Machine Learning. Graphical Models and Exact Inference. Eric Xing , Fall Lecture 17, November 7, Receptor A X 2.
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1 Min Lrning Fll 2016 Grpil Mols n Ext Inrn Eri Xing Rptor Kins X 1 Rptor X 2 X 3 Kins D X 4 Kins E X 5 Ltur 17 Novr TF F X 6 Gn G X 7 Gn H X 8 Ring: p. 8. ook 1
2 Rp o si ro. onpts Rprsnttion: wt is t joint proility ist. on ultipl vrils? X X X X X X X X 8 How ny stt onigurtions in totl? r ty ll n to rprsnt? Do w gt ny sintii/il insigt? Lrning: wr o w gt ll tis proilitis? Mxil-liklioo stition? ut ow ny t o w n? r tr otr st. prinipls? Wr o w put oin knowlg in trs o plusil rltionsips twn vrils n plusil vlus o t proilitis? G F D H E Inrn: I not ll vrils r osrvl ow to oput t onitionl istriution o ltnt vrils givn vin? oputing ph woul rquir suing ovr ll 2 6 onigurtions o t unosrv vrils 2
3 Wt is Grpil Mol? --- Multivrit Distriution in Hig-D Sp possil worl or llulr signl trnsution: Rptor Rptor X 1 X 2 Kins X 3 Kins D X 4 Kins E X 5 TF F X 6 Gn G X 7 Gn H X 8 3
4 GM: Strutur Sipliis Rprsnttion Dpnnis ong vrils Rptor X 1 Rptor X 2 Mrn Kins X 3 Kins D X 4 Kins E X 5 ytosol TF F X 6 Gn G X 7 Gn H X 8 4
5 roilisti Grpil Mols I X i 's r onitionlly inpnnt s sri y GM t joint n tor to prout o siplr trs.g. Rptor Kins X 1 Rptor X 2 X 3 Kins D X 4 Kins E X 5 TF F X 6 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 = X 1 X 2 X 3 X 1 X 4 X 2 X 5 X 2 X 6 X 3 X 4 X 7 X 6 X 8 X 5 X 6 Gn G X 7 Gn H X 8 Sty tun or wt r ts inpnnis! Wy w y vor GM? Inorportion o oin knowlg n usl logil struturs =18 16-ol rution ro 2 8 in rprsnttion ost! 5
6 GM: Dt Intgrtion Rptor X 1 Rptor X 2 Kins X 3 Kins D X 4 Kins E X 5 TF F X 6 Gn G X 7 Gn H X 8 6
7 Mor Dt Intgrtion Txt + Ig + Ntwork Holisti Soil Mi Gno + roto + Trnsrito + no + noi iology 7
8 roilisti Grpil Mols I X i 's r onitionlly inpnnt s sri y GM t joint n tor to prout o siplr trs.g. Rptor Kins X 1 Rptor X 2 X 3 Kins D X 4 Kins E X 5 TF F X 6 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 = X 2 X 4 X 2 X 5 X 2 X 1 X 3 X 1 X 6 X 3 X 4 X 7 X 6 X 8 X 5 X 6 Gn G X 7 Gn H X 8 Wy w y vor GM? Inorportion o oin knowlg n usl logil struturs =36 n 8-ol rution ro 2 8 in rprsnttion ost! Moulr ointion o trognous prts t usion 8
9 Rtionl Sttistil Inrn T ys Tor: ostrior proility p H Liklioo p p p p rior proility Su ovr sp o ypotss Tis llows us to ptur unrtinty out t ol in prinipl wy ut ow n w spiy n rprsnt oplit ol? Typilly t nur o gns n to ol r in t orr o tousns! 9
10 GM: MLE n ysin Lrning roilisti sttnts o is onition on t vlus o t osrv vrils os n prior p p D E D E F F G H G H DE =TFFTF = DE =TFTTF.. DE =FTTTF ys Θ ΘΘp postrior liklioo priorθ D F D ΘΘΘp ; p p ; 10
11 roilisti Grpil Mols I X i 's r onitionlly inpnnt s sri y GM t joint n tor to prout o siplr trs.g. Rptor Kins X 1 Rptor X 2 X 3 Kins D X 4 Kins E X 5 TF F X 6 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 = X 1 X 2 X 3 X 1 X 4 X 2 X 5 X 2 X 6 X 3 X 4 X 7 X 6 X 8 X 5 X 6 Gn G X 7 Gn H X 8 Wy w y vor GM? Inorportion o oin knowlg n usl logil struturs =36 n 8-ol rution ro 2 8 in rprsnttion ost! Moulr ointion o trognous prts t usion ysin ilosopy Knowlg ts t 11
