lkelhood arg K. rc g rc U Z L Paraeer s. for coleel observed s of gve srucure The daa: { N N } rc g rc U

4 4 rc g rc U Lkelhood for ow le's assue ha he srucure s gve: Log-Lkelhood: Daa -lkelhood L 3 ; L l DT l 3 3 arg a } { DT l L 3 arg a arg a a arg * * * The basc dea uderlg L rc g rc U The coleel observed odel: Z s a class dcaor vecor s a codoal aussa varable wh a class-secfc ea Z 0 ad ] [ where Z Z Z Z Z Z K ll ece oe of hese ers wll be oe { } - e - / µ σ σ µ σ N σ µ σ µ ad a dau s class w.. ale : codoal aussa

5 ale : codoal aussa Daa -lkelhood L l D µ σ µ σ * N σ - µ + * * µ arg a l D µ he average of rc g rc U µ σ * arg a l D l D 0 s.. N N Z he fraco of sales of class sales of class ale : H: wo scearos Suervsed learg: esao whe he rgh aswer s kow ales: IVN: IVN: a geoc rego where we have good eereal aoaos of he slads he caso laer allows us o observe h oe eveg as he chages dce ad roduces 0000 rolls Usuervsed learg: esao whe he rgh aswer s ukow ales: IVN: IVN: he orcue geoe; we do kow how freque are he slads here eher do we kow her cooso 0000 rolls of he caso laer bu we do see whe he chages dce QUSTION: Udae he araeers of he odel o ae P --- aal lkelhood L esao rc g rc U

6 Recall defo of H Traso robables bewee a wo saes 3... T a 3... T or a a K a. ~ uloal I Sar robables ~ uloal K. sso robables assocaed wh each sae b b K b. ~ uloal K I or geeral:. ~ f I rc g rc U Suervsed L esao ve N for whch he rue sae ah N s kow Defe: B k # es sae raso occurs # es sae es k We ca show ha he au lkelhood araeers are: a b L L k # # # k # T T : : T : N Wha f s couous? We ca rea as N T observaos of e.g. a aussa ad al learg rules for aussa ' ' T k Bk T B k ' k ' { } rc g rc U

7 Suervsed L esao cd. Iuo: Whe we kow he uderlg saes he bes esae of s he average frequec of rasos & essos ha occur he rag daa Drawback: ve lle daa here a be overfg: P s aed bu s ureasoable 0 robables VRY BD ale: ve 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. rc g rc U Pseudocous Soluo for sall rag ses: dd seudocous B k # es sae raso occurs + R # es sae es k + S k R S are seudocous rereseg our ror belef Toal seudocous: R Σ R S Σ k S k --- "sregh" of ror belef --- oal uber of agar saces he ror Larger oal seudocous srog ror belef Sall oal seudocous: us o avod 0 robables --- soohg Ths s equvale o Baesa es. uder a ufor ror wh "araeer sregh" equals o he seudocous rc g rc U

8 L for geeral BN araeers If we assue he araeers for each PD are globall deede ad all odes are full observed he he lkelhood fuco decooses o a su of local ers oe er ode: l ; D D rc g rc U ale: decoosable lkelhood of a dreced odel osder he dsrbuo defed b he dreced acclc : Ths s eacl lke learg four searae sall BNs each of whch cosss of a ode ad s ares rc g rc U

9 .g.: L for BNs wh abular PDs ssue each PD s rereseed as a able uloal where def k Noe ha case of ulle ares wll have a coose sae ad he PD wll be a hgh-desoal able The suffce sascs are cous of fal cofguraos The -lkelhood s Usg a Lagrage uller o eforce we ge: k k k def k k l ; D k k k L k ' k k k ' k k rc g rc U Z Learg arall observed s The daa: { 3... N } rc g rc U

10 Wha f soe odes are o observed? osder he dsrbuo defed b he dreced acclc : Need o coue H V ferece rc g rc U Recall: lgorh wa of ag lkelhood fuco for lae varable odels. Fds L of araeers whe he orgal hard roble ca be broke u o wo eas eces:. sae soe ssg or uobserved daa fro observed daa ad curre araeers.. Usg hs colee daa fd he au lkelhood araeer esaes. lerae bewee fllg he lae varables usg he bes guess oseror ad udag he araeers based o hs guess: -se: -se: q + arg af q q + + arg af q I he -se we oe a lower boud o he lkelhood. I he - se we close he ga akg boudlkelhood. rc g rc U

11 for geeral BNs whle o coverged % -se for each ode SS 0 % rese eeced suffce sascs for each daa sale do ferece wh H for each ode SS SS % -se for each ode : LSS + H H rc g rc U ale: H Suervsed learg: esao whe he rgh aswer s kow ales: IVN: a geoc rego where we have good eereal aoaos of he slads IVN: he caso laer allows us o observe h oe eveg as he chages dce ad roduces 0000 rolls Usuervsed learg: esao whe he rgh aswer s ukow ales: IVN: IVN: he orcue geoe; we do kow how freque are he slads here eher do we kow her cooso 0000 rolls of he caso laer bu we do see whe he chages dce QUSTION: Udae he araeers of he odel o ae P - -- aal lkelhood L esao rc g rc U

