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					Statistics and The War on Spam
David Madigan Rutgers University
Abstract Text categorization algorithms assign texts to predeﬁned categories. The study of such algorithms has a rich history dating back at least forty years. In the last decade or so, the statistical approach has dominated the literature. The essential idea is to infer a text categorization algorithm from a set of labeled documents, i.e., documents with known category assignments. The algorithm, once learned, automatically assigns labels to new documents. Motivated by a successful application to “spam” or “unsolicited bulk e-mail” ﬁltering, this chapter will present the “Naive Bayes” approach to statistical text categorization.



“Are you an Inventor? Have a Great Idea? We can help” “Now available - cash maker” “Find out if your mortgage rate is too high, NOW. Free Search” “Instantaneously Attract Women” “Conﬁrming your FREE PDA” These are the subject lines from unsolicited e-mail messages I received in a one hour period. Once a mild annoyance, unsolicited bulk e-mail–also known as spam– comprises close to half of all message traﬃc on the Internet. Spam is a big problem for several reasons: Spam imposes a signiﬁcant burden on Internet Service Providers; the resulting costs ultimately trickle down to individual subscribers. Spam supports pyramid schemes, dubious health products and remedies, and other fraudulent activities. Spam exposes minors to innappropriate material. Spam results in a loss of 1

productivity; the cumulative costs add up quickly when all e-mail users spend a few minutes a day dealing with and disposing of spam. Defenders of spam draw comparisons with print catalogs. Catalog companies use regular mail to send catalogs to potential new customers. Why is it OK to send unsolicited catalogs, and not OK to send unsolicited e-mails? One key diﬀerence is cost - spammers can send millions of messages for little or no cost. Paul Graham, an anti-spam activist, proposed the following thought experiment1 :

“Suppose instead of getting a couple print catalogs a day, you got a hundred. Suppose that they were handed to you in person by couriers who arrived at random times all through the day. And ﬁnally suppose they were all just ugly photocopied sheets advertising pornography, diet drugs, and mortgages.”

Given the central role of e-mail in our professional lives, the analogy with spam is not so farfetched. The problem of spam is attracting increasing media attention and a sort of War on Spam has begun. In what follows I describe the critical role Statistics plays in this War.


Spam Filters

Systemic progress against spam may require complex worldwide legislation. Individual users, however, can protect themselves with so-called “spam ﬁlters.” A spam ﬁlter is a computer program that scans incoming e-mail and sends likely spam to a special spam folder. Spam ﬁlters can make two kinds of mistakes. A ﬁlter might fail to detect that a particular message is spam and incorrectly let it through to the inbox (a false negative). Equally a ﬁlter might incorrectly route a perfectly good message to the spam folder (a false positive). An eﬀective spam ﬁlter will have a low false negative rate, and perhaps more criticially, a low false positive rate.


Early spam ﬁlters used hand-crafted rules to identify spam. Here are the antecedents for some typical spam rules: • <Subject> contains “FREE” in CAPS • <Body> contains “University Diploma” • <Body> contain an entire line in CAPS • <From:> starts with numbers This approach can produce eﬀective spam ﬁlters. In fact, merely looking for the word “FREE” for instance, catches about 60% of the e-mails in my collection of spam, with less than 1% false positives. However, the hand-crafting approach suﬀers from two signiﬁcant drawbacks. First, creating rules is a tedious, expensive, and error-prone process. Second, humans are unlikely to spot more obscure indicators of spam such as the presence of particular HTML tags or the use of certain font colors. In the last couple of years, the statistical approach to constructing spam ﬁlters has emerged as a method-of-choice and provides the core technology for several leading commercial spam ﬁlters. The statistical approach starts with a collection of e-mail messages that have been hand-labeled as spam or not-spam; creating these “training data” is the hard part. Next, a statistical algorithm scans the collection and identiﬁes discriminating features along with associated weights. Finally, the trained algorithm scans new e-mail messages as they arrive and automatically labels each one as spam or not-spam. Spam ﬁlter builders use many diﬀerent statistical algorithms including decision trees, nearest-neighbor methods, and support vector machines. In what follows, we describe “Naive Bayes,” a straightforward and popular approach, that has achieved excellent results on some standard test collections.


