Probabilistic Graphical Models

Transcription

1 School of Computer Scence Prolstc Grphcl Models Vrtonl Inference Erc ng Lecture 13, Ferury 24, 2014 Redng: See clss weste Erc CMU,

2 Inference Prolems Compute the lelhood of oserved dt Compute the mrgnl dstruton over prtculr suset of nodes Compute the condtonl dstruton for dsont susets A nd B Compute mode of the densty Methods we hve Brute force Elmnton Messge Pssng Forwrd-cwrd, M-product /BP, Juncton Tree Indvdul computtons ndependent Shrng ntermedte terms Erc CMU,

3 Sum-Product Revsted Tree-structured GMs Messge Pssng on Trees: On trees, converge to unque fed pont fter fnte numer of tertons Erc CMU,

4 Juncton Tree Revsted Generl Algorthm on Grphs wth Cycles Steps: => Trngulrzton => Construct JTs => Messge Pssng on Clque Trees B S C Erc CMU,

5 Locl Consstency Gven set of functons ssocted wth the clques nd seprtor sets They re loclly consstent f: For uncton trees, locl consstency s equvlent to glol consstency! Erc CMU,

6 An Isng model on 2-D mge Nodes encode hdden nformton ptchdentty. They receve locl nformton from the mge rghtness, color. Informton s propgted though the grph over ts edges. Edges encode comptlty etween nodes. Erc CMU, r or wter?? 6

7 Why Appromte Inference? Why cn t we ust run uncton tree on ths grph? 1 ep Z p 0 If NN grd, tree wdth t lest N N cn e huge numer~1000s of pels If N~O1000, we hve clque wth entres Erc CMU,

8 Approches to nference Ect nference lgorthms The elmnton lgorthm Messge-pssng lgorthm sum-product, elef propgton The uncton tree lgorthms Appromte nference technques Vrtonl lgorthms Loopy elef propgton Men feld ppromton Stochstc smulton / smplng methods Mrov chn Monte Crlo methods Erc CMU,

9 Loopy Belef Propogton Erc CMU,

10 M M M, Comptltes nterctons eternl evdence M Recp: Belef Propgton BP Messge-updte Rules BP on trees lwys converges to ect mrgnls cf. Juncton tree lgorthm 10 Erc CMU,

11 Belefs nd messges n FG N m f elefs messges \ N N c c m f m m N m f m \ \ 11 Erc CMU,

12 Wht f the grph s loopy? Erc CMU,

13 M Belef Propgton on loopy grphs BP Messge-updte Rules My not converge or converge to wrong soluton M M, Comptltes nterctons eternl evdence M 13 Erc CMU,

14 A fed pont terton procedure tht tres to mnmze F ethe Strt wth rndom ntlzton of messges nd elefs Whle not converged do At convergence, sttonrty propertes re gurnteed However, not gurnteed to converge! Loopy Belef Propgton N m N m f N c c new m m \ N new m f m \ \ 14 Erc CMU,

15 Loopy Belef Propgton If BP s used on grphs wth loops, messges my crculte ndefntely But let s run t nywy nd hope for the est Emprclly, good ppromton s stll chevle Stop fter fed # of tertons Stop when no sgnfcnt chnge n elefs If soluton s not osclltory ut converges, t usully s good ppromton Loopy-elef Propgton for Appromte Inference: An Emprcl Study Kevn Murphy, Yr Wess, nd Mchel Jordn. UAI '99 Uncertnty n AI. ] Erc CMU,

16 So wht s gong on? Is t drty hc tht you et your luc? Erc CMU,

17 Appromte Inference Let us cll the ctul dstruton P We wsh to fnd dstruton Q such tht Q s good ppromton to P P 1/ Z f f F Recll the defnton of KL-dvergence KL Q 1 Q 2 Q Q1 log Q 1 2 KLQ 1 Q 2 >=0 KLQ 1 Q 2 =0 ff Q 1 =Q 2 We cn therefore use KL s scorng functon to decde good Q But, KLQ 1 Q 2 KLQ 2 Q 1 Erc CMU,