12 So Wt Is GM tr ll? In nutsll: GM = Multivrit Sttistis + Strutur GM = Multivrit Oj. Fun. + Strutur 12
13 So Wt Is GM tr ll? T inorl lur: It is srt wy to writ/spiy/opos/sign xponntilly-lrg proility istriutions witout pying n xponntil ost n t t s ti now t istriutions wit strutur sntis or orl sription: It rrs to ily o istriutions on st o rno vrils tt r optil wit ll t proilisti inpnn propositions no y grp tt onnts ts vrils F G H E D F G H E D F G H E D F G H E D F G H E D X X X X X X X X : X X X X X X X X X X X X X X X X X X 13
14 Two typs o GMs Dirt gs giv uslity rltionsips ysin Ntwork or Dirt Grpil Mol: X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 = X 1 X 2 X 3 X 1 X 4 X 2 X 5 X 2 X 6 X 3 X 4 X 7 X 6 X 8 X 5 X 6 Rptor Kins Gn G X 1 Rptor X 2 X 3 Kins D X 4 Kins E X 5 TF F X 6 X 7 Gn H X 8 Unirt gs siply giv orrltions twn vrils Mrkov Rno Fil or Unirt Grpil ol: X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 = 1/Z xp{ex 1 +EX 2 +EX 3 X 1 +EX 4 X 2 +EX 5 X 2 + EX 6 X 3 X 4 +EX 7 X 6 +EX 8 X 5 X 6 } Rptor Kins Gn G X 1 Rptor X 2 X 3 Kins D X 4 Kins E X 5 TF F X 6 X 7 Gn H X 8 14
15 Towrs struturl spiition o proility istriution Sprtion proprtis in t grp iply inpnn proprtis out t ssoit vrils For t grp to usul ny onitionl inpnn proprtis w n riv ro t grp soul ol or t proility istriution tt t grp rprsnts T Equivln Tor For grp G Lt D 1 not t ily o ll istriutions tt stisy IG Lt D 2 not t ily o ll istriutions tt tor oring to G Tn D 1 D 2. 15
16 ysin Ntworks Strutur: DG Mning: no is onitionlly inpnnt o vry otr no in t ntwork outsi its Mrkov lnkt Lol onitionl istriutions D n t DG opltly trin t joint ist. Y 1 Y 2 il X nstor rnt Giv uslity rltionsips n ilitt gnrtiv pross ilrn's o-prnt Dsnnt 16
17 Mrkov Rno Fils Strutur: unirt grp Mning: no is onitionlly inpnnt o vry otr no in t ntwork givn its Dirt nigors Lol ontingny untions potntils n t liqus in t grp opltly trin t joint ist. Y 1 Y 2 X Giv orrltions twn vrils ut no xpliit wy to gnrt spls 17
18 GMs r your ol rins Dnsity stition rtri n nonprtri tos Rgrssion Linr onitionl ixtur nonprtri lssiition Gnrtiv n isriintiv ppro lustring X s X Q X X Y Q X 18
19 n inoplt gnlogy o grpil ols itur y Zouin Grni n S Rowis 19
20 Fnir GMs: in trnsltion SMT T HM-iTM ol. Zo n E. Xing L
21 Fnir GMs: soli stt pysis Ising/otts ol 21
22 Gnrtiv S or ol sign 22
23 Wy grpil ols lngug or ounition lngug or oputtion lngug or vlopnt Origins: Wrigt 1920 s Inpnntly vlop y Spiglltr n Luritzn in sttistis n rl in oputr sin in t lt 1980 s 23
24 Wy grpil ols roility tory provis t glu wry t prts r oin nsuring tt t syst s wol is onsistnt n proviing wys to intr ols to t. T grp torti si o grpil ols provis ot n intuitivly ppling intr y wi uns n ol igly-intrting sts o vrils s wll s t strutur tt lns itsl nturlly to t sign o iint gnrl-purpos lgorits. Mny o t lssil ultivrit proilisti systs stui in ils su s sttistis systs nginring inortion tory pttrn rognition n sttistil nis r spil ss o t gnrl grpil ol orlis T grpil ol rwork provis wy to viw ll o ts systs s instns o oon unrlying orlis. --- M. Jorn 24