Representing E-mail for Statistical Algorithms

All statistical algorithms for spam ﬁltering begin with a vector representation of individual e-mail messages. Researchers have studied many diﬀerent representations 3

but most applications use the so-called “bag-of-words” representation. Bag-of-words represents each e-mail message by a “term vector.” The length of the term vector is the number of distinct words in all the e-mail messages in the training data. The entry for a particular word in the term vector for a particular e-mail message is usually the number of occurences of the word in the e-mail message. For example, Table 1 presents toy training data comprising four e-mail messages. These data contain ten distinct words: the, quick, brown, fox, rabbit, ran, and, run, at, and rest. Table 2 shows the corresponding set of term vectors. # 1 2 3 4 Message the quick brown fox the quick rabbit ran and ran rabbit run run run rabbit at rest Spam no yes no yes

Table 1: Training data comprising four labeled e-mail messages. # 1 2 3 4 and 0 1 0 0 at brown 0 0 0 1 1 0 0 0 fox 1 0 0 0 quick 1 1 0 0 rabbit 0 1 1 1 ran 0 2 0 0 rest run 0 0 0 1 0 0 3 0 the 1 1 0 0

Table 2: Term vectors corresponding to the training data. If the training data comprise thousands of e-mail messages, the number of distinct words often exceeds 10,000. Two simple strategies to reduce the size of the term vector somewhat are to remove “stop words” (words like and, of, the, etc.) and to reduce words to their root form, a process known as stemming (so, for example, “ran” and “run” reduce to “run”). Table 3 shows the reduced term vectors along with the spam label. The bag-of-words representation has some limitations. In particular, this representation contains no information about the order in which the words appear in the e-mail messages! 4

# 1 2 3 4

X1 brown 1 0 0 0

X2 fox 1 0 0 0

X3 quick 1 1 0 0

X4 rabbit 0 1 1 1

X5 rest 0 0 0 1

X6 run 0 2 3 0

Y Spam 0 1 0 1

Table 3: Term vectors after stemming and stopword removal with the Spam label, coded as 0=no, 1=yes.


Naive Bayes for Spam

Let X = (X1 , . . . , Xd ) denote the term vector for a random e-mail message, where d is the number of distinct words in the training data, after stemming and stopword removal. Let Y denote the corresponding spam label. The Naive Bayes model seeks to build a model for: P r(Y = 1|X1 = x1 , . . . , Xd = xd ). From Bayes theorem, we have: P r(Y = 1|X1 = x1 , . . . , Xd = xd ) = or, on the log odds scale, P r(Y = 1) P r(X1 = x1 , . . . , Xd = xd |Y = 1) P r(Y = 1|X1 = x1 , . . . , Xd = xd ) = log +log . P r(Y = 0|X1 = x1 , . . . , Xd = xd ) P r(Y = 0) P r(X1 = x1 , . . . , Xd = xd |Y = 0) (2) The log odds scale avoids the normalizing constant in the denominator of the right hand side of (1). log The ﬁrst term on the right hand side of (2) involves the prior probability that a random message is spam. The second term on the right hand side of (2) involves two conditional probabilities, namely the conditional probability of a term vector given that the term vector’s message is spam, and the conditional probability of a term vector given that the term vector’s message is not spam. This is problematic insofar as it involves the joint probability distribution of d random variables, X1 , . . . , Xd , 5 P r(Y = 1) × P r(X1 = x1 , . . . , Xd = xd |Y = 1) , P r(X1 = x1 , . . . , Xd = xd ) (1)

where d is potentially large. The key assumption of the Naive Bayes model is that these random variables are conditionally independent given Y . That is:

P r(X1 = x1 , . . . , Xd = xd |Y = 1) = and P r(X1 = x1 , . . . , Xd = xd |Y = 0) = leading to: log


P r(Xi = xi |Y = 1)

d i=1

P r(Xi = xi |Y = 0)

d P r(Y = 1|X1 = x1 , . . . , Xd = xd ) P r(Y = 1) P r(Xi = xi |Y = 1) = log + log . P r(Y = 0|X1 = x1 , . . . , Xd = xd ) P r(Y = 0) i=1 P r(Xi = xi |Y = 0) (3)

This independence assumption is unlikely to reﬂect reality. For instance, in my collection of of spam messages, the word “mortgage” is strongly associated with the word “interest,” a violation of the conditional independence assumption. Nonetheless, the independence assumption provides a drastic reduction in the number of distinct probabilities that we need to estimate from the training data and yet often performs well in practice.


