Probability-Theoretic Junction Trees

Transcription

1 Probablty-Theoretc Juncton Trees Payam Pakzad, (wth Venkat Anantharam, EECS Dept, U.C. Berkeley EPFL, ALGO/LMA Semnar 2/2/2004

2 Margnalzaton Problem Gven an arbtrary functon of many varables, fnd (some of ts margnals. For eample β ( = 2, L, n β (, L, or even γ = ma β (, L, (, L, 2 e.g. β (... could be the jont densty on varables, then β ( = margnal probablty of, and n γ ( = probablty of most lkely conf. of rest of varables n n 2

3 Applcatons Error Correctng Codes Image Processng Computer Vson Networkng Statstcal Physcs Problem: Computatons ntractable. State space s eponental n number of varables. 3

4 4 Product Functons and Factorzaton Suppose β factorzes as product of functons of smaller subsets of varables. e.g. = = n n p p 0 0 ( (,, β ( L Markov chan model: Then certan condtonal ndependences may hold whch can smplfy the margnalzaton task. E.g. ( 0,, s cond. ndep. of ( +2,, n gven +, hence = = + = },, { 2 },, { 0 }\,, { ( ( ( n n n j j j j j j p p p L L L β

5 Product Functons - Notaton Gven state vector: a collecton of local domans, R: = (,..., N r R, r {, L, N} and local functons (kernels: α (, r r r R 5 Objectve functon factorzes: Fnd margnals: β ( α ( r R r r β r ( r β ( \ r

6 Independency Graphs A graph wth nodes correspondng to domans r R, s.t. separaton on graph mples (condtonal ndependence. A tree s called a Juncton Tree f whenever a node r 0 separates nodes r and r 2 on tree, then r r 2 r 0, 2, 3, 2, 3, 2 2, 3, 2, 4 4, 2, 4 4, 3 6

7 Juncton Tree Algorthm (a.k.a. GDL, Belef Propagaton, sum-product, Defne messages µ r,u ( u r for each edge (r, u of the ndependency graph. Intalze to. t µ t,r µ Update messages as: r, u u r αr ( r µ t, r ( r t \ t N ( r \{ u} ( t k r u r µ tk,r µ r,u u 7 Defne Belefs b r ( r α r ( r t N ( r µ t, r ( r t

8 Juncton Trees If graph s juncton tree, algorthm converges n fnte tme, and then belefs b r ( r are the margnals of the product functon. Gven regons, can quckly determne f a juncton tree ests. Involves fndng a mamal-weght spannng tree of regons. If none est, can always epand regons n a way to create a juncton tree, but complety s eponental n the sze of regons. 8

9 Graphs Wth Cycles - Perspectve B.P. on graphs wth loops s used as appromaton to the margnals, e.g. n Turbo codes and LDPC decodng. Attempts at justfcaton of appromaton on loopy graphs: One-loop case (Wess, Horn et. al. Gaussan case (Van Roy, Wess Fed ponts and stablty (Rchardson LDPC studes (Gallager, McKay & Neal, Luby et. al., Urbanke & Rchardson, Shokrollah. Tree-based Reparametrzaton (Wanwrght 9

10 Probablty-theoretc Vew Motvaton: Relyng on varables can conceal partal structure n statespace; can only recognze/create rectangular decompostons. Generalze the problem to arbtrary state-spaces; State-space need not be represented by varables. 0

11 Eample Let,, M take value n (a fnte sem-group A and let µ(,, M = f( + + M, p ( be real functons. We would lke to calculate E = M µ ( M M, L, =, L, p (

12 Eample, cntd. GDL Soluton: Or,, M M,, M- 2,, M E = p 2 ( p2( 2 L µ (, L, M M p M ( M Requres O( A M addtons and multplcatons. 2

13 Eample, cntd. In comparson, wth S + + M A S, S 2 M-2, S M- M-, S M S M s an ndependency tree, suggestng E = S f ( S p( L pm ( SM pm ( M + S2 = S M + SM = SM Ths requres only O(M A 2 addtons and multplcatons. 3

14 Probablty-theoretc GDL A systematc theory to recognze and eplot such structures. Reformulate the MPF problem as takng condtonal epectatons n an arbtrary sample space. 4 GDL Varables,, N Local domans r R Local functons α r, r R Margnals β r = \r ΠX r PGDL Arbtrary representaton σ-felds F,, F M Random varables X,, X M Cond. Epectatons E[ΠX F j ]

