1. Web-Mining Agents Part: Data Mining
Prof. Dr. Ralf Möller
Dr. Özgür Özcep
Universität zu Lübeck
Institut für Informationssysteme
Tanya Braun (Exercises)

2. Raymond J. Mooney University of Texas at Austin
Discriminative Training Markov Networks and Conditional Random Fields (CRFs)
Raymond J. Mooney
University of Texas at Austin 2 2

3. Probabilistic Classification
Let Y be the random variable for the class which takes values {y1,y2,…ym}.
Let X be the random variable describing an instance consisting of a vector of values for n features <X1,X2…Xn>, let xk be a possible vector value for X and xij a possible value for Xi.
For classification, we need to compute P(Y=yi | X=xk) for i = 1…m
Could be done using joint distribution but this requires estimating an exponential number of parameters.

4. Bayesian Categorization
Determine category of xk by determining for each yi
P(X=xk) can be determined since categories are complete and disjoint.

5. Bayesian Categorization (cont.)
Need to know:
Priors: P(Y=yi)
Conditionals: P(X=xk | Y=yi)
P(Y=yi) are easily estimated from data.
If ni of the examples in D are in yi then P(Y=yi) = ni / |D|
Too many possible instances (e.g. 2n for binary features) to estimate all P(X=xk | Y=yi).
Still need to make some sort of independence assumptions about the features to make learning tractable.

6. Naïve Bayes Generative Model
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7. Naïve Bayes Inference Problem
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8. Naïve Bayesian Categorization
If we assume features of an instance are independent given the category (conditionally independent).
Therefore, we then only need to know P(Xi | Y) for each possible pair of a feature-value and a category.
If Y and all Xi and binary, this requires specifying only 2n parameters:
P(Xi=true | Y=true) and P(Xi=true | Y=false) for each Xi
P(Xi=false | Y) = 1 – P(Xi=true | Y)
Compared to specifying 2n parameters without any independence assumptions.

9. Generative vs. Discriminative Models
Generative models are not directly designed to maximize the performance of classification. They model the complete joint distribution P(X,Y).
Classification is then done using Bayesian inference given the generative model of the joint distribution.
But a generative model can also be used to perform any other inference task, e.g. P(X1 | X2, …Xn, Y)
“Jack of all trades, master of none.”
Discriminative models are specifically designed and trained to maximize performance of classification. They only model the conditional distribution P(Y | X).
By focusing on modeling the conditional distribution, they generally perform better on classification than generative models when given a reasonable amount of training data.

10. Logistic Regression (for discrete vars)
Assumes a parametric form for directly estimating P(Y | X). For binary concepts, this is:
Equivalent to a one-layer backpropagation neural net.
Logistic regression is the source of the sigmoid function used in backpropagation.
Objective function for training is somewhat different.

11. Logistic Regression as a Log-Linear Model
Logistic regression is basically a linear model, which is demonstrated by taking logs.
Also called a maximum entropy model (MaxEnt) because it can be shown that standard training for logistic regression gives the distribution with maximum entropy that is consistent with the training data.

12. Logistic Regression Training
Weights are set during training to maximize the conditional data likelihood :
where D is the set of training examples and Yd and Xd denote, respectively, the values of Y and X for example d.
Equivalently viewed as maximizing the conditional log likelihood (CLL)

13. Logistic Regression Training
Like neural-nets, can use standard gradient descent to find the parameters (weights) that optimize the CLL objective function.
Many other more advanced training methods are possible to speed up convergence.

14. Preventing Overfitting in Logistic Regression
To prevent overfitting, one can use regularization (a.k.a. smoothing) by penalizing large weights by changing the training objective:
Where λ is a constant that determines the amount of smoothing
This can be shown to be equivalent to MAP parameter estimation assuming a Gaussian prior for W with zero mean and a variance related to 1/λ.

