STATISTICAL MECHANICS OF THE INVESE ISING MODEL Muro Cro Supervsors: rof. Mchele Cselle rof. ccrdo Zecchn uly 2009
INTODUCTION SUMMAY OF THE ESENTATION Defnton of the drect nd nverse prole Approton ethods of the drect prole: vrtonl pproches Bethe Overvew of drect nd nverse lgorths Sultons GOALS OF THE THESIS Use of lgorths whch generlze Bethe pproton Gp n order to solve the nverse prole Upgrde of the Gp lgorth to get ore ccurte clculton of the correltons nd pplcton to the nverse prole
ISING MODEL We cn represent the usul Boltznn prolty dstruton wth fctor grph: > < h H e e e } { } { β β β > < h H } {
ISING MODEL H { } < > h h > < > < Inverse role!
EXAMLES OF ALICATIONS Neuron networks reconstructon [E.Schnedn M..Berry.Segev W.Blek 2006] Genes networks reconstructon [A.Brusten A.gn M.Wegt.Zecchn 2008] roten networks recontructons [G.Tkck 2007] Why Isng odel? Mu entropy prncple
VAIATIONAL AOACHES Men feld pprotons G[ ] < E > T S G[ ] ~ F Gs Free Energy Boltznn Dstr. Heloltz Free Energy Men feld pproton Mnzton of G In sudon We chose for for nd we nze the functonl G
Locl consstency: BELIEFS: 1 1 VAIATIONAL AOACHES Bethe pproton [ ] > < q p 1 } { [ ] [ ] log 1 log E q E G > < Mnu equtons Constrnts equtons 1 Couplngs f 2 Couplngs f Messges equtons t the f pont
B Drect Isng B equtons t t \ 1 η ν... k k t k t \ \ ν ψ η..k..
B Drect Isng * η t t s s \ * \ * η η ψ
GB [.S.Yedd W.T. Freen Y.Wess 2001] Drect Isng Let s defne: c c U E S Log : vrles close to functon node c 1 c U U A Generlzton of regons {} : ng[{}] Bethe pproton New for of G GB equtons
GB Drect Isng \ \ D I I N I I F ψ
GB Drect Isng \ \ D I I N I I F ψ Ψ \ ' ' D D E D D D A
GB [.S.Yedd W.T. Freen Y.Wess 2001] Drect Isng Is G vld? depends on regons c Condton: c I c I 1 I 1 f We wnt to count every node only once
INVESE ISING Solvng the nverse prole through n tertve ethod Coplete Grph χ Itertve process Updte rules for h Grph wth: < σ > < σσ > χ
B/GB Inverse Isng Self-consstent equtons n the essges Self consstent equtons n the nputs χ f M h χ g M h Messges fed h f 1 M χ g M 1 χ
B/GB Inverse Isng χ Otnng eperentl vlues of Intlze coplete grph wth rndo h For t 1T Itertons Bp/Gp sve essges Functon nodes updte n the grph h f M 1 g M 1 χ χ End
B Inverse Isng EXAMLE ~ h h Tnh h ATnh h ~ ν ν ν ν ~ ~ ~ ~ h h h h h h h h e e e e
B Inverse Isng EXAMLE χ e ν ν 2 1 ν 2 1 ν 1 Tnh Tnh χ ATnh χ χ 1
Sus.rop. [M.Mezrd T.Mor 2008] Drect nd Inverse Isng B lttons χ only f < > Fluctuton-esponse theore < σ σ > < σ >< σ > h h h wth η We hve to know dervtves of essges! B equtons ν η B dervtves ν η ν η χ f h Invertng f New updte rule for
GS Drect Isng A: prove ccurcy n χ n the GB schee B Sus.rop. Flutt.- esp. T. GB GS Fnd equtons for dervtves of essges n GB Gsp equtons Do the dervtves of the 1-elefs equtons Correltons
GS Isng Inverso Etrctng couplngs fro GS wth rtrry regons? Contrnt: t lest ll the 1 nd 2 regons 1-regons see Bethe ppro. rtl reuse of Sus.rop. forls
Sll correltons epnson [.Monsson V.Sessk 2008] Lkelhood zton Let s suppose to hve n copes of the syste Let s ze the prolty to hve these copes gven the theory: 1 [ Log { σ} n h ] n n where : S Log Z h ns h c Let s solve the prole for c 0 c β c Anltcl epressons for e h
Sultons eples 2 Coplete 2 2 N N 1 < Grph N 20 Bp Gp Nve M.F. Sus.rop. Gsp S.C.E.
Sultons eples 2 2 2 N N 1 < Tree N 20 Bp Gp Nve M.F. Sus.rop. Gsp S.C.E.
Conclusons nd future B Sus.rop. GB GS OBLEMS Bg correltons Glssy phse Loopy B Sll Correlton Epnson Necessty to redefne the prole Iterton ethods n g correltons rege Cvty ethods for GB n 1SB [T.zzo et l. 2009] New pprotons for the free energy
Acronys & Notton B GB Sus.rop. GS S.C.E. Nve M.F. GB 3 GS 3 Belef ropgton Generlzed Belef ropgton Susceptlty ropgton Generlzed Susceptlty ropgton Sll Correlton Epnson Men Feld Theory Fluctuton esponse Theore GB wth ll regons wth 1 or 2 or 3 spns GS wth ll regons wth 1 or 2 or 3 spns \ Identty true ecept for norlzton fctor Set of nodes whch re connected to wth lnk Set wthout the node node representng vrle leled y k node representng functon leled y c