18 Whch KL? Computng KLP Q requres nference! But KLQ P cn e computed wthout performng nference on P Usng KL Q P H Q Q log P Q log Q Q log P Q E log P P 1/ Z f f F KL Q P H Q EQ log1/ Z F f F f H log1/ Z E log f Q Q f Q Erc CMU,

19 Optmzton functon KL Q P H E log f Q f F Q log Z F P, Q We wll cll F P, Q the Free energy * F P, P =? FP,Q >= FP,P Erc CMU, *Gs Free Energy 19

20 The Energy Functonl Let us loo t the functonl Q cn e computed f we hve mrgnls over ech f f F E log f H Q Q log Q s hrder! Requres summton over ll possle vlues Computng F, s therefore hrd n generl. Approch 1: Appromte F P, Q wth esy to compute F P, Q H E log f Q f F Q F P, Q Erc CMU,

21 Tree Energy Functonls Consder tree-structured dstruton The prolty cn e wrtten s: H F tree Tree F ln d 1 ln ln 1 f 1d nvolves summton over edges nd vertces nd s therefore esy to compute d ln 12 F23.. F67 F78 F1 F5 F2 F6 F3 F7 Erc CMU,

22 Bethe Appromton to Gs Free Energy For generl grph, choose H F Bethe Bethe F P, Q F Beth Clled Bethe ppromton fter the physcst Hns Bethe ln d 1 ln ln d f H eth f 1 ln F ethe F 2 F.. F 12 F23.. F67 F78 F1 F5 F2 2 Equl to the ect Gs free energy when the fctor grph s tree In generl, H Bethe s not the sme s the H of tree Erc CMU,

23 Bethe Appromton Pros: Esy to compute, snce entropy term nvolves sum over prwse nd sngle vrles Cons: F P, Q my or my not e well connected to F P, Q F ethe It could, n generl, e greter, equl or less thn Optmze ech 's. For dscrete elef, constrned opt. wth Lgrngn multpler For contnuous elef, not yet generl formul Not lwys converge F P, Q Erc CMU,

24 Bethe Free Energy for FG Beth d f F ln ln 1 Bethe d H ln ln 1 eth Bethe H f F 24 Erc CMU,

25 Mnmzng the Bethe Free Energy Set dervtve to zero N Bethe F L \ } {1 25 Erc CMU,

26 N Bethe F L \ 1} { 0 L 1 1 ep N d 0 L ep N E Constrned Mnmzton of the Bethe Free Energy 26 Erc CMU,

27 Bethe = BP on FG We hd: Identfy to otn BP equtons: N m f elefs messges \ N N c c m f The elef s the BP ppromton of the mrgnl prolty. 1 1 ep N d log ep N f N m m log log 27 Erc CMU,

28 Usng, \ we get N N m f m \ \ \ = BP Messge-updte Rules A sum product lgorthm 28 Erc CMU,

29 Summry so fr F f f Z P 1/ F f Q Q f E H Q P F log, d f Q P F log 1 log, 1 1 ep N d log ep N f 29 Erc CMU,

30 The Theory Behnd LBP For dstruton p ssocted wth comple grph, computng the mrgnl or condtonl prolty of rtrry rndom vrles s ntrctle Vrtonl methods formultng prolstc nference s n optmzton prolem: q * rg mn qs F Beth p, q F Bethe f ln 1 d ln f H ethe q : trctle prolty dstruton Erc CMU,

31 The Theory Behnd LBP But we do not optmze q eplctly, focus on the set of elefs e.g., {,,, } Rel the optmzton prolem ppromte oectve: reled fesle set: H q F M M M M o o The loopy BP lgorthm: * rg mn M fed pont terton procedure tht tres to solve * o E F Erc CMU,