15. Multinomial Logistic Regression
Logistic regression can be generalized to multi-class problems (where Y has a multinomial distribution).
Create feature functions for each combination of a class value y´ and each feature Xj and another for the “bias weight” of each class.
f y´, j (Y, X) = Xj if Y= y´ and 0 otherwise
f y´ (Y, X) = 1 if Y= y´ and 0 otherwise
The final conditional distribution is :
Note: k in {1, ..., K} enumerates all possible combinations of class values with Xj plus the class values
(λk are weights)
(normalizing constant)

16. Multinomial Logistic Regression
Formula boils down to the following:
Assume Y has n class values
Let βk (1 <= k <= n) be feature weight vectors
Can set βn = 0 (zero vector)
P(Y = k | X) = e(βk * X) / (1 + Σn-1k = 1 e(βk * X) )
Special case of bolean class follows when taking one class value as pivot and setting one weight vector to zero

17. Graphical Models
If no assumption of independence is made, then an exponential number of parameters must be estimated for sound probabilistic inference.
No realistic amount of training data is sufficient to estimate so many parameters.
If a blanket assumption of conditional independence is made, efficient training and inference is possible, but such a strong assumption is rarely warranted.
Graphical models use directed or undirected graphs over a set of random variables to explicitly specify variable dependencies and allow for less restrictive independence assumptions while limiting the number of parameters that must be estimated.
Bayesian Networks: Directed acyclic graphs that indicate causal structure.
Markov Networks: Undirected graphs that capture general dependencies.

18. Bayesian Networks Directed Acyclic Graph (DAG)
Nodes are random variables
Edges indicate causal influences
Burglary
Earthquake
Alarm
JohnCalls
MaryCalls

19. Conditional Probability Tables
Each node has a conditional probability table (CPT) that gives the probability of each of its values given every possible combination of values for its parents (conditioning case).
Roots (sources) of the DAG that have no parents are given prior probabilities.
P(B)
.001
P(E)
.002
Burglary
Earthquake B E
P(A) T
.95 F
.94
.29
.001
Alarm A
P(J) T
.90 F
.05 A
P(M) T
.70 F
.01
JohnCalls
MaryCalls

20. Joint Distributions for Bayes Nets
A Bayesian Network implicitly defines a joint distribution.
Example

21. Naïve Bayes as a Bayes Net
Naïve Bayes is a simple Bayes Net Y … X1 X2 Xn
Priors P(Y) and conditionals P(Xi|Y) for Naïve Bayes provide CPTs for the network.

22. Markov Networks
Undirected graph over a set of random variables, where an edge represents a dependency.
The Markov blanket of a node, X, in a Markov Net is the set of its neighbors in the graph (nodes that have an edge connecting to X).
Every node in a Markov Net is conditionally independent of every other node given its Markov blanket.

23. Distribution for a Markov Network
The distribution of a Markov net is most compactly described in terms of a set of potential functions (a.k.a. factors, compatibility functions), φk, for each clique, k, in the graph.
For each joint assignment of values to the variables in clique k, φk assigns a non-negative real value that represents the compatibility of these values.
The joint distribution of a Markov network is then defined by:
Where x{k} represents the joint assignment of the variables in clique k, and Z is a normalizing constant that makes a joint distribution that sums to 1.

24. Sample Markov Network Burglary Earthquake Alarm JohnCalls MaryCalls E2 T 50 F 10 1
200 B A 1 T
100 F 1
200
Burglary
Earthquake
Alarm
JohnCalls
MaryCalls M A 4 T 50 F 1 10
200 J A 3 T 75 F 10 1
200

25. Logistic Regression as a Markov Net
Logistic regression is a simple Markov Net Y … X1 X2 Xn
But only models the conditional distribution, P(Y | X) and not the full joint P(X,Y)

26. Markov Nets vs. Bayes Nets

27. Markov Nets vs. Bayes Nets
Property
Markov Nets
Bayes Nets
Form
Prod. potentials
Potentials
Arbitrary
Cond. probabilities
Cycles
Allowed
Forbidden
Partition func.
Z = ? global
Normalization local
Indep. check
Graph separation
D-separation
Indep. props.
Some
Inference
MCMC etc.