25 ysin Ntwork: Ftoriztion Tor Rptor Kins X 1 Rptor X 2 X 3 Kins D X 4 Kins E X 5 TF F X 6 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 = X 1 X 2 X 3 X 1 X 4 X 2 X 5 X 2 X 6 X 3 X 4 X 7 X 6 X 8 X 5 X 6 Gn G X 7 Gn H X 8 Tor: Givn DG T ost gnrl or o t proility istriution tt is onsistnt wit t proilisti inpnn proprtis no in t grp tors oring to no givn its prnts : X X i X i i wr X i is t st o prnts o xi. is t nur o nos vrils in t grp. 25
26 Expl: pigr o popl Gnti igr g g F0 Fg F1 M 0 M 1 S g g
27 Spiition o N Tr r two oponnts to ny GM: t qulittiv spiition t quntittiv spiition D E G F H D F D
28 Qulittiv Spiition Wr os t qulittiv spiition o ro? rior knowlg o usl rltionsips rior knowlg o oulr rltionsips ssssnt ro xprts Lrning ro t W siply link rtin rittur.g. lyr grp 28
29 ysin Ntwork: Ftoriztion Tor Rptor Kins X 1 Rptor X 2 X 3 Kins D X 4 Kins E X 5 TF F X 6 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 = X 1 X 2 X 3 X 1 X 4 X 2 X 5 X 2 X 6 X 3 X 4 X 7 X 6 X 8 X 5 X 6 Gn G X 7 Gn H X 8 Tor: Givn DG T ost gnrl or o t proility istriution tt is onsistnt wit t proilisti inpnn proprtis no in t grp tors oring to no givn its prnts : X X i X i i wr X i is t st o prnts o xi. is t nur o nos vrils in t grp. 29
30 Lol Struturs & Inpnnis oon prnt Fixing oupls n "givn t lvl o gn t lvls o n r inpnnt" s Knowing oupls n "givn t lvl o gn t lvl gn provis no xtr prition vlu or t lvl o gn " V-strutur Knowing oupls n us n "xplin wy" w.r.t. "I orrlts to tn n or to lso orrlt to will rs" T lngug is opt t onpts r ri! 30
31 sipl justiition 31
32 Grp sprtion ritrion D-sprtion ritrion or ysin ntworks D or Dirt gs: Dinition: vrils x n y r D-sprt onitionlly inpnnt givn z i ty r sprt in t orliz nstrl grp Expl: 32
33 Lol Mrkov proprtis o DGs Strutur: DG Mning: no is onitionlly inpnnt o vry otr no in t ntwork outsi its Mrkov lnkt Lol onitionl istriutions D n t DG opltly trin t joint ist. Y 1 Y 2 il X nstor rnt Giv uslity rltionsips n ilitt gnrtiv pross ilrn's o-prnt Dsnnt 33
34 Glol Mrkov proprtis o DGs X is -sprt irt-sprt ro Z givn Y i w n't sn ll ro ny no in X to ny no in Z using t "ysll" lgorit illustrt llow n plus so ounry onitions: Dn: IGll inpnn proprtis tt orrspon to - sprtion: I G X Z Y : sp G X ; Z Y D-sprtion is soun n oplt 34
35 Expl: x 4 oplt t IG o tis grp: x 1 x 3 x 2 Essntilly: N is ts o r. Inpnn sttnts ong vrils. 35
36 Towrs quntittiv spiition o proility istriution Sprtion proprtis in t grp iply inpnn proprtis out t ssoit vrils For t grp to usul ny onitionl inpnn proprtis w n riv ro t grp soul ol or t proility istriution tt t grp rprsnts T Equivln Tor For grp G Lt D 1 not t ily o ll istriutions tt stisy IG Lt D 2 not t ily o ll istriutions tt tor oring to G Tn D 1 D 2. 36
37 onitionl proility tls Ts = D
38 onitionl proility nsity un. Ds ~Nμ Σ ~Nμ Σ. = ~N+ Σ D D D~Nμ + Σ D 38
39 onitionl Inpnnis Y Ll X 1 X 2 X n-1 X n Fturs Wt is tis ol 1. Wn Y is osrv? 2. Wn Y is unosrv? 39
40 onitionlly Inpnnt Osrvtions Mol prtrs X 1 X 2 X n-1 X n Dt = {y 1 y n } 40
41 lt Nottion Mol prtrs X i Dt = {x 1 x n } i=1:n lt = rtngl in grpil ol vrils witin plt r rplit in onitionlly inpnnt nnr 41