Binary Naive Bayes

For now, let us simplify the representation to binary term vectors. That is, let Xi = 1 if word i is present one or more times in a message and Xi = 0 otherwise, i = 1, . . . , d. Without the Naive Bayes assumption, the model needs estimates for 2×2d parameters for the term vectors (one probability for every possible term vector for both spam and not-spam). With the Naive Bayes assumption, the model needs 2 × d estimates. The extreme nature of the core independence assumption earns the model its “Naive” name. Some earlier literature refers to the model as “Idiot’s Bayes.” There remains the issue of how to estimate the model’s probabilities from the training data. Speciﬁcally, sticking with the binary representation, the model needs estimates for log P r(Y =1) and for log P r(Xi =1|Y =1) , i = 1, . . . , d (the literature refers to these latter P r(Y =0) P r(Xi =1|Y =0) terms as “weights of evidence”). The obvious approach is to estimate each probability 6

by its observed fraction in the training data and plug these estimated probabilities into the expressions for the log odds. For the example in Table 3 this leads to, for instance, 2/4 P r(Y = 1) = log =0 log 2/4 P r(Y = 0) and P r(X3 = 1|Y = 1) 2/2 log = log 2 = log 1/2 P r(X3 = 1|Y = 0) where P r denotes an estimated probability. These correspond to the maximum likelihood estimates of the probabilities. This approach does not provide unbiased estimates for the weights of evidence, and also runs into practical diﬃculties when it estimates some probabilities as zero. For example, in the example, P r(X5 = 1|Y = 0) = 0, leading to an undeﬁned estimate for the corresponding weight of evidence. The standard solution to this problem is to “smooth” the estimates by adding a small positive constant to each numerator and denominator of each probability estimate. The Appendix describes one particular rationale for choosing the value of the small positive constant, focusing on reducing bias (and suggests setting the constant to 0.5). Table 4 shows the resulting estimated weights of evidence for the example. X1 brown Term Present Term Absent -1.10 0.51 X2 fox -1.10 0.51 X3 quick 0 0 X4 rabbit 0.51 -1.10 X5 rest 1.10 -0.51 X6 run 0 0

Table 4: Estimated Weights of evidence for the example. Now suppose a new e-mail message arrives: The quick rabbit rests Does the Naive Bayes model predict that this message is spam? After stemming and stopword removal, Table 5 shows the term vector for this message along with the associated weights of evidence. 7

X1 brown Term Vector Weight of Evidence 0 0.51

X2 fox 0 0.51

X3 quick 1 0

X4 rabbit 1 0.51

X5 X6 rest run 1 1.10 0 0

Table 5: Making predictions for a new e-mail message. According to the model, the log-odds that this message is spam is the prior log odds (zero in this case) plus the sum of the relevant weights of evidence: 0.51 + 0.51 + 0 + 0.51 + 1.10 + 0 = 2.63. This corresponds to a predicted probability of 0.93. The spam ﬁlter builder chooses a threshold and then compares this predicted probability to the threshold. If the probability exceeds the threshold, the ﬁlter routes the message to the spam folder. Otherwise the ﬁlter lets the message through to the inbox. The choice of the threshold must reﬂect the builder’s relative concerns about the two kinds of errors mentioned earlier. A low threshold may result in a higher false positive rate more legitimate messages ending up in the spam folder. A high threshold may result in a higher false negative rate - more spam messages ending up in the inbox. Most commercial spam ﬁlters allow the user to select the threshold.


