Rome - Feb. 2010

Lesson 6

Belief propagation

Sergio Barbarossa

Rome - Feb. 2010

Pairwise Markov Random Fields

Scene: (for example, the pixels of an image)

-  The scene of interest must have a structure

-  A general structure model is the statistical dependency among the pixels of the scene

-  The field is pairwise Markov if the statistical dependency can be expresses through functions of pairs of nodes

Observation:

-  The observation is itself related to the scene through a statistical model

Observation model

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The best way to model a pairwise statistical dependency is a graph representation, where each vertex represents a random variable

There is an edge between the nodes representing the random variables and if and only if the compatibility function of the pair is different from zero

The overall statistical dependency is

The goal is to recover the underlying field from the observations

Observation model

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Examples

x2 x1

x3

x4

y1 y2

y3

y4

x1

y1 y2 y3

y4

The number of observation could be equal to the number of unknowns

The number of observation could be greater than the number of unknowns

Observation model

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Standard belief propagation

Goal: allow each node to compute its belief (a posteriori probability), given the observations of all the nodes, in a totally distributed way

Each variable exchanges messages only with its neighbors

In a MRF with discrete rv’s, the messages are vectors of size equal to the number of values that the rv can assume

The message sent from node i to node j is about what state node j should be in, according to node i:

Belief propagation

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The belief computed at node i is proportional to the product of the local evidence and the messages coming into node i

Belief propagation

Flow of messages

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Belief propagation

x2 x1

x3

x4

y1 y2

y3

y4

Belief at node 1:

Example

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Belief propagation

BP is a distributed way to compute a marginal pdf

If the graph has no loops, BP provides the exact marginal at every node in a finite number of steps

The whole computation takes a time proportional to the number of links, which is significantly less than the exponential growth resulting from the straightforward saturation of variables

Substituting

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Belief propagation

BP is indeed a way to organize global computations of marginal beliefs in terms of local (simpler) computations

The BP algorithm described above does not make any reference to the graph topology. However, if there are loops and they are ignored, the messages could circulate forever and the process might not converge or it might converge to an incorrect value

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Consensus estimation via belief propagation

Consensus estimation can be interpreted as BP over the following graph, where yi denote the observation variables and x is the common parameter to be estimated

x

y1 y2 y3

y4

The evidence function in this case is

The compatibility function is

The message is

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Consensus estimation via belief propagation

Taking logarithms

Distributed consensus ends up into a (local) linear combination of logarithms of the messages

In the Gaussian case, the nodes must exchange only mean vector and covariance matrices

The convergence depends on the topology of the graph

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Consensus estimation via belief propagation

Given

we can define

and use the alternative notation

Let

and consider the product

and

and

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Consensus estimation via belief propagation

Then

with

Observation model

with

and

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Consensus estimation via belief propagation

Messages at step n

After a number of steps proportional to the number of links, every node has

And then it is able to compute the globally optimal ML estimate

and

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References

[1] J. Yedidia, W. Freeman, Y. Weiss, “Understanding Belief Propagation and its Generalizations”

[2] Y. Weiss, W. Freeman, “Correctness of belief propagation in Gaussian graphical models of arbitrary topology”