Belief Propagation on Markov Random Fields

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Belief Propagation on Markov Random Fields

• Aggeliki Tsoli

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Outline

• Graphical Models
• Markov Random Fields (MRFs)
• Belief Propagation

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Graphical Models

• Diagrams
• Nodes random variables
• Edges statistical dependencies among random
  variables
• Advantages
• Better visualization
• conditional independence properties
• new models design
• Factorization

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Graphical Models types

• Directed
• causal relationships
• e.g. Bayesian networks
• Undirected
• no constraints imposed on causality of events
  (weak dependencies)
• Markov Random Fields (MRFs)

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Example MRF Application Image Denoising
Noisy image e.g. 10 of noise
Original image (Binary)

• Question How can we retrieve the original image
  given the noisy one?

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MRF formulation

• Nodes
• For each pixel i,
• xi latent variable (value in original image)
• yi observed variable (value in noisy image)
• xi, yi ? 0,1

y1
y2
x1
x2
yi
xi
yn
xn
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MRF formulation

• Edges
• xi,yi of each pixel i correlated
• local evidence function ?(xi,yi)
• E.g. ?(xi,yi) 0.9 (if xi yi) and ?(xi,yi)
  0.1 otherwise (10 noise)
• Neighboring pixels, similar value
• compatibility function ?(xi, xj)

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MRF formulation
P(x1, x2, , xn) (1/Z) ?(ij) ?(xi, xj) ?i ?(xi,
yi)

• Question What are the marginal distributions for
  xi, i 1, ,n?

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Belief Propagation

• Goal compute marginals of the latent nodes of
  underlying graphical model
• Attributes
• iterative algorithm
• message passing between neighboring latent
  variables nodes
• Question Can it also be applied to directed
  graphs?
• Answer Yes, but here we will apply it to MRFs

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Belief Propagation Algorithm

• Select random neighboring latent nodes xi, xj
• Send message mi?j from xi to xj
• Update belief about marginal distribution at node
  xj
• Go to step 1, until convergence
• How is convergence defined?

yi
yj
xi
xj
mi?j
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Step 2 Message Passing

• Message mi?j from xi to xj what node xi thinks
  about the marginal distribution of xj

yi
yj
N(i)\j
xi
xj
mi?j(xj) ?(xi) ?(xi, yi) ?(xi, xj) ?k?N(i)\j
mk?i(xi)

• Messages initially uniformly distributed

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Step 3 Belief Update

• Belief b(xj) what node xj thinks its marginal
  distribution is

N(j)
yj
xj
b(xj) k ?(xj, yj) ?q?N(j) mq?j(xj)
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Belief Propagation Algorithm

1. Select random neighboring latent nodes xi, xj
2. Send message mi?j from xi to xj
3. Update belief about marginal distribution at node
   xj
4. Go to step 1, until convergence

yi
yj
xi
xj
mi?j
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Example

• - Compute belief at node 1.

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m3?2
Fig. 12 (Yedidia et al.)
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1
m2?1
m4?2
4
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Does graph topology matter?

• BP procedure the same!
• Performance
• Failure to converge/predict accurate beliefs
  Murphy, Weiss, Jordan 1999
• Success at
• decoding for error-correcting codes Frey and
  Mackay 1998
• computer vision problems where underlying MRF
  full of loops Freeman, Pasztor, Carmichael 2000

vs.
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How long does it take?

• No explicit reference on paper
• My opinion, depends on
• nodes of graph
• graph topology
• Work on improving the running time of BP (for
  specific applications)
• Next time?

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Questions?
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Next time ?

• BP on directed graphs
• Improve running time of BP
• More about loopy BP
• Can an initial estimation of messages
  (non-uniform) alleviate the problem?