UNDERSTANDING BELIEF PROPOGATION AND ITS GENERALIZATIONS

Transcription

1 UNDERSTANDING BELIEF PROPOGATION AND ITS GENERALIZATIONS JONATHAN YEDIDIA, WILLIAM FREEMAN, YAIR WEISS 2001 MERL TECH REPORT Kristin Branson and Ian Fasel June 11, 2003

2 1. Inference Inference problems in networks include: Conditional Probability Query: What is p(x i X E = x E )? Most Probable Explanation: What is argmax xu p(x U = x U X E = x E )? Inference problems also arise in statistical physics. In general, these problems are intractable.

3 1.1. Inference Approximations Pearl s Belief Propagation (BP) algorithm solves inference Exactly for tree networks and Approximately for other networks. BP is not well-understood for loopy networks. BP is closely connected to the Bethe approximation of statistical physics.

4 1.2. Purpose Our goal in this paper is to Gain understanding of the BP approximation and How to make it more exact by exploring equivalent approximations in statistical physics. This has lead to an improved inference algorithm, Generalized BP.

5 Outline The Settings. Pairwise Markov Random Fields. The Potts and Ising Models. The Approximations. Belief Propagation. The Mean-Field Approximation. The Bethe Approximation. The connection between the Bethe and BP approximations. The Kikuchi Approximation. Generalized Belief Propagation.

6 2. The Settings

7 2.1. Pairwise Markov Random Fields A pairwise MRF is an undirected network with cliques of size two, e.g.: observed node hidden node With hidden variables x and observed variables y, p(x y) = 1 ψ ij (x i, x j ) φ i (x i, y i ). Z (i,j) i Any MRF can be converted into this form (Weiss, 2001). We want to calculate p(x i y) or argmax x p(x y).

8 2.2. The Potts Model The pairwise MRF can be brought into a form recognizable to physicists as the Potts model: p(x y) = 1 Z exp J ij (x i, x j ) + h i (x i ) /T (i,j) i by setting The interaction J ij (x i, x j ) = lg ψ ij (x i, x j ), The field h i (x i ) = lg φ i (x i, y i ), and the temperature T = 1.

9 2.3. The Ising Model The Ising model is a special case of the Potts model in which Each variable is binary, Interactions J ij are symmetric, The distribution [( can be expressed as: p(x) = 1 Z exp (i,j) J ijx i x j + ) ] i h ix i /T -1 +1

10 Use of the Ising Model The Ising model was invented to describe phase transitions in magnetics. J encourages neighboring particle to have equal values. h represents an external magnetic field. The magnetization of a particle is its expected value, m i = p(x i = +1) p(x i = 1). Phase transitions (Veytsman and Kotelyanskii, 1997).

11 3. The Approximations

12 3.1. Standard Belief Propogation i Beliefs: b i (x i ) Messages: m ji (x i ) b i (x i ) = kφ i (x i ) m ji (x i ) j N(i) The belief is the BP approximation of the marginal probability.

13 BP Message-update rules To get marginal beliefs b i (x i ), sum over other variables: b i (x i ) = X a\x i b a (X a ) So we write the messages as: m ij (x j ) φ i (x i )ψ ij (x i, x j ) x i m ki (x i ) k N(i)\j i j = i j

14 BP is Exact for Trees b i (x i ) m 21 (x 1 ) ψ 12 (x 1, x 2 )m 23 (x 3 )m 24 (x 4 ) x 2 ψ 12 (x 1, x 2 )ψ 23 (x 2, x 3 )ψ 24 (x 2, x 4 ) x 2,x 3,x 4

15 3.2. Mean-Field Approximations The true distribution p(x y) is approximated by the closest distribution b(x) in the family Q. What is the definition of closest? The Maximum-Likelihood definition is b ( ) = argmin b( ) KL(p( y) b( )) For reasonable Q, computing b ( ) is intractable. The Mean-Field definition is b( ) = argmin b( ) KL(b( ) p( y)).

