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

SLP defines prior over BN structures

MCMC sampling BN structures

New Sampling Method

An SLP Defining prior BN structuresbn([],[],[]). 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 RV1/3:: which_edge([H|T],RV,[RV-H|Rest]) :-choose_edges(T,RV,Rest).% No edge1/3:: which_edge([_H|T],RV,Rest) :-choose_edges(T,RV,Rest).

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.

1,)(),|(),(

)(),|(),(min),(

*

****

iii

ii

TpTXYpTTq

TpTXYpTTqTT

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.

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.

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.

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?

Adaptive Sampling Strategy: New Acceptance Ratio

If we use the following acceptance ratio:

1,),,()),(1)(,()(),|(

),,()),(1)(,()(),|(min

),,(

2102

10101

00

0122

12121

22

210

TTTqTTTTqTpTXYp

TTTqTTTTqTpTXYp

TTT

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

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.

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

Modeling Relational Text Data

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

Unrolled Bayesian Network

Transduction: Train and Testing together

The test data are also included in the model

Transduction: EM AlgorithmE step: Belief propagation

M step: Maximum Likelihood Re-estimation

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.

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 .

New Modeling and Bayesian Training

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

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.

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