12 rc g rc U The Bau Welch algorh The colee lkelhood The eeced colee lkelhood The se The se "sbolcall" decal o L T T c ; l + + T k k T c b a ; l γ ξ T T L a γ ξ T k T L k b γ γ N L γ rc g rc U Usuervsed L esao ve N for whch he rue sae ah N s ukow PTTION IIZTION 0. Sarg wh our bes guess of a odel araeers :. sae B k he rag daa How?. Udae accordg o B k Now a "suervsed learg" roble 3. Reea & ul covergece Ths s called he Bau-Welch lgorh We ca ge o a rovabl ore or equall lkel araeer se each erao k k B

13 3 rc g rc U L Srucural Learg for coleel observed s R B Daa K K K K rc g rc U Iforao Theorec Ierreao of L cou D D ˆ ; l Fro su over daa os o su over cou of varable saes

14 4 rc g rc U Iforao Theorec Ierreao of L co'd H I D D ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ; l Decoosable score ad a fuco of he grah srucure rc g rc U Srucural Search How a grahs over odes? How a rees over odes? Bu urs ou ha we ca fd eac soluo of a oal ree uder L! Trck: a ree each ode has ol oe are! how-lu algorh O O!

15 how-lu ree learg algorh Obeco fuco: l ; D ˆ D how-lu: Iˆ Hˆ Iˆ For each ar of varable ad oue ercal dsrbuo: oue uual forao: cou ˆ Iˆ ˆ ˆ ˆ ˆ Defe a grah wh ode dge I ges wegh Iˆ rc g rc U how-lu algorh co'd Obeco fuco: l ; D ˆ D Iˆ Hˆ Iˆ how-lu: Oal ree BN oue au wegh sag ree Dreco BN: ck a ode as roo do breadh-frs-search o defe drecos I-equvalece: D B D B D B I B + I + I D + I rc g rc U

16 Srucure Learg for geeral grahs Theore: The roble of learg a BN srucure wh a os d ares s NP-hard for a fed d os srucure learg aroaches use heurscs lo score decooso Two heurscs ha elo decooso dffere was rc g reed search hrough sace of ode-orders Local search of grah srucures rc U ee resso Proflg b croarras Receor Kase Receor B 3 Kase D Kase 4 Tras. Facor 6 F ee rc g 7 rc U ee H 5 F F 8 3 6

17 croarra Daa hr hr 3hr 4hr rc g rc U Srucure Learg lgorhs Srucural Freda 998 The orgal algorh resso daa Learg lgorh B R Sarse addae lgorh Freda e al. Dscreg arra sgals Hll-clbg search usg local oeraors: add/delee/swa of a sgle edge Feaure eraco: arkov relaos order relaos Re-asseble hgh-cofdece sub-eworks fro feaures odule ework learg Segal e al. Heursc search of srucure a "odule grah" odule assge Paraeer sharg Pror kowledge: ossble regulaors TF gees rc g rc U

18 Scorg Neworks B D Lear R resale B D resale D Lear R resale... B D Lear R rc g rc U Learg srucure Learg of bes PDs gve D s eas collec sascs of values of each ode gve secfc assge o s ares Learg of he grah oo srucure s NP-hard heursc search us be aled geerall leads o a locall oal ework Overfg I urs ou ha rcher srucures gve hgher lkelhood PD o he daa addg a edge s alwas referable B B P P B ore araeers o f > ore freedo > alwas es ore "oal" PD We refer sler ore elaaor eworks Praccal scores regulare he lkelhood rovee cole eworks. rc g rc U

19 Learg sarse ulvarae aussa over all couous eressos [... ] Σ The recso ar KΣ reveals he oo of he udreced ework K / K dge ~ K > 0 - e - T - { } r - µ Σ r - µ Learg lgorh: ovarace seleco Wa a sarse ar Regresso for each ode wh degree cosra Dobra e al. Regresso for each ode wh herarchcal Baesa ror L e al rahcal Lasso we wll descrbe shorl rc g rc U Learg Isg odel.e. arwse RF ssug he odes are dscree ad edges are weghed he for a sale d we have rah lasso has bee used o oba a sarse esae of wh couous We ca use grahcal L_ regulared sc regresso o oba a sarse esae of wh dscree rc g rc U

20 Recall lasso rc g rc U rah Regresso Lasso: rc g rc U

21 rah Regresso rc g rc U rah Regresso rc g rc U

22 ossec Theore: for he grahcal regresso algorh uder cera verfable codos oed here for slc: Noe he fro hs heore oe should see ha he regularer s o acuall used o roduce a arfcal sars bas bu a devse o esure cossec uder fe daa ad hgh deso codo. rc g rc U Learg Learg of bes PDs gve D s eas collec sascs of values of each ode gve secfc assge o s ares Learg of he grah oo srucure s NP-hard heursc search us be aled geerall leads o a locall oal ework We refer sler ore elaaor eworks Regulared grah regresso rc g rc U