Markov Networks

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Markov Networks
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Markov Networks

• Undirected graphical models

B
A
D
C

• Potential functions defined over cliques

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Markov Networks

• Undirected graphical models

B
A
D
C

• Potential functions defined over cliques

Weight of Feature i
Feature i
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Hammersley-Clifford Theorem

• If Distribution is strictly positive (P(x) gt 0)
• And Graph encodes conditional independences
• Then Distribution is product of potentials over
  cliques of graph
• Inverse is also true.
• (Markov network Gibbs distribution)

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Markov Nets vs. Bayes Nets
Property Markov Nets Bayes Nets
Form Prod. potentials Prod. potentials
Potentials Arbitrary Cond. probabilities
Cycles Allowed Forbidden
Partition func. Z ? Z 1
Indep. check Graph separation D-separation
Indep. props. Some Some
Inference MCMC, BP, etc. Convert to Markov
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Inference in Markov Networks

• Goal compute marginals conditionals of
• Exact inference is P-complete
• Conditioning on Markov blanket is easy
• Gibbs sampling exploits this

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Markov Chain Monte Carlo

• Gibbs Sampler
• 1. Start with an initial assignment to nodes
• 2. One node at a time, sample node given
  others
• 3. Repeat
• 4. Use samples to compute P(X)
• Convergence Burn-in Mixing time
• Many modes ? Multiple chains

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Other Inference Methods

• Belief propagation (sum-product)
• Mean field / Variational approximations

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MAP Inference

• Iterated conditional modes
• Simulated annealing
• Graph cuts
• Belief propagation (max-product)

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Learning Weights

• Maximize likelihood (or posterior)
• Convex optimization gradient ascent,
  quasi-Newton methods, etc.
• Requires inference at each step (slow!)

Feature count according to data
Feature count according to model
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Pseudo-Likelihood

• Likelihood of each variable given its Markov
  blanket in the data
• Does not require inference at each step
• Very fast numerical optimization
• Ignores non-local dependences

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Learning Structure

• Feature search
• 1. Start with atomic features
• 2. Form conjunctions of pairs of features
• 3. Select best and add to feature set
• 4. Repeat until no improvement
• Evaluation
• Likelihood, K-L divergence
• Approximation Previous weights dont change