42 Expl: Gussin Mol Gnrtiv ol: x i i=1:n px 1 x n = px i = pt prtrs = pd wr = { } Liklioo = pt prtrs = p D = L Liklioo tlls us ow likly t osrv t r onition on prtiulr stting o t prtrs Otn sir to work wit log L 42
43 ysin ols x i i=1:n 43
44 Sury Rprsnt pnny strutur wit irt yli grp No <-> rno vril Egs no pnnis sn o g -> onitionl inpnn lt rprsnttion GM is ts o pro. Inpnn sttnt on vrils T toriztion tor o t joint proility Lol spiition glolly onsistnt istriution Lol rprsnttion or xponntilly oplx stt-sp It is srt wy to writ/spiy/opos/sign xponntilly-lrg proility istriutions witout pying n xponntil ost n t t s ti now t istriutions wit strutur sntis Support iint inrn n lrning 44
45 Inrn n Lrning W now v opt rprsnttions o proility istriutions: N N M sris uniqu proility istriution Typil tsks: Tsk 1: How o w nswr quris out? W us inrn s n or t pross o oputing nswrs to su quris Tsk 2: How o w stit plusil ol M ro t D? i. W us lrning s n or t pross o otining point stit o M. ii. ut or ysin ty sk pm D wi is tully n inrn prol. iii. Wn not ll vrils r osrvl vn oputing point stit o M n to o inrn to iput t issing t. 45
46 Inrntil Qury 1: Liklioo Most o t quris on y sk involv vin Evin x v is n ssignnt o vlus to st X v o nos in t GM ovr vril st X={X 1 X 2 X n } Witout loss o gnrlity X v ={X k+1 X n } Writ X H =X\X v s t st o in vrils X H n or X Siplst qury: oput proility o vin x v X x H H X v x x x x 1 x k 1 k v tis is otn rrr to s oputing t liklioo o x v 46
47 Inrntil Qury 2: onitionl roility Otn w r intrst in t onitionl proility istriution o vril givn t vin X H X V x V XH x x V V x H X X H H x x V H x V tis is t postriori li in X H givn vin x v W usully qury sust Y o ll in vrils X H ={YZ} n "on't r" out t rining Z: Y x Y Z z V x V z t pross o suing out t "on't r" vrils z is ll rginliztion n t rsulting Yx v is ll rginl pro. 47
48 pplitions o postriori li rition: wt is t proility o n outo givn t strting onition? t qury no is snnt o t vin Dignosis: wt is t proility o iss/ult givn syptos t qury no n nstor o t vin Lrning unr prtil osrvtion? ill in t unosrv vlus unr n "EM" stting or ltr T irtionlity o inortion low twn vrils is not rstrit y t irtionlity o t gs in GM proilisti inrn n oin vin or ll prts o t ntwork 48
49 Inrntil Qury 3: Most rol ssignnt In tis qury w wnt to in t ost prol joint ssignnt M or so vrils o intrst Su rsoning is usully pror unr so givn vin x v n ignoring t vlus o otr vrils Z: Y * x V rg x y Y x V rg x y z Y Z z x V tis is t xiu postriori onigurtion o Y. 49
50 oplxity o Inrn T: oputing X H =x H x v in n ritrry GM is N-r Hrnss os not n w nnot solv inrn It iplis tt w nnot in gnrl prour tt works iintly or ritrry GMs For prtiulr ilis o GMs w n v provly iint prours 50
51 ppros to inrn Ext inrn lgorits T liintion lgorit li propgtion T juntion tr lgorits ut will not ovr in til r pproxit inrn tniqus Vritionl lgorits Stosti siultion / spling tos Mrkov in Mont rlo tos 51
52 oo w: Qury: y in oposition w gt Mrginliztion n Eliintion 52 g g D E F G H nïv sution ns to nurt ovr n xponntil nur o trs Wt is t proility tt wks r lving givn tt t grss onition is poor? g g
