Text Categorization Moshe Koppel Lecture 2: Naïve Bayes

1. Text CategorizationMoshe KoppelLecture 2: Naïve Bayes Slides based on Manning, Raghavan and Schutze

2. Naïve Bayes: Why Bother? • Tightly tied to text categorization • Interesting theoretical properties. • A simple example of an important class of learners based on generative models that approximate how data is produced • For certain special cases, NB is the best thing you can do.

3. Bayes’ Rule

4. Maximum a posteriori Hypothesis As P(D) is constant

5. Maximum likelihood Hypothesis If all hypotheses are a priori equally likely,we only need to consider the P(D|h) term:

6. Naive Bayes Classifiers Task: Classify a new instance D based on a tuple of attribute values into one of the classes cj C

7. Naïve Bayes Classifier: Naïve Bayes Assumption • P(cj) • Can be estimated from the frequency of classes in the training examples. • P(x1,x2,…,xn|cj) • O(|X|n•|C|) parameters • Could only be estimated if a very, very large number of training examples was available. Naïve Bayes Conditional Independence Assumption: • Assume that the probability of observing the conjunction of attributes is equal to the product of the individual probabilities P(xi|cj).

8. Smoothing to Avoid Overfitting # of values ofXi overall fraction in data where Xi=xi,k • Somewhat more subtle version extent of “smoothing”

9. Naive Bayes for Text Categorization • Attributes are text positions, values are words. • Still too many possibilities • Assume that classification is independent of the positions of the words • Use same parameters for each position • Result is bag of words model (over tokens not types)

10. Naïve Bayes: Learning • From training corpus, extract Vocabulary • Calculate required P(cj)and P(xk | cj)terms • For each cj in C do • docsjsubset of documents for which the target class is cj • Textj single document containing all docsj • for each word xkin Vocabulary • nk number of occurrences of xkin Textj

11. Naïve Bayes: Classifying • positions  all word positions in current document which contain tokens found in Vocabulary • Return cNB, where

12. Underflow Prevention • Multiplying lots of probabilities, which are between 0 and 1 by definition, can result in floating-point underflow. • Since log(xy) = log(x) + log(y), it is better to perform all computations by summing logs of probabilities rather than multiplying probabilities. • Class with highest final un-normalized log probability score is still the most probable.

13. multiply Naïve Bayes as Stochastic Language Models • Model probability of generating strings (each word in turn) in the language (commonly all strings over ∑). E.g., unigram model Model M 0.2 the 0.1 a 0.01 man 0.01 woman 0.03 said 0.02 likes … the man likes the woman 0.2 0.01 0.02 0.2 0.01 P(s | M) = 0.00000008

14. the class pleaseth yon maiden 0.2 0.01 0.0001 0.0001 0.0005 0.2 0.0001 0.02 0.1 0.01 Naïve Bayes as Stochastic Language Models • Model probability of generating any string Model M1 Model M2 0.2 the 0.0001 class 0.03 sayst 0.02 pleaseth 0.1 yon 0.01 maiden 0.0001 woman 0.2 the 0.01 class 0.0001 sayst 0.0001 pleaseth 0.0001 yon 0.0005 maiden 0.01 woman P(s|M2) > P(s|M1)

15. P ( ) P ( ) P ( ) P ( ) P ( ) P ( ) P ( | ) P ( | ) P ( | ) P ( | ) = P ( ) P ( | ) P ( | ) Unigram and higher-order models • Unigram Language Models • Bigram (generally, n-gram) Language Models Easy. Effective!

16. Smoothing and Backoff • Suppose we’re using a trigram model. We need to estimate P(w3 | w1,w2) • It will often be the case that the trigram w1,w2,w3 is rare or non-existent in the training corpus.(Similar to problem we saw above with unigrams.) • First resort: backoff. Estimate P(w3 | w1,w2) using P(w3 | w2) • Alternatively, use some very large backup corpus. • Various combinations have been tried.

17. Multinomial Naïve Bayes = class conditional language model Cat • Think of wi as the ith word in the document • Effectively, the probability of each class is done as a class-specific unigram language model w1 w2 w3 w4 w5 w6

18. But Wait! Another Approach Cat • Now think of wi as the ith word in the dictionary (not the document) • Each value is either 1 (in the doc) or 0 (not) This is very different than the multinomial method. McCallum and Nigam (1998) observed that the two were often confused. w1 w2 w3 w4 w5 w6

19. Binomial Naïve Bayes • One feature Xw for each word in dictionary • Xw = true in document d if w appears in d • Naive Bayes assumption: Given the document’s topic, appearance of one word in the document tells us nothing about chances that another word appears

20. Parameter Estimation • Binomial model: • Multinomial model: • Can create a mega-document for topic j by concatenating all documents in this topic • Use frequency of w in mega-document fraction of documents of topic cj in which word w appears fraction of times in which word w appears across all documents of topic cj

21. Experiment: Multinomial vs Binomial • M&N (1998) did some experiments to see which is better • Determine if a university web page is {student, faculty, other_stuff} • Train on ~5,000 hand-labeled web pages • Cornell, Washington, U.Texas, Wisconsin • Crawl and classify a new site (CMU)

22. Multinomial vs. Binomial

23. Conclusions • Multinomial is better • For Binomial, it’s really important to do feature filtering • Other experiments bear out these conclusions

24. Feature Filtering • If irrelevant words mess up the results, let’s try to use only words that might help • In training set, choose k words which best discriminate the categories. • Best way to choose: for each category build a list of j most discriminating terms

25. Infogain • Use terms with maximal Mutual Information with the classes: • For each word w and each category c (This is equivalent to the usual two-class Infogain formula.)

26. Chi-Square Feature Selection X2 = N(AD-BC)2 / ( (A+B) (A+C) (B+D) (C+D) ) For complete independence of term and category: AD=BC

27. Feature Selection • Many other measures of differentiation have been tried. • Empirical tests suggest Infogain works best. • Simply eliminating rare terms is easy and usual doesn’t do much harm. • Be sure not to use test data when you do feature selection. (This is tricky when you’re using k-fold cross-validation.)

28. Naïve Bayes: Conclusions • Classification results of naïve Bayes (the class with maximum posterior probability) are usually fairly accurate, though not nearly as good as, say, SVM. • However, due to the inadequacy of the conditional independence assumption, the actual posterior-probability numerical estimates are not. • Output probabilities are generally very close to 0 or 1.

29. Some Good Things about NB • Theoretically optimal if the independence assumptions hold • Fast • Sort of robust to irrelevant features (but not really) • Very good in domains with many equally important features • Probably only method useful for very short test documents (Why?)