16 3.3. Why This Measure? For some reasonable Q, we can compute b( ). If p Q, we will exactly compute b(x) = p(x). Minimizing KL(b( ) p( y)) yields the best lower bound on the log-likelihood l(p; y): l(p; y) = x b(x) lg p(y) = p(x,y)b(x) x b(x) lg b(x)p(x y) = p(x,y) x b(x) lg b(x) + b(x) x b(x) lg p(x y) = lg p(x,y) b(x) + KL(b( ) p( y)) b lg p(x,y) b(x) (Tanaka, 2001; Jaakkola, 2001) b

17 3.4. The Gibbs Free Energy In physics, this distance is the Gibbs Free Energy: G(b( ), y) = KL(b( ) p( y)) = x b(x)e(x, y) + x b(x) lg b(x) + lg Z where the energy E(x, y) = (i,j) lg ψ ij (x i, x j ) i lg φ i (x i, y i ) (thus p(x y) = 1 Z exp[ E(x, y)]).

18 The Helmholtz Free Energy Minimizing the Gibbs Free Energy is equivalent to minimizing the variational free energy F (b( ), y) = x b(x)e(x, y) + x b(x) lg b(x). If b( ) = p( y), then the variational free energy achieves the Helmholtz Free Energy F (y) = lg Z = x b(x)e(x, y) + x b(x) lg b(x). Otherwise, F (b( ), y) is an upperbound on F (y). Minimizing G(b( ), y) thus minimizes this upperbound (Yedidia, 2001).

19 3.5. The Naive MF Approximation The naive MF approximation restricts Q to be the set of distributions factorizable as b(x) = i b i (x i ). The Gibbs free energy is then G(b i, y) = b i (x i )b j (x j ) lg ψ ij (x i, x j ) (i,j) x i,x j b i (x i ) lg φ i (x i, y i ) i x i + b i (x i ) lg b i (x i ) + lg Z i x i

20 3.6. The Naive MF Algorithm G MF is minimized (subj to x i b i (x i ) = 0) by setting the derivative wrt b i (x i ) to 0, yielding b i (x i ) = αφ i (x i, y i ) exp b j (x j ) lg ψ ij (x i, x j ), j Ni x j where α is a normalization constant. Each estimate b i (x i ) is iteratively updated until convergence. p(x i y) is approximated by the steady-state b i (x i ) (Weiss, 2001).

21 4. Region Based Inference Approximations Naive MF restricts Q to distributions expressible in terms of single-node beliefs. A better approximation is to restrict Q to distributions expressible in terms of node-cluster beliefs.

22 4.1. The Bethe Approximation The Bethe approximation restricts Q to b(x) that can be expressed in terms of the one- and two-node beliefs: b(x) = (i,j) b ij (x i, x j ) i b i (x i ) 1 q i The Gibbs free energy is then: G Bethe = b ij (x i, x j )(E ij (x i, x j ) + ln b ij (x i, x j )) x i,x j x i,x j + (1 q i ) b i (x i ) ln b i (x i ) i x i

23 Minimizing the Bethe Free Energy Minimize by constrained optimization using Lagrange multipliers. L = G Bethe + { } λ i b i (x i ) = 1 i x i + { } λ ai (x i ) b a (X a ) = b i (x i ) a i N(a) x i X a\x i Results in belief equations: L b i (x i ) = 0 b i(x i ) exp 1 λ ai (x i ) d i 1 a N(i) L b ai (x a ) = 0 b a(x a ) exp E a (X a ) + λ ai (x i ) a N(i)

24 Equivalence of BP to the Bethe Free Energy Identify: λ i,j (x j ) = ln m kj (x j ) To obtain BP equations: k N(j)\i i b i (X i ) a N(i) m a i(x i ) i j = i j m ij (x j ) x i φ i (x i )ψ ij (x i, x j ) k N(i)\j m ki(x i )

25 4.2. Generalizing the Bethe Approximation The Bethe approximation restricts Q to b(x) that can be expressed in terms of the one- and two-node beliefs: b(x) = (i,j) b ij(x i, x j ) i b i(x i ) 1 q i. Another way to write the Bethe Approximation: Let C = {{i, j}}

