Why BP Works STAT 232B

Transcription

1 Why BP Works STAT 232B Free Energes Helmholz & Gbbs Free Energes 1 Dstance between Probablstc Models - K-L dvergence b{ KL b{ p{ = b{ ln { } p{ Here, p{ s the eact ont prob. b{ s the appromaton, called belef. Boltzmann s law for computng ont prob. 1 p{ = ep E{ / T Z

2 Free Energes Helmholz & Gbbs Free Energes 2 b{ KL b{ p{ = b{ ln { } p{ = b{ ln b{ + b{ E{ / T + ln Z Snce KL.. >=, { } { } b{ ln b{ + b{ E{ / T + ln Z { } { } Defne F = T ln Z Tb{ ln b{ + b{ E{ F { } { } F s called the Helmholz free energy, whch s the lower bound of the above nequalty. Free Energes Helmholz & Gbbs Free Energes 3 Let s defne, G b{ = b{ E{ + Tb{ ln b{ = U b{ TS b{ { } { } 1 where Gb{ s called Appromate Gbbs free energy, U s called average energy, and S s called entropy. The Eact Gbbs free energy s defned as G eact p{ = p{ E{ + T { } { } = U p{ TS p{ = F p{ ln p{ The Eact Gbbs free energy s equal to the Helmholtz free energy at equlbrum.

3 Free Energes Mean-feld free energy a varatonal approach 1 Let s ntroduce an arbtrary tral energy functon Snce We have Z = = p ep E where <.> s the epectaton. = ep E / T ep E / T Z = ep E E ep E / T / T ep E / T E, a tral prob. s constructed as: / T ep E ep E E / T p ep E / T ep =< ep E E / T > E / T / T Free Energes Mean-feld free energy a varatonal approach 2 By the property of the convety of the eponental functon : < ep > ep < > We have Z ep < E E / T > ep E Then / T E / T + < E E > Fvar F T ln ep After a few more steps manpulate, we have where F var S =< E > TS F = p ln p p Ths suggests us a useful varatonal arguments: look for the tral prob. func. whch gves us the lowest varatonal free energy. The closer the tral prob. to the eact ont prob., the better the varatonal appromaton.

4 Free Energes Mean-feld free energy 3 Mean-feld theory assumes a tral probablty func. bearng the factorzed form where p { = b b = 1 The energy of a confguraton of a parwse MRF s E { = lnψ, lnφ, Pluggng ths energy nto 1, we obtan mean-feld Gbbs free energy G MF = U MF TS MF Free Energes Mean-feld free energy 4 where U MF = b{ E{ { } = b b lnψ,,, b lnφ and S = b{ ln b{ = b ln b MF { } Note the eact Gbbs free energy s a func. of full ont prob Helmholz free energy lowest bound of KL dvergence. But the mean-feld Gbbs free energy s only a func. of the one-node belefs. To obtan the best appromaton of p{, we need to search for b{ whch mnmze. G MF

5 Free Energes The Bethe free energy 1 For tree-lke topology MRF, the eact ont prob. can be factorzed nto a form that only depends on one-node and two-node margnal prob. where s the number of nodes that are connected to node. We defne S U q and We obtan B B E b{ = b,, [ b ] 1 q, = lnψ, lnφ lnφ E = lnφ = b, ln b, q 1 b ln b,, = b, E, q 1 b E,, Free Energes The Bethe free energy 2 Then the Bethe free energy s G B = b, E, +,, 1 b E + ln b, q ln b Together wth a few normalzaton and margnalzaton constrants, the Lagrangan L s as follows: λ [ b b, ] + β [1 L = G + b ] + B,, [ b b, ] + β [1, λ b, ] 2,,,

