6.899 Relational Data Learning

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6.899 Relational Data Learning

• Yuan Qi
• MIT Media Lab
• yuanqi_at_media.mit.edu
• May 7, 2002

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Outline

• Structure Learning Using Stochastic Logic
  Programming (SLP)
• Text Classification Using Probabilistic
• Relational Models (PRM)

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Part 1Structure Learning Using SLP

• SLP defines prior over BN structures
• MCMC sampling BN structures
• New Sampling Method

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An SLP Defining prior BN structures

• bn(,,).
• bn(RVRVs,BN,AncBN)- bn(RVs, BN2, AncBN2),
• connect_no_cycles(RV,BN2,AncBN2,BN,AncBN).
• An edge RV parent of H
• 1/3 which_edge(HT,RV,H-RVRest)-
• choose_edges(T,RV,Rest).
• An edge H parent of RV
• 1/3 which_edge(HT,RV,RV-HRest) -
• choose_edges(T,RV,Rest).
• No edge
• 1/3 which_edge(_HT,RV,Rest) -
• choose_edges(T,RV,Rest).

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Metropolis-Hasting Sampling

• p(T) specifies a tree prior for BN structures.
• Sampling T from the transition distribution
  q(Ti,T).
• Set Ti T with the acceptance ratio
• else set Ti1 Ti.

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The Transition Kernel (1)

• The transition kernel can be implemented by
  generating a new derivation(yielding a new model
  M) from the derivation which yields the current
  model Mi. To be specific, we have
• Backtrack one step to the most recent choice
  point in the SLD-tree (i.e., the probability
  tree)
• If at the top of the tree, stop. Otherwise,
  backtrack one more step to the next choice point
  with a predefined backtrack probability pb.

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The Transition Kernel (2)

• Once stopped backtracking, choose a new leaf M
  from the choice point by selecting branches
  according to their probabilities attached to them
  (loglinear sampling). However, we may not choose
  the branch that leads back to Mi.

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Sampling Problems

• Inefficiency of the previous Metropolis-Hasting
  sampling. pb 0.8, Acceptance ratio 4.
• lf pb is small, slow movement of the samples,
  higher acceptance ratio
• lf pb is large, large movement of the samples,
  lower acceptance ratio
• Fixed pb the balance between local jumps to
  neighboring models and big jumps to distant ones.
• An Improvement Cyclic transition kernel pb
  1-2-n for n 1,.28.

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Adaptive Sampling Strategy Re-Try the Proposals

• Suppose a proposal T1 from the proposal
  distribution q1(T, T1) is tried and rejected.
  The rejection suggests that this proposal
  distribution may not be good and a different
  proposal could be tried. Suppose a new sample T2
  is drawn from a new proposal q2(T, T1 , T2).
• But how to get a valid Markov sampling chain?

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Adaptive Sampling Strategy New Acceptance Ratio

• If we use the following acceptance ratio

then we have a valid MCMC sampler for the target
distribution, that is, the posterior of BN
structures.
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Part 1 Conclusion

• To adaptively sample BN structures, we can start
  with large backtrack probability pb, and if get
  rejected samples, we reduce pb and draw a
  sampled structure using the new backtrack
  probability. This process can be repeated.
• Adaptive proposal distribution allows the SLP
  sampler to locally tune its parameter to achieve
  a good balance between local jumps to neighboring
  models and big jumps to distant ones. Therefore,
  we expect a much more efficient sampling result.

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Part 2 Text Classification Using Probabilistic
Relational Models (PRM)

• Why using PRMs?
• SLP Discrete R.V.s
• PRM Discrete and Continuous R.V.s
• Why relational modeling of text?
• Author relation
• Citation relation

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Modeling Relational Text Data

• Figure 1.PRM modeling of text. By Taskar, Segal,
  and Koller

Unrolled Bayesian Network
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Transduction Train and Testing together

• The test data are also included in the model
• Transduction EM Algorithm
• E step Belief propagation
• M step Maximum Likelihood Re-estimation

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Several Problems of Modeling in Figure 1

• Naïve Bayes (Independence) assumption on
  generating words
• Wrong edge direction between words and topic
  nodes
• Wrong edge direction between a paper and its
  citations.

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Drawback of EM training and Transductions

• High dimensional data, relatively limited
  training points
• Transduction helps training, but is very
  expensive for testing, since we need retraining
  the whole model for a new data point .

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New Modeling and Bayesian Training

• The new node, h, models a classifier which takes
  input from words, aggregated citation and
  aggregated author.

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Training the new PRM

• Unrolling this new PRM, we get a Bayesian network
  modeling the text data.
• Training Extension of belief propagation,
  expectation propagation.
• We can also easily incorporate the kernel trick
  like in SVM or Gaussian processes into the
  classifier h. Note that h models the conditional
  relation between the text class and words,
  citations, and authors.

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Part 2 Conclusion

• Benefit of the new approach
• No overfitting like ML approaches
• Choice of using transduction or not.
• Much more powerful classifier, Bayesian Point
  Machine with Kernel Expansion, compared to Naïve
  Bayes method
• Better Relation modeling