53 Qury: N to liint: DEFGH Initil tors: oos n liintion orr: HGFED Stp 1: onitioning ix t vin no i.. on its osrv vlu i.. : Tis stp is isoorpi to rginliztion stp: D E F G H g ~ p ~ p ~ D E F G Vril Eliintion 53
54 Qury: N to liint: DEFG Initil tors: Stp 2: Eliint G oput D E F G H g g 1 g g g p D E F g Expl: Vril Eliintion 54
55 Qury: N to liint: DEF Initil tors: Stp 3: Eliint F oput D E F G H Expl: Vril Eliintion g g p D E 55
56 D E Qury: N to liint: DE Initil tors: Stp 4: Eliint E oput D E F G H Expl: Vril Eliintion g g p D 56
57 Qury: N to liint: D Initil tors: Stp 5: Eliint D oput D E F G H Expl: Vril Eliintion g g p 57
58 Qury: N to liint: Initil tors: Stp 6: Eliint oput D E F G H Expl: Vril Eliintion p g g 58
59 Qury: N to liint: Initil tors: Stp 7: Eliint oput D E F G H Expl: Vril Eliintion g g p 59
60 Qury: N to liint: Initil tors: Stp 8: Wrp-up D E F G H Expl: Vril Eliintion g g ~ p p p p ~ p p ~ 60
61 oplxity o vril liintion Suppos in on liintion stp w oput Tis rquirs x y1 yk ' x x y1 yk ' x x x k 1 yk i x y i i1 k Vl X Vl Y ultiplitions i y i For vlu o x y 1 y k w o k ultiplitions Vl X Vl Y i itions i For vlu o y 1 y k w o VlX itions oplxity is xponntil in nur o vrils in t intrit tor 61
62 Eliintion liqu Inu pnny uring rginliztion is ptur in liintion liqus Sution <-> liintion Intrit tr <-> liintion liqu D D E F n tis l to n gnri inrn lgorit? G E E E H F 62
63 Fro Eliintion to Mssg ssing Eliintion ssg pssing on liqu tr D p g G E g E D E E H F F Mssgs n rus 63
64 Fro Eliintion to Mssg ssing Eliintion ssg pssing on liqu tr notr qury... D D E F G E g E E H F Mssgs n r rus otrs n to roput 64
65 Fro liintion to ssg pssing Rll ELIMINTION lgorit: oos n orring Z in wi qury no is t inl no l ll potntils on n tiv list Eliint no i y roving ll potntils ontining i tk su/prout ovr x i. l t rsultnt tor k on t list For TREE grp: oos qury no s t root o t tr Viw tr s irt tr wit gs pointing towrs ro Eliintion orring s on pt-irst trvrsl Eliintion o no n onsir s ssg-pssing or li ropgtion irtly long tr rns rtr tn on so trnsor grps tus w n us t tr itsl s t-strutur to o gnrl inrn!! 65
66 Mssg pssing or trs Lt ij x i not t tor rsulting ro liinting vrils ro llow up to i wi is untion o x i : i Tis is rinisnt o ssg snt ro j to i. j k l ij x i rprsnts "li" o x i ro x j! 66
67 Eliintion on trs is quivlnt to ssg pssing long tr rns! i j k l 67
68 T ssg pssing protool: two-pss lgorit: X 1 21 X 1 12 X 2 32 X 2 X 2 42 X 2 X 3 X 4 24 X 4 23 X 3 68
69 li ropgtion S-lgorit: Squntil iplnttion 69
70 Inrn on gnrl GM Now wt i t GM is not tr-lik grp? n w still irtly run ssg ssg-pssing protool long its gs? For non-trs w o not v t gurnt tt ssg-pssing will onsistnt! Tn wt? onstrut grp t-strutur ro tt s tr strutur n run ssg-pssing on it! Juntion tr lgorit 70
71 Sury T sipl Eliint lgorit pturs t ky lgoriti Oprtion unrlying proilisti inrn: --- Tt o tking su ovr prout o potntil untions T oputtionl oplxity o t Eliint lgorit n ru to purly grp-torti onsirtions. Tis grp intrprttion will lso provi ints out ow to sign iprov inrn lgorits Wt n w sy out t ovrll oputtionl oplxity o t lgorit? In prtiulr ow n w ontrol t "siz" o t suns tt ppr in t squn o sution oprtion. 71
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