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6.899 Relational Data Learning

6.899 Relational Data Learning. Yuan Qi MIT Media Lab yuanqi@media.mit.edu May 7, 2002. Outline. Structure Learning Using Stochastic Logic Programming (SLP) Text Classification Using Probabilistic Relational Models (PRM). Part 1:Structure Learning Using SLP.

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6.899 Relational Data Learning

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  1. 6.899 Relational Data Learning Yuan Qi MIT Media Lab yuanqi@media.mit.edu May 7, 2002

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

  3. Part 1:Structure Learning Using SLP • SLP defines prior over BN structures • MCMC sampling BN structures • New Sampling Method

  4. An SLP Defining prior BN structures bn([],[],[]). bn([RV|RVs],BN,AncBN):- bn(RVs, BN2, AncBN2), connect_no_cycles(RV,BN2,AncBN2,BN,AncBN). % An edge: RV parent of H 1/3:: which_edge([H|T],RV,[H-RV|Rest]):- choose_edges(T,RV,Rest). % An edge: H parent of RV 1/3:: which_edge([H|T],RV,[RV-H|Rest]) :- choose_edges(T,RV,Rest). % No edge 1/3:: which_edge([_H|T],RV,Rest) :- choose_edges(T,RV,Rest).

  5. 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 Ti+1 = Ti.

  6. 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.

  7. 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.

  8. 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.

  9. 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?

  10. 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.

  11. 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.

  12. 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

  13. Modeling Relational Text Data Figure 1.PRM modeling of text. By Taskar, Segal, and Koller Unrolled Bayesian Network

  14. 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

  15. 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.

  16. 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 .

  17. New Modeling and Bayesian Training The new node, h, models a classifier which takes input from words, aggregated citation and aggregated author.

  18. 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.

  19. 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

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