Exploiting Structure in Probability Distributions Irit Gat-Viks

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1 Explotng Structure n rolty Dstrutons Irt Gt-Vks Bsed on presentton nd lecture notes of Nr Fredmn, Herew Unversty

2 Generl References: D. Koller nd N. Fredmn, prolstc grphcl models erl, rolstc Resonng n Intellgent Systems Jensen, An Introducton to Byesn Networks Heckermn, A tutorl on lernng wth Byesn networks

3 Exmple I Reltonshps n oesty Trglycerds n lowdensty lpoprotens LDLs Cholesterol n lowdensty lpoprotens LDLs Hert rte

4 Exmple II The currently ccepted consensus network of humn prmry CD4 T cells, downstrem of CD3, CD8, nd LFA- 1 ctvton 10 erturtons y smll molecules 11 Colored nodes: Mesured sgnlng protens Cusl roten-sgnlng Networks Derved from Multprmeter Sngle-Cell Dt. Kren Schs, et l. 005.

5 Byesn network results KC->KA ws vldted expermentlly. Akt ws not ffected y Erk n ddtonl experments

6 Exmple III: Byesn network models for trnscrpton fctor-dna ndng motf wth 5 postons SSM Byesn network

7 Exmple IV: Dgnostc Byesn network model

8 Bsc rolty Defntons roduct Rule: A,BA B*B B A*A Independence etween A nd B: A,BA*B, or lterntvely: ABA, BAB. Totl prolty theorem: n U B Ω, j 1 j BI B φ n A A, B B * A B n 1 1

9 Bsc rolty Defntons Byes Rule: A B B A A B A B, C B A, C A C B C Chn Rule:,..., 1 n,...,,...,,...,... 1 n 3 n 3 4 n n 1 n n

10 Explotng Independence roperty G: whether the womn s pregnnt D: whether the doctor s test s postve The jont dstruton representton g,d: G D G,D Fctorl representton Usng condtonl prolty: g,dg*dg. The dstruton of g, dg: G g0 g dg d0 d1 D g0 g Exmple: g0,d10.06 vs. g0*d1g00.6*

11 Explotng Independence roperty H: home test Independence ssumpton: IndH;DG.e., gven G, H s ndependent of D. Jont dstruton d,h,gd,hg*gdg* hg*g Fctorl representton roduct rule IndH;DG G g0 g h0 h1 d0 d1 g H D g g g

12 Explotng Independence roperty No. of prmeters Addng new vrle H representton of d,g,h jont dstruton 7 chngng the dstruton entrely fctored dstruton 5 Modulrty: reuse the locl prolty model. Only new locl prolty model for H. G g0 g Locl prolty g0 h0 0.9 h1 0.1 H D g0 d0 0.9 d1 0.1 g g > Byesn networks: Explotng ndependence propertes of the dstruton n order to llow compct nd nturl representton.

13 Outlne Introducton Byesn Networks» Representton & Semntcs Inference n Byesn networks Lernng Byesn networks

14 Representng the Uncertnty Erthquke Burglry A story wth fve rndom vrles: Burglry, Erthquke, Alrm, Neghor Cll, Rdo Announcement Rdo Specfy jont dstruton wth 5 3 prmeters Alrm Cll mye An expert system for montorng ntensve cre ptents Specfy jont dstruton over 37 vrles wth t lest 37 prmeters no wy!!!

15 rolstc Independence: Key for Representton nd Resonng Recll tht f nd Y re ndependent gven Z then In our story f urglry nd erthquke re ndependent lrm sound nd rdo re ndependent gven erthquke urglry nd rdo re ndependent gven erthquke then nsted of 15 prmeters we need 8,,,,,, B B E B E R B E R A B E R A,,,, B E E R B E A B E A R, Y Y Z versus Need lnguge to represent ndependence sttements

16 Mrkov Assumpton Ancestor Generlzng: A chld s condtonlly ndependent from ts non-descendents, gven the vlue of ts prents. Ind ; NonDescendnt Y 1 Y rent It s nturl ssumpton for mny cusl processes Descendent Non-descendent Descendent

17 Mrkov Assumpton cont. Exmples: R s ndependent of A, B, C, gven E A s ndependent of R, gven B nd E C s ndependent of B, E, R, gven A Erthquke Rdo Burglry Alrm Cll

18 Byesn Network Semntcs R E A C B Qulttve prt condtonl ndependence sttements n BN structure Quntttve prt locl prolty + Models e.g., multnoml, lner Gussn Unque jont dstruton over domn Compct & effcent representton: nodes hve k prents O k n vs. O n prms prmeters pertn to locl nterctons C,A,R,E,B B*EB*RE,B*AR,B,E*CA,R,B,E versus C,A,R,E,B B*E * RE * AB,E * CA In generl: x1,..., x x n x 1,.., n

19 Byesn networks Effcent representton of prolty dstrutons v condtonl ndependence Qulttve prt: sttstcl ndependence sttements Drected cyclc grph DAG Nodes - rndom vrles of nterest exhustve nd mutully exclusve sttes Edges - drect nfluence Erthquke Rdo Burglry Alrm Cll E e e e e B A E,B Quntttve prt: Locl prolty models. Set of condtonl prolty dstrutons.