26 4.2. Generalizing the Bethe Approximation The Bethe approximation restricts Q to b(x) that can be expressed in terms of the one- and two-node beliefs: b(x) = (i,j) b ij(x i, x j ) i b i(x i ) 1 q i. Another way to write the Bethe Approximation: Let C = {{i, j}}

27 4.2. Generalizing the Bethe Approximation The Bethe approximation restricts Q to b(x) that can be expressed in terms of the one- and two-node beliefs: b(x) = (i,j) b ij(x i, x j ) i b i(x i ) 1 q i. Another way to write the Bethe Approximation: Let C = {{i, j}}

28 4.2. Generalizing the Bethe Approximation The Bethe approximation restricts Q to b(x) that can be expressed in terms of the one- and two-node beliefs: b(x) = (i,j) b ij(x i, x j ) i b i(x i ) 1 q i. Another way to write the Bethe Approximation: Let C = {{i, j}}

29 4.2. Generalizing the Bethe Approximation The Bethe approximation restricts Q to b(x) that can be expressed in terms of the one- and two-node beliefs: b(x) = (i,j) b ij(x i, x j ) i b i(x i ) 1 q i. Another way to write the Bethe Approximation: Let C = {{i, j}} and R = {C i } {C i C j }

30 4.2. Generalizing the Bethe Approximation The Bethe approximation restricts Q to b(x) that can be expressed in terms of the one- and two-node beliefs: b(x) = (i,j) b ij(x i, x j ) i b i(x i ) 1 q i. Another way to write the Bethe Approximation: Let C = {{i, j}} and R = {C i } {C i C j }

31 4.2. Generalizing the Bethe Approximation The Bethe approximation restricts Q to b(x) that can be expressed in terms of the one- and two-node beliefs: b(x) = (i,j) b ij(x i, x j ) i b i(x i ) 1 q i. Another way to write the Bethe Approximation: Let C = {{i, j}} and R = {C i } {C i C j }. Define c r = 1 s super(r) c s and super(r) = {s r s}. Thus, If r = {i, j}, super(r) = {}, so c r = 1. If r = {i}, super(r) = {{i, }}, so c r = 1 q i. b(x) = r R b r(x r ) c r,

32 4.3. Kikuchi Approximations The Kikuchi approximation allows C to be any set of clusters of nodes s.t. every edge and node of the MRF is in some cluster, and sets Example: R = C {C i C j } {(C i C j ) (C k C l )} Original MRF C {C i C j } {(C i C j ) (C k C l )}

33 4.3. Kikuchi Approximations The Kikuchi approximation restricts Q to b(x) that can be expressed in terms of cluster beliefs: b(x) = r R b r (x r ) 1 c r. p( y) is approximated by b( ) = argmin b( ) Q KL(b( ) p( y)). The Kikuchi approximation can be made exact, in which case it is equivalent to the junction tree algorithm.

34 4.4. Other Approximations In general, it is possible to define a set of rules for constructing valid region approximations. Bethe restricts regions to neighboring pairs Kikuchi restricts regions to be connected clusters of nodes Example of a region that cannot be created using other methods: Many others...!

35 4.4. Other Approximations Other region based approximations exist, given a few rules for constructing valid approximations (involving counting numbers, etc.)

36 5. Generalizing BP Given a valid region-based approximation, we can construct a generalized belief propagation (GBP) algorithm. Belief in a region is the product of: Local information (factors in a region) Messages from parent regions Messages into descendant regions from parents who are not descendants Message update rule obtained by enforcing marginalization constraints. We do this for Cluster Variational Method (Kikuchi) as an example...

37 5.1. Constructing clusters First, construct clusters and find intersection regions (assign counting numbers at this point)

38 5.2. Construct region graph A hierarchy of regions and their direct sub-regions (This example is the region graph using Kikuchi method.)