32 The Theory Behnd LBP But we do not optmze q eplctly, focus on the set of elefs e.g., {,,, } Rel the optmzton prolem ppromte oectve: reled fesle set: H H,, M o Beth 0 1,, The loopy BP lgorthm: * rgmn M fed pont terton procedure tht tres to solve * o E F Erc CMU,

33 Men Feld Appromton Erc CMU,

34 Nïve Men Feld Fully fctorzed vrtonl dstruton Erc CMU,

35 Nïve Men Feld for Isng Model Optmzton Prolem Updte Rule resemles messge sent from node to forms the men feld ppled to from ts neghorhood Erc CMU,

36 Men feld methods Optmze q H n the spce of trctle fmles.e., sugrph of G p over whch ect computton of H q s fesle Tghtenng the optmzton spce ect oectve: tghtened fesle set: H q Q T T Q q * rg mn qt E q H q Erc CMU,

37 Cluster-sed ppro. to the Gs free energy Wegernc 2001, ng et l 03,04 Ect: G[ p ] ntrctle Clusters: G[{ q c c }] Erc CMU,

38 Men feld ppro. to Gs free energy Gven dsont clusterng, {C 1,, C I }, of ll vrles Let Men-feld free energy G MF q q C, E q ln q q C C C q q q ln q e.g., GMF q C C C nïve men feld Wll never equl to the ect Gs free energy no mtter wht clusterng s used, ut t does lwys defne lower ound of the lelhood Optmze ech q c 's. Vrtonl clculus Do nference n ech q c usng ny trctle lgorthm Erc CMU,

39 The Generlzed Men Feld theorem Theorem: The optmum GMF ppromton to the cluster mrgnl s somorphc to the cluster posteror of the orgnl dstruton gven nternl evdence nd ts generlzed men felds: * q H, C p H, C E, C, H, MB q GMF lgorthm: Iterte over ech q Erc CMU,

40 A generlzed men feld lgorthm [ng et l. UAI 2003] Erc CMU,

41 A generlzed men feld lgorthm [ng et l. UAI 2003] Erc CMU,

42 Convergence theorem Theorem: The GMF lgorthm s gurnteed to converge to locl optmum, nd provdes lower ound for the lelhood of evdence or prtton functon the model. Erc CMU,

43 The nve men feld ppromton Appromte p y fully fctorzed q=p q For Boltzmnn dstruton p=ep{ < q +q o }/Z : men Gs feld predctve equton: dstruton: pq ep ep 0 0 N N p p { { : : N} }} { qq q A q q { q : : N N } resemles messge sent from node to { q } forms the men feld ppled to from ts neghorhood Erc CMU,

44 Emple 1: Generlzed MF ppromtons to Isng models Cluster mrgnl of squre loc C : q C ep 0, C C C, MB, ' MBC q C ' Vrtully reprmeterzed Isng model of smll sze. Erc CMU,

45 GMF ppromton to Isng models GMF 44 GMF 22 BP Erc CMU, Attrctve couplng: postvely weghted Repulsve couplng: negtvely weghted 45

46 Emple 2: Sgmod elef networ GMF GMF r BP Erc CMU,

47 Emple 3: Fctorl HMM Erc CMU,

48 Automtc Vrtonl Inference S 1 S 2 S 3... S N y 11 y 12 y y 1N... y 1... y 2... y y N A 1 A A A N fhmm Men feld ppro. Structured vrtonl ppro. Currently for ech new model we hve to derve the vrtonl updte equtons wrte pplcton-specfc code to fnd the soluton Ech cn e tme consumng nd error prone Cn we uld generl-purpose nference engne whch utomtes these procedures? Erc CMU,

49 Cluster-sed MF e.g., GMF generl, tertve messge pssng lgorthm clusterng completely defnes ppromton preserves dependences flele performnce/cost trde-off clusterng utomtle recovers model-specfc structured VI lgorthms, ncludng: fhmm, LDA vrtonl Byesn lernng lgorthms esly provdes new structured VI ppromtons to comple models Erc CMU,