20 Outlne Introducton Byesn Networks Representton & Semntcs» Inference n Byesn networks Lernng Byesn networks

21 Inference n Byesn networks A Byesn network represents prolty dstruton. Cn we nswer queres out ths dstruton? Exmples: YZz Most prole estmton Mxmum posteror ME W Z z rg mx w w, z MA Y Z z rg mx y z y

22 Inference n Byesn networks Gol: compute Ee,A n the followng Byesn network: A B C D E Usng defnton of prolty, we hve, e,, c, d, e c d c d c d c e d

23 Elmntng d, we get A B C E D c c d c d c e F c d e c d c d e c d c e, 1, Inference n Byesn networks ec

24 Elmntng c, we get A B C E D c c e F c e c c e c e,, Inference n Byesn networks pe

25 Fnlly, we elmnte A B C E D, 3, e F e p e p e Inference n Byesn networks pe

26 Vrle Elmnton Algorthm Generl de: Wrte query n the form x L x p 1 x x x k 3 Itertvely Move ll rrelevnt terms outsde of nnermost sum erform nnermost sum, gettng new term Insert the new term nto the product In cse of evdence x 1 evdence x j, use: x x x, x / x j j j

27 Complexty of nference Nïve exct nference exponentl n the numer of vrles n the network Vrle elmnton complexty exponentl n the sze of lrgest fctor polynoml n the numer of vrles n the network Vrle elmnton computton depend on order of elmnton mny heurstcs, e.g., clque tree lgorthm.

28 Outlne Introducton Byesn Networks Representton & Semntcs Inference n Byesn networks» Lernng Byesn networks rmeter Lernng Structure Lernng

29 Lernng rocess Input: dtset nd pror nformton Output: Byesn network E B Dt + ror nformton Lernng R A C

30 The Lernng rolem Complete Dt Incomplete Dt Known Structure Sttstcl prmetrc estmton closed-form eq. rmetrc optmzton EM, grdent descent... Unknown Structure Dscrete optmzton over structures dscrete serch Comned Structurl EM, mxture models We wll focus on complete dt for the rest of the tlk The stuton wth ncomplete dt s more nvolved

31 Outlne Introducton Byesn Networks Representton & Semntcs Inference n Byesn networks Lernng Byesn networks»rmeter Lernng Structure Lernng

32 Lernng rmeters Key concept: the lkelhood functon L θ : D D θ x[ m] θ mesures how the prolty of the dt chnges when we chnge prmeters m Estmton: MLE: choose prmeters tht mxmze lkelhood Byesn: tret prmeters s n unknown quntty, nd mrgnlze over t

33 MLE prncple for Bnoml Dt Dt: H, H, T, H, H. Θs the unknown prolty H. Nk Lkelhood functon: L Θ : D θk L θ : D θ θ 1 θ θ θ k 0,1 Estmton tsk: Gven sequence of smples x[1], x[] x[m], we wnt to estmte the prolty H θ nd T1-θ. MLE prncple: choose prmeter tht mxmze the lkelhood functon. Applyng the MLE prncple we get NH θ N + N H MLE for H s 4/5 0.8 T Lθ :D θ 33

34 MLE prncple for Multnoml Dt Suppose cn hve the vlues 1,,,k. We wnt to lern the prmeters θ 1,, θ k. N 1,,N k - The numer of tmes ech outcome s oserved. Lkelhood functon: The MLE s: θ L Θ : D N l 1,..., k N l K k 1 θ N k k Count of k th outcome n D rolty of k th outcome

35 MLE prncple for Byesn networks E B Trnng dt hs the form: Assume..d. smples D E [1] E [ M] B[1] B[ M] A[1] A[ M] C [1] C [ M] A C L Θ: D E[ m], B[ m], A[ m], C[ m] : Θ m By defnton of network m E[ m]: Θ B[ m]: Θ A[ m] B[ m], E[ m]: Θ C[ m] A[ m]: Θ m m m E[ m]: Θ B[ m]: Θ A[ m] B[ m], E[ m]: Θ m C[ m] A[ m]: Θ E[1] E [ M] B[1] B[ M] A[1] A[ M] C[1] C[ M] E [1] E [ M ] B [1] B [ M ] A[1] A[ M ] C [1] C [ M ]

36 MLE prncple for Byesn networks Generlzng for ny Byesn network: L Θ : D x [ m] [ m]: Θ L Θ : D m L θ : D x [ m] [ m]: θ m x p : θ p x N x N x, p, p x p p x θ The lkelhood decomposes ccordng to the network structure. Decomposton Independent estmton prolems If the prmeters for ech fmly re not relted For ech vlue p of the prent of we get ndependent multnoml prolem. The MLE s ˆ θx p N x, p N p

