Undirected graphical models

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Undirected graphical models

• CS2750 Project report
• Tomas Singliar
• tomas_at_cs.pitt.edu

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Outline

• Graphical models
• directed and undirected models
• Representation and problems
• Boltzmann machines
• Inference
• Monte Carlo methods
• Variational methods (mean-field)
• Loopy belief propagation
• Learning
• Applications

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Graphical models NLP example

• We want to represent how words appear together in
  text
  
• Lots of ambiguity involved
• Would be difficult to represent with a directed
  model

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Graphical models NLP example

• Split the nodes according to semantics
• The Rolling Stones spokesman agreed that the
  rock stars prefer beer to lime juice.
  
• Disambiguation of lime take the node that
  achieves the higher probability

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Representation

• Markov condition
• Each node is independent of all other given its
• neighbors (the Markov blanket)

• The joint probability is expressed as product of
  local potentials

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Boltzmann machines

• Graph of stochastic units, take on values 0, 1
• Always includes hidden units
• Potentials are pairwise defined weights wij and
  bias weights hi
  
• Energy (optimized in Metropolis-Hastings or
  simulated annealing)
  
• Gibbs distribution low energy high
  probability
  
• Update rule - stochastic

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Inference

• Inference task given evidence nodes E, compute
• Boltzmann machine MC method
• Gibbs sampler
• Choose i so that every i appears often enough
• sample from
• converges to true distribution under condition of
  positivity
  
• Variational methods
• choose approximating distribution
• e.g. all independent (mean-field)
• minimize KL divergence KL(PQ)

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Inference Boltzmann machine mean field

• assume product of independent Bernoulli
  distributions
  
• minimize KL divergence
• compute derivatives w.r.t. ?i and set equal to 0
• we get sigma-shaped fixpoint equations

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Inference

• Variational methods
• mean field all nodes disconnected
• can be imprecise
• allow more edges, e.g. acyclic graphs are easy
  junction tree
  
• cluster trade speed for precision
• Loopy belief propagation
• Belief propagation applied to loopy graphs
• Not guaranteed to converge
• Works out well for some graphs in practice
• If converges, then to Bethe/Kikuchi free energy
  minima
  

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Learning - MCMC

• Monte Carlo methods
• Boltzmann machines
• clamp evidence nodes
• run network (minimize energy, Metropolis
  algorithm)
  
• increment weights where both on capture
  dependencies
  
• release evidence nodes
• run network
• decrement where both on remove spurious
  correlations
  
• loop
• comes from minimizing KL-divergence between free
  and clamped
  

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Learning - IPF

• by maximizing data likelihood we get the
  condition
  
• model marginals equal empirical marginals
• IPF goes after that rescale potentials so that
  the above holds
  
• yields iterative equation
• performs log-likelihood ascent in coordinates fC
• can also be derived by minimizing KL

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Applications

• Magnetic properties material physics
• Probabilistic image modeling
• Image transferred trough noisy channel
• MRF encodes channel noise properties
• Restored image MAP estimate of image before
  noise
  
• Natural language processing
• word/lexeme distribution in corpus
• Network reliability analysis
• percolation phenomena
• failure of one component puts other under stress

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Undirected graphical models