15 Prelmnares - Notaton λ λ A dscrete state-space Ω A measure µ(. on Ω σ-felds F on Ω And r.v. s X F E Want to calculate: M = X F j ( f = ì( f M ω f = X ( ωì( ω 5 for each (nonzero measure atom f of F j

16 Prelmnares, cntd. Augmentaton: F {,2} F F 2 Condtonal Independence: Say F G H f for each atom h of H: If µ(h 0 then f F, g G, ì ( f, g, hì ( h = ì ( f, hì ( g, h If µ(h = 0 then f F, g G, ì ( f, g, h = 0 6

17 Margnalzaton Problem and Juncton Trees Gven a collecton (Ω, {F,, F M }, µ of meas. spaces, and r.v. s X F : MPF Problem: For one or more {,,M} fnd E[Π j X j F ] A tree wth nodes {,, M} s called a juncton tree f: A, B {,, M } and {,, M } s.t. separates A and B on tree, we have F A F B F 7 As before, a juncton tree captures the ndependences

18 Probablstc Juncton Tree Algorthm The followng algorthm solves the MPF problem (c.f. GDL 8 Defne messages Y,j F j for each edge (, j of the juncton tree. Intalze to. Update messages as: Y = X Defne Belefs B X At termnaton [ Y F, j E k, j k N ( \{ j} Y k N ( B k, = E [ M X j j= ] F ] k k m Y k, j Y,j Y km,

19 Estence of Juncton Trees 9 Man results: Gven a collecton (Ω, {F,,F M }, µ of meas. spaces: λ =,, M, there ests a partton P of {,, M}\{} (called Fnest Vald Partton w.r.t., and If a juncton tree ests, then for each =,, M there ests a J.T. compatble wth P. Eample: Suppose M=7 and P 2 ={{} {3,6,7} {4,5}} then:

20 Lftng Let (Ω, {F,, F M }, µ be a collecton of meas. spaces. Another collecton (Ω, {F,, F M }, µ s a lftng of λ (Ω, {F,, F M }, µ f there s a map f : Ω Ω s.t.: µ s consstent wth µ under f, and for all =,, M, f s (F, F -measurable. Ω Ω 20 F f F - (F

21 Lftng to Create Condtonal Independency Gven F, F 2 and F 3 we lke to lft n a way so to have: F F 3 F 2. Corresponds to certan matrces of jont measures beng rank one. If some of these matrces are not rank-one, fnd a rank-one decomposton. lftng obtaned by splttng the atom correspondng to the matr. 2

22 Algorthm to Create a Juncton Tree Ths algorthm wll fnd a J.T. on {F,, F M } f one ests: Pck any node {,, M} as the root, Fnd any vald partton w.r.t., say {c,, c l } For each j = to l Pck a node t c j and splt atoms of F t s.t. F F c j F t. Fnd a J.T. on c j, wth t as the root. Attach ths tree by addng the edge (,t. End 22

23 Eample 2: Eact Decodng of LDPC Codes H GF(2 m n: Party check matr of an LDPC code wth m checks and block sze n Codewords satsfy H =0, wth a posteror probabltes P * n m = * ( P( P( y ( H j Z = Objectve: fnd margnals of P * : P * ( = P \ * ( j= = 0 23

24 Eample 2, cntd. Eact nave GDL soluton: trangulate the graph, and run J.T. algorthm on the tree of clques. Problem: graph hghly connected, so clques are large, algorthm neffcent. PGDL soluton: state-space = the codebook; unform measure; * P( = j P( y = random varables = orgnal σ-felds = drectons of varables s j 24 Lft usng the automatc Matlab algorthm to form a chan.

25 25 Eample 2, Results:

26 Concludng Remarks 26 λ As wth GDL, our algorthm generalzes to any sem-feld (e.g. ma-product. Cost of lftng s hgh, but a juncton tree can be used over all sets of observatons. The complete sample space Ω s very large, but the resultng algorthm only requres storage of the matrces of jont denstes of atoms of neghborng σ-felds. The conventonal method corresponds to the case of a product space, wth a unform measure. Measure theory s transparent to the end user.