# Why the Junction Tree Algorithm? The Junction Tree Algorithm. Clique Potential Representation. Overview. Chris Williams 1.

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1 Why th Juntion Tr lgorithm? Th Juntion Tr lgorithm hris Willims 1 Shool of Informtis, Univrsity of Einurgh Otor 2009 Th JT is gnrl-purpos lgorithm for omputing (onitionl) mrginls on grphs. It os this y rting tr of liqus, n rrying out mssg-pssing prour on this tr Th st thing out gnrl-purpos lgorithm is tht thr is no longr ny n to pulish sprt ppr xplining how to l with h nw mol th JT gnrliss nrly ll th populr prvious spil s lgorithms. Ring: Jorn hptr 17 1 s on slis y vi rr 1 / 28 2 / 28 Ovrviw liqu Potntil Rprsnttion liqu Potntil Rprsnttion onstruting Juntion Tr Morliztion Tringultion ssmling liqus into juntion tr Mssg Pssing Introuing Evin Propgtion on Juntion Tr Osrv tht for oth irt n unirt grphs, th joint proility is in prout form. W n intrprt th PTs in irt grphs s potntil funtions. si i is to rprsnt proility istriution orrsponing to ny grph s prout of liqu potntils: p(x) = 1 Ψ (x ) Z whr x is th st of vrils orrsponing to liqu. liqu is fully-onnt sust of nos in grph 3 / 28 4 / 28

2 n xmpl f f f Morliztion Tringultion p(,,,,, f ) = p()p( )p( )p( )p( )p(f, ) 5 / 28 6 / 28 liqu Trs n Sprtors f liqu tr is n (unirt) tr of liqus,,,,,,,f Th liqu potntil rprsnttion is p(,,,,, f ) = Ψ(,, )Ψ(, )Ψ(,, )Ψ(,, f ) vli ssignmnt of lustr potntils is Ψ(,, ) = p()p( )p( ), Ψ(, ) = p( ), Ψ(,, ) = p( ), Ψ(,, f ) = p(f, ) n Z = 1 7 / 28 Vrils shr y nighouring liqus r rwn in th sprtor sts in lu. Th potntil rprsnttion of liqu tr is th prout of th liqu potntils, ivi y th prout of th sprtor potntils. p(x) = Ψ (x ) S Φ S(x S ) 8 / 28

3 onstruting Juntion Tr from G Initilly, ll sprtor potntils r st to 1. ftr running th JT, w will hv Ψ(x ) = p(x, Φ(x S ) = p(x S, whr nots thos vrils in tht r not in E, n similrly for S. 1 Morliz th grph 2 Tringult th grph 3 onstrut juntion tr 9 / / 28 Morl Grphs Lt s rprsnt th following G s prout of liqu potntils: p() p() p(,) = Ψ (,) = Ψ(,,) Ψ(,) To nsur tht no n its prnts r in th sm liqu, w hv to mrry th prnts morlistion. Morl Exmpl to us ll ftr morlistion, w gt th following unirt grph Th prout of liqu potntils is p(,,,,, f ) = Ψ(,, )Ψ(,, )Ψ(,, f ) whr Ψ(,, ) = p()p()p(, ), Ψ(,, ) = p( )p( ), Ψ(,, f ) = p(f, ) E E F F 11 / / 28

4 Th n for tringultion onsir th following grph n orrsponing liqu tr,, Tringultion In tringult grph, ll loops ontining 4 or mor nos ontin hor:,,,, pprs in two non-nighouring liqus. Thr is no gurnt tht mrginl on in ths two liqus shoul qul, i. Ψ(, ) = Ψ(, ) Tht is, lol onsistny os not nssrily imply glol onsistny. Tringultion provis solution.,, On wy to rt tringult grph is vi th limintion lgorithm (s Jorn 3.2) 13 / / 28 onstruting Juntion Tr liqu tr is juntion tr if it hs th following juntion tr proprty: if no pprs in two liqus, it pprs vrywhr on th pth twn th liqus. For vry tringult grph thr xists liqu tr whih oys th juntion tr proprty Thus lol onsistny implis glol onsistny Not ll liqu trs r juntion trs Thorm liqu tr is juntion tr iff it is mximl spnning tr, whr th wight is givn y th sum of th rinlitis of th sprtor sts 15 / / 28

