CS774. Markov Random Field : Theory and Application
Lecture 04
Kyomin JungKAIST
Sep 15 2009
Basic Idea of Belief Propagation (BP)
Let be the marginal prob. of the MRF on the subtree rooted at j, and so on.
i
j k
… …
)1()1,0()0()0,0()1()1,0()0()0,0()0()0( kikkikjijjijii ZZZZZ
)( jj xZ
)( jj xZ )( kk xZ
)( ii xZ
Belief Propagation (BP)
i j k
)( jj x
),( ijji xx
ijNkjjk
tjj
xijjiiij
t xmxxxxmj \)(
1 )()(),()(
)( jj xZ
)(
)()()(iNj
itijiii
ti xmxxb
i j∏
Belief Propagation (BP)
ijNkjjk
t
xjjijjiiij
t xmxxxxmj \)(
1 )()(),()(
Belief at node i at time t:
Ni
For t>n, and)()(1i
tijiij
t xmxm )()( 1
itii
ti xbxb
Properties of BP (and MP)
Exact for trees Each node separates Graph into 2 disjoint
components
On a tree, the BP algorithm converges in time proportional to diameter of the graph – at most linear
For general Graphs Exact inference is NP-hard Constant Approximate inference is hard
Loopy Belief Propagation
Approaches for general graphsExact Inference
Computation tree based approach (for graph with large girth)
Junction Tree algorithm (for bounded tree width graph)
Graph cut algorithm (for submodular MRF)Approximate Inference
Loopy BP Sampling based algorithm Graph decomposition based approximation
Loopy Belief Propagation
If BP is used on graphs with loops, messages may circulate indefinitely
Empirically, a good approximation is still achievableStop after fixed # of iterationsStop when no significant change in be-
liefs If solution is not oscillatory but con-
verges, it usually is a good approxima-tion
Example: LDPC Codes
Fixed point of BP
Messages of BP at time t forms a di-mensional real vector. Let M(t) be this vector.
If we normalize , the output of BP(marginal probabilities) is the same.
BP algorithm is a continuous function that maps M(t) to M(t+1). BP:
Hence, by Brouwer Fixed Point Theorem, BP has at least one fixed point. (since the domain is a convex, compact set)
||||2 E
||2||)(|| 1 EtM
}2||:||{}2||:||{ 1||||
1|||| ExRxExRx EE
Fixed point of BP
Now important questions are “Is there a unique fixed point ?”“Does BP converges to a fixed point ?”“If it does, how fast ?”
Studying these questions are of current re-search topics. Ex, studying them for restricted class of MRF
(ex graphs with large girth) Studying relations of BP fixed point with other
values (ex Minima of the Bethe Free energy)
Girth of a Graph
For a graph G=(V,E), the girth of G is the length of a shortest cycle contained in G.
If G has girth, and bounded de-gree, and the MRF satisfies exponential (s-patial) correlation decay, then BP com-putes good approximation of the solution.Proof: By considering computation tree
of BP It can be used to design a system based
on MRF Ex: LDPC code
)(logn
Computation Tree of BP
Graph G Computation tree of G at x1
(Temporal) Decay of correlations in Markov chains
A Markov chain with transition matrix satisfies decay of correlation (mixes)
if and only if it is aperiodic
(Spatial) Decay of correlations
Same thing, but time is replaced by a “spatial” distance
Correlation Decay
A sequence of spatially (graph) related random variables
exhibits a correlation decay(long-range independence), if when is large
Principle motivation - statistical phyisics. Uniqueness of Gibbs measures on infinite lattices, Dobrushin [60s].
Correlation Decay
),( VvX v
],[]|[ vvgvv xXPyBxXP y
Weitz [05]. Independent sets - graph
Goldberg, Martin & Paterson [05]. Coloring. General graphs
Jonasson [01]. Coloring. Regular trees
• is the maximum vertex degree of G.
• in the independent set is the weight for each vertex.
(i.e. weight for an independent set of size I is )
• q in the coloring problem is the number of possible colors.
||I
What is known about correlation decay ?