39 5.3. Construct Belief Equations Belief equations for every region r are proportional to the product of local information (factors in a region) messages from parent regions messages into descendant regions from parents who are not descendants

40 Constructing Belief Equations Step 3: Construct belief equations for each region b 5 (x 5 ) = k[φ 5 ][m 2 5 m 4 5 m 6 5 m 8 5 ]

41 Constructing Belief Equations Step 3: Construct belief equations for each region b 45 = k[φ 4 φ 5 ψ 45 ][m m m 2 5 m 6 5 m 8 5 ]

42 Constructing Belief Equations Step 3: Construct belief equations for each region b 1245 = k[φ 1 φ 2 φ 4 φ 5 ψ 12 ψ 14 ψ 25 ψ 45 ][m m m 6 5 m 8 5 ]

43 5.4. Message Update Rule Use marginalization constraints to define message update rules. b 5 (x 5 ) = b 45 (x 4, x 5 ) x 4 Combining previous belief equations: k 1 [φ 5 ][m 2 5 m 4 5 m 6 5 m 8 5 ] = k 2 [φ 4 φ 5 ψ 45 ][m m m 2 5 m 6 5 m 8 5 ] x 4 Gives us: m 4 5 (x 5 ) = k x 4 φ 4 (x 4 )ψ 45 (x 4, x 5 )m (x 4, x 5 )m (x 2, x 5 )

44 Message Update Rule Use marginalization constraints to define message update rules: = b 5 = x 4 b 45 (x 4, x 5 )

45 Message Update Rule Use marginalization constraints to define message update rules: = b 5 = x 4 b 45 (x 4, x 5 )

46 Message Update Rule Use marginalization constraints to define message update rules: = b 5 = x 4 b 45 (x 4, x 5 )

47 Message Update Rule Use marginalization constraints to define message update rules: = m 4 5 (x 5 ) = k x 4 φ 4 (x 4 )ψ 45 (x 4, x 5 )m (x 4, x 5 )m (x 2, x 5 )

48 5.5. Run GBP algorithm The GBP algorithm runs in the same way as BP: 1. Initialize messages to unbiased states 2. Iterate through message update rules until (hopefully) convergence Occasionally, helpful to move only part-way to new values of messages at each iteration.

49 5.6. Generalized Belief Propagation Theorems: Stationary points of Bethe approximation are fixed points of BP Stationary points of Kikuchi approximation are fixed points of GBP. When region graph is a tree, message-passing algorithm is exact Empirically, more likely to converge than BP Can be nearly as fast, but much more accurate (depending on details of region graph)

50 6. Conclusions Standard BP is equivalent to minimizing Bethe Free Energy. The Bethe and Junction Tree approximations are both sub-classes of the Kikuchi approximation. GBP is equivalent to minimizing the Kikuchi Free Energy. GBP is exact when region graph is a tree.

51 References Jaakkola, T. (2001). Tutorial on variational approximation methods. In Opper, M. and Saad, D., editors, Advanced Mean Field Methods: Theory and Practice, chapter 10, pages MIT Press. Kappen, H. and Wiegerninck, W. (2001). Mean field theory for graphical models. In Opper, M. and Saad, D., editors, Advanced Mean Field Methods: Theory and Practice, chapter 4, pages MIT Press. Opper, M. and Winther, O. (2001). From naive mean field theory to the tap equations. In Opper, M. and Saad, D., editors, Advanced Mean Field Methods: Theory and Practice, chapter 2, pages MIT Press. Tanaka, T. (2001). Information geometry of mean-field approximation. In Opper, M. and Saad, D., editors, Advanced Mean Field Methods: Theory and Practice, chapter 17, pages MIT Press. Veytsman, B. and Kotelyanskii, M. (1997). Website: Ising model and its applications at www/matsc597c-1997/phases/lecture3/.

52 Weiss, Y. (2001). Comparing the mean field method and belief propagation for approximate inference in mrfs. In Opper, M. and Saad, D., editors, Advanced Mean Field Methods: Theory and Practice, chapter 15, pages MIT Press. Yedidia, J. (2001). An idiosynchratic journey beyond mean field theory. In Opper, M. and Saad, D., editors, Advanced Mean Field Methods: Theory and Practice, chapter 3, pages MIT Press.