5 Mssg Pssing sorption In orr tht th liqus ontin ll informtion rquir for mrginls of th vrils in th liqu, w n to nfor onsistny. Tht is, if liqu V (ontining st of vrils) n liqu W shr vrils S, th mrginls on thir sprtors must qul. Ψ( V) Φ( S) Ψ( W) W n V \S Ψ(V ) = Φ(S) = W \S Ψ(W ). sorption psss mssg from on no to nothr: * * Ψ( V) Φ( S) Ψ( W) W sors from V Ψ (W ) = Ψ(W ) Φ (S) Φ(S), whr Φ (S) = V \S Ψ(V ) Similrly, ftr pssing mssg on wy, w pss it th othr: ** ** * Ψ( V) Φ( S) Ψ( W) V sors from W Ψ (V ) = Ψ (V ) Φ (S) Φ (S), whr Ψ (V ) = Ψ(V ) n Φ (S) = W \S Ψ (W ) 17 / / 28 This nsurs onsistny: V \S Ψ (V ) = Φ (S) = W \S Ψ (W ). lso Ψ(V )Ψ(W ) Φ(S) = Ψ (V )Ψ (W ) Φ (S) = Ψ (V )Ψ (W ) Φ (S) Introuing Evin p(x) = Ψ (x ) Split nos into H (hin) n E (vin) p(x H, = Ψ (x, x E ) Ψ (x ) whr Ψ (W ) = Ψ (W ), thus mintining th liqu tr rprsnttion of th grph. Show tht Ψ (V ) n Ψ (W ) hv th sm mrginls on S This is prout of slis of potntil funtions. Thus to introu vin, w moify th potntils in th originl grph, stting ny nos to thir vintil vlus. On n lso us th vin potntil pproh y stting Ψ (x ) = Ψ (x )δ(x E, x E ) ut this fills th liqu potntils with lots of zros thus n wsts storg n omputtion 19 / / 28

6 Propgtion on Juntion Tr No V n sn xtly on mssg to nighour W, n it my only snt whn V hs riv mssg from ll of its othr nighours hoos on liqu (ritrrily) s root of th tr; ollt mssgs to this no n thn istriut mssgs wy from it ftr olltion n istriution phss, w hv in h liqu tht Ψ(x ) = p(x, olltevin istriutevin 21 / / 28 Summry of JT Proof of orrtnss of JT onvrt lif ntwork into JT Initiliz potntils n sprtors Inorport vin (JT is inonsistnt) olltevin n istriutevin (to giv onsistnt JT) Otin liqu mrginls y mrginliztion/normliztion Thorm Lt th proility p(x H, rprsnt y th liqu potntils of juntion tr. Whn th juntion tr lgorithm trmints, th liqu potntils n sprtor potntils r proportionl to th lol mrginl proilitis. In prtiulr: Ψ = p(x,, Φ S = p(x S, Proof Osrv tht th sprtors r susts of th liqus whih r onsistnt with th liqus. Thus w only n to prov th rsult for th liqus. 23 / / 28

7 Throughout th propgtion pross w hv mintin th rprsnttion p(x H, = Ψ (x ) S Φ S(x S ) ftr th ollt- n istriut-vin stgs th juntion tr is onsistnt (i.. th mrginliztion of th potntils of th liqus t ithr n of sprtor giv th sm sprtor potntil). W now show tht mrginliztion of th joint p(x H, givs th sir rsult. R S hoos liqu tht is lf of th JT with sprtor S. Lt = \E n S = S\E. Lt R = \ S, n th rmining non-vin nos not T. W now rmov liqu y summing out R from p(x H, = p(x R, x S, x T, V 25 / / 28 JT xmpl p(x T, x S, = R = R = R p(x H, Ψ (x ) S Φ S(x S) Ψ (x ) Φ S(x S) R = Ψ (x ) Φ S(x S) S S = Ψ (x ) S S Φ S (x S ) Ψ (x ) Φ S (x S ) Ψ (x ) S S Φ S (x S ) omput p() p( = 0, = 1) p( = 1) pplying this pross rptly w otin p(x, = Ψ (x, 27 / / 28

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