28. Generative vs. Discriminative Sequence Labeling Models
HMMs are generative models and are not directly designed to maximize the performance of sequence labeling. They model the joint distribution P(O,Q).
HMMs are trained to have an accurate probabilistic model of the underlying language, and not all aspects of this model benefit the sequence labeling task.
Conditional Random Fields (CRFs) are specifically designed and trained to maximize performance of sequence labeling. They model the conditional distribution P(Q | O)
HMM = Hidden Markov Model

29. … … Classification Naïve Bayes Discriminative Logistic Regression Y X1Xn
Generative
Discriminative
Conditional Y
Logistic
Regression … X1 X2 Xn

30. .. … .. … Sequence Labeling HMM Discriminative Linear-chain CRF Y1 Y2YT
HMM … X1 X2 XT
Generative
Discriminative
Conditional .. Y1 Y2 YT
Linear-chain CRF … X1 X2 XT

31. Simple Linear Chain CRF Features
Modeling the conditional distribution is similar to that used in multinomial logistic regression.
Create feature functions fk(Yt, Yt−1, Xt)
Feature for each state transition pair i, j
fi,j(Yt, Yt−1, Xt) = 1 if Yt = i and Yt−1 = j and 0 otherwise
Feature for each state observation pair i, o
fi,o(Yt, Yt−1, Xt) = 1 if Yt = i and Xt = o and 0 otherwise
Note: number of features grows quadratically in the number of states (i.e. tags).

32. Conditional Distribution for Linear Chain CRF
Using these feature functions for a simple linear chain CRF, we can define:

33. Adding Token Features to a CRF
Can add token features Xi,j … Y1 Y2 YT … … … …
X1,1
X1,m
X2,1
X2,m
XT,1
XT,m
Can add additional feature functions for each token feature to model conditional distribution.

34. Features in POS Tagging
For POS Tagging, use lexicographic features of tokens.
Capitalized?
Start with numeral?
Ends in given suffix (e.g. “s”, “ed”, “ly”)?

35. Enhanced Linear Chain CRF (standard approach)
Can also condition transition on the current token features. … Y1 Y2 YT …
X1,1
X2,1
XT,1 … … …
X1,m
X2,m
XT,m
Add feature functions:
fi,j,k(Yt, Yt−1, X) 1 if Yt = i and Yt−1 = j and Xt −1,k = and 0 otherwise

36. Supervised Learning (Parameter Estimation)
As in logistic regression, use L-BFGS optimization procedure, to set λ weights to maximize CLL of the supervised training data.
See paper for details.

37. Sequence Tagging (Inference)
Variant of Viterbi algorithm can be used to efficiently, O(TN2), determine the globally most probable label sequence for a given token sequence using a given log-linear model of the conditional probability P(Y | X).
See paper for details.

38. Skip-Chain CRFs
Can model some long-distance dependencies (i.e. the same word appearing in different parts of the text) by including long-distance edges in the Markov model. Y1 Y2 Y3
Y100
Y101 … X1 X2 X3
X100
X101
Michael Dell said
Dell bought
Additional links make exact inference intractable, so must resort to approximate inference to try to find the most probable labeling.

39. CRF Results
Experimental results verify that they have superior accuracy on various sequence labeling tasks.
Part of Speech tagging
Noun phrase chunking
Named entity recognition
Semantic role labeling
However, CRFs are much slower to train and do not scale as well to large amounts of training data.
Training for POS on full Penn Treebank (~1M words) currently takes “over a week.”
Skip-chain CRFs improve results on IE. 40 40

40. CRF Summary
CRFs are a discriminative approach to sequence labeling whereas HMMs are generative.
Discriminative methods are usually more accurate since they are trained for a specific performance task.
CRFs also easily allow adding additional token features without making additional independence assumptions.
Training time is increased since a complex optimization procedure is needed to fit supervised training data.
CRFs are a state-of-the-art method for sequence labeling.