6 Free Energes The Bethe free energy 3 Takng dervatves of the L wrt the belefs and those Lagrange multplers, we have margnal prob. appromaton: λ 1 E b = ep[ + ] Z T T q 1 b, 1 E, λ λ = ep[ + + ] Z T T T 3 4 The Bethe appromaton s a much better appromaton to the eact Gbbs free energy than the mean feld appromaton. The dffculty les n the computatonal part. The Belef Propagaton algorthm provdes a good soluton. Bref of Belef PropagatonBP 1 For parwse MRF s, the ont prob. dstrbuton for {} can be factorzed 1 p { = ψ, φ Z, where ψ, tells nternal bound between node and, and φ ndcates eternal evdence at node. φ 1 φ 2 φ ψ 1, 2 ψ 2 3,

7 Bref of Belef PropagatonBP 2 Messages m are ntroduced to pass nformaton between nodes n BP network. The belef margnal posteror at a node s computed as follows: and the ont belef ont margnal posteror of a par of neghborng nodes and s: = N m b φ =,, N l l N k k m m b φ φ βψ the message from nodes to s: N k k m m \, ψ φ 5 6 Equvalence of BP to the Bethe Appromaton By defnng t s easly to show that 3 and 4 derved for the Bethe appromaton are equvalent to the BP equaton 5 and 6. = N k k m T \ ln λ

8 Equvalence of BP to Dynamc Programmng To get MAP soluton ma product from a belef network, e.g. 3-node graph, the BP algorthm s equvalent to the dynamc programmng. ˆ = arg ma φ ma φ ψ, ma φ ψ, ˆ = arg ma φ ma φ ψ, ma φ ψ, ˆ = arg ma φ ma φ ψ, ma φ ψ, Loopy & non-loopy graph BP works for sngly connected networks. It s guaranteed to converge to the correct answers. BP does not always work for loopy networks. Because same evdence s passed around the network multple tmes and mstaken for new evdence.

9 Loopy graph works sometmes Although evdence s double counted, all evdence may be double counted. It s proved to be correct n ths stuaton. Sngle loop BP s guaranteed to generate the most lkely state sequence. Multple loops Balanced network wll work. BP Vstng Order Reschedule In tradtonal BP algorthm, messages beng passed and updated between nodes are wthout any prorty. Ths s not effcent because notes wth weak evdence provdng less useful nformaton to ther neghbors. Messages from these nodes should be passed at later stage compared wth those nodes wth strong evdence. We desgn a new node vstng order to effectvely passng messages between graph nodes. 1. Rank the nodes accordng to the belef of ther local evdence breadth frst search. Most nformatve node passes ts message frst. 2. Reverse the order n step 1, pass messages back.

10 Toy Problem 1 - Isng model 1 Defnton E { h = J, 1 p{ = e Z Specfcatons 1, J, =.1, h = 1 E{ T = Toy Problem 1 - Isng model 2 Bounce back vstng order potental BP Iteratons

11 Toy Problem 2 - Rectangle Matchng 1 Matchng black rectangle n two mages match Buld BP graph 7 7 lattce φ φ 1 2 φ 4 φ 3 ψ, 1 2 ψ, ψ, ψ, 3 4 Toy Problem 2 - Rectangle Matchng 2 Ordnary vstng order potental BP Iteratons

12 Toy Problem 2 - Rectangle Matchng 3 Flush vstng order startng from 4 corners potental BP Iteratons Real Data Peps Can

13 Real Data Peps Can Real Data Car

14 Real Data Car Real Data Flower Garden

15 Real Data Flower Garden Real Data Beverly

16 Real Data Beverly Real Data Prntng Room

17 Real Data Prntng Room Real Data Nagara Fall

18 Real Data Nagara Fall Reference J. Yedda, W. T. Freeman and Y. Wess, Understandng belef propagaton and ts generalzatons Internatonal Jont Conference on Artfcal Intellgence IJCAI 21. Yedda, J.S., "An Idosyncratc Journey Beyond Mean Feld Theory", Advanced Mean Feld Methods, Theory and Practce, ISBN: , pps 21-36, February 21.