37 Contnuous Gussn vrles Dscrete L θ : D x [ m] [ m]: θ m Y Y ~ N µ, σ L Θ : D x [ m] [ m]: Θ L Θ : D m ~ N µ, σ,, The lkelhood decomposes ccordng to the network structure. Decomposton Independent estmton prolems If the prmeters for ech fmly re not relted For ech vlue p of the prent of we get ndependent mxmzton prolem. The MLE s µ σ,, m: [ m] p N p x [ m] x m: [ m] p [ m] µ N p,

38 Contnuous Gussn vrles ~ N µ, σ Y ~ N x +, σ Y ~ N µ, σ ~ N p +, σ L Θ : D x [ m] [ m]: Θ L Θ : D m L θ : D x [ m] [ m]: θ m The lkelhood decomposes Independent estmton prolems The MLE s ~ N p +, σ

39 Sttstcl ckground - regresson Assume Y + + Z, σ, µ re populton vrnce nd men of. Thus : 1.. Y ~ N Usng lest - squres estmton of nd :, rg mn Solvng mx lkelhood estmton of Y : x y x µ µ + x, rg mx log LY n 1 rg mx ln πσ rg mn +, σ. y y z x x where Z + y + ~ N0, σ, x + z / σ Ind,Z.

40 1... E usng populton prmeters Express Y E EY -, cov rg mn, xy y x x y xy y x y x y xy x y x y x y x y xy x y x xy x Y Vr Y Y x y ρ σ µ σ σ ρ µ µ µ µ µ ρ σ σ µ µ µ µ σ σ ρ σ σ σ ρ σ Sttstcl ckground - regresson

41 Contnuous Gussn vrles 1 E xy y x x y xy y Y Vr Y ρ σ µ σ σ ρ µ +, ~ σ p N + Y, ~ N σ µ 1 E Vr ρ σ µ σ σ ρ µ +, ~ σ x N Y +, ~ N σ µ

42 Lernng rmeters Key concept: the lkelhood functon L θ : D D θ x[ m] θ mesures how the prolty of the dt chnges when we chnge prmeters m Estmton: MLE: choose prmeters tht mxmze lkelhood Byesn: tret prmeters s n unknown quntty, nd mrgnlze over t

43 The Byesn Approch to lernng Fnd the posteror! [ M + 1] H D [ M + 1] H θ, D θ D dθ [ M + 1] H θ θ D dθ θ D θ [ M + 1] H θ dθ D [ M + 1] H θ θ D θ dθ θ D θ dθ θ θ D θ dθ θ D θ dθ

44 Byesn pproch for Bnoml Dt H θ. ror: unform for θ n [0,1]. therefore, θ1 Dt: N H,N T 4,1 MLE for H s N H /N H+ N T 4/5 0.8 Byesn predcton s: θ θ D θ dθ x[ M + 1] H D θ D θ dθ NH NT θ 1 θ 1 θ dθ 5 NH NT 1 θ 1 θ dθ K

45 Byesn pproch for Multnoml Dt Recll tht the lkelhood functon s L Θ : D K k 1 θ N k k Drchlet pror wth hyperprmeters α 1,,α K α 1! j K j 1 αk 1 θk 1 k k 1 Θ α 1! α 1!... α 1! k the posteror: Drchlet wth hyperprmeters α 1 +N 1,,α K +N K Θ D Θ c α Nk Θ D θk D D k α + N 1! K K αk 1 θk k 1 k 1 j j K j 1 αk + Nk 1 θk 1 + N1 + N k + Nk k 1 α 1! α 1!... α 1!

46 Byesn pproch for Multnoml Dt If Θ s Drchlet wth hyperprmeters α 1,,α K The posteror s lso Drchlet: ΘD s Drchlet wth hyperprmeters α 1 +N 1,α K + N k nd thus we get [ M + 1] k D θ θ D dθ k α k + l l N k α + N l

47 Lernng rmeters for Byesn networks : Summry For multnomls: counts Nx,p rmeter estmton ˆ θx p N x, p N p ~ θ x p α x, p + N x, p α p + N p MLE Byesn Drchlet Both cn e mplemented n n on-lne mnner y ccumultng counts.

48 Outlne Introducton Byesn Networks Representton & Semntcs Inference n Byesn networks Lernng Byesn networks rmeter Lernng»Structure Lernng

49 Lernng Structure: Motvton Erthquke Alrm Set Burglry Sound Addng n rc Mssng n rc Erthquke Alrm Set Burglry Erthquke Alrm Set Burglry Sound Sound

50 Optmzton rolem Input: Trnng dt Scorng functon ncludng prors Set of possle structures Output: A network or networks tht mxmze the score Key roperty: Decomposlty: the score of network s sum of terms.

51 Scores For exmple. The BDE score: Score G : D G D D G G D G, θ θ G dθ G When the dt s complete, the score s decomposle: G Score G : D Score : D

52 Heurstc Serch cont. Typcl opertons: S C S C Add C D E E D Remove C E D Reverse C E S C S C E D E D

53 Heurstc Serch We ddress the prolem y usng heurstc serch Trverse the spce of possle networks, lookng for hgh-scorng structures Serch technques: Greedy hll-clmng Smulted Annelng...

54 Outlne Introducton Byesn Networks Representton & Semntcs Inference n Byesn networks Lernng Byesn networks Concluson Applctons