Multinomial Naive Bayes

The binary representation ignores the number of occurences of each term in a message, yet these term frequencies may provide useful useful clues for identifying spam. Incorporation of term frequencies requires a model for P r(Xi = xi ), i = 1, . . . , d, xi ∈ {0, 1, 2, . . .}. One can apply standard models for non-negative integers such as Poisson and Negative Binomial in this context but the literature reports mixed success (see, for example, Eyheramendy et al., 2003, and the classic text by Mosteller and Wallace, 1984). A number of researchers have reported success with the so-called multinomial Naive Bayes model. This model assumes that message length is independent of spam status. We refer the interested reader to Eyheramendy et al. (2003), Lewis (1998), and McCallum and Nigam (1998) for details.



Naive Bayes Eﬀectiveness

The Naive Bayes model is simple to implement and scales well to large-scale training data. The literature on spam ﬁltering reports a number of experiments where Naive Bayes delivers competitive false positive and false negative rates. Sahami et al. (1998) conducted experiments with a corpus of 1,789 e-mail messages, almost 90% of which were spam. They randomly split this corpus into 1,538 training messages and 251 testing message. A Naive Bayes classiﬁer, as described above, achieved a false negative rate of 12% and a false positive rate of 3%. Extending the term vector representation to include hand-crafted phrases and some domain-speciﬁc features reduced the false negative rate to 4% and the false positive rate to 0%. Androutsopoulos et al. (2000) report experiments using the “Ling-spam” corpus of 2,893 e-mail messages, 481 of which are spam. They reported a false positive rate of 1% computed via 10-fold cross-validation. Their Naive Bayes algorithm outperformed a set of rules built into a widely used commerical e-mail client. Carreras and Marquez (2001) on yet another corpus (PU1) report a Naive Bayes false positive rate of 5%. Neither Androutsopoulos et al. nor Carreras and Marquez report false negative rates. Newer statistical algorithms such as regularized logistic regression, support vector machines, and boosted decision trees can achieve higher eﬀectiveness. Carreras and Marquez (2001), for example, reported experiments in which boosted decision trees out-performed Naive Bayes by a few percentage points on the Ling-spam corpus.


Discussion: Beyond Spam Filtering

Spam ﬁltering has provided the primary focus for this chapter and we have highlighted the central role of Statistics. In fact spam ﬁltering is one particular example of the broader topic of “text categorization.” Text categorization algorithms assign texts to predeﬁned categories. The study of such algorithms has a rich history dating back at least forty years. In the last decade or so, the statistical approach has dominated the literature. The essential idea, as with statistical spam ﬁltering, is to infer a text 9

categorization algorithm from a set of labeled documents, i.e., documents with known category assignments, where a feature vector represents the documents. Sebastiani (2002) provides an overview. Applications of statistical text categorization include: • Building web directories like the dmoz open directory and Yahoo! • Routing incoming customer service e-mail • Identifying “interesting” new stories for intelligence analysts • Classifying ﬁnancial news stories as “signiﬁcant” or “insigniﬁcant” While Naive Bayes is rarely the top-performing algorithm in any of these applications, it invariably provides reasonable eﬀectiveness with minimal computational eﬀort.


Appendix: Smoothing Estimated Weights of Evidence to Reduce Bias

We consider a binomial sampling model for a single term. Let N1 denote the number of spam e-mail messages in the training data and N2 the number of not-spam messages. Let R1 denote the number of spam messages containing the term and R2 the number of not-spam messages containing the term (see Table 6). The binomial sampling model assumes that Ni is ﬁxed, and that Ri is binomially distributed with parameters Ni and θi , i = 1, 2. Let W (S : T ) = log θ1 denote the target weight of evidence and θ2 W (S : T ) the corresponding estimated weight of evidence. As in Spiegelhalter and Knill-Jones (1984), we want to choose a, b, c, d ∈ + such that: E(W (S : T )) = E(log R1 + a R2 + c / ) N1 + b N2 + d

is as close as possible to W (S : T ). That is, we want to minimize the bias of W (S : T ). Following Cox (1970, p.33) let: √ Ri = Ni θi + Ui N i , i = 1, 2, 10

Spam Term Present R1 Term Absent N1 − R1

Not Spam R2 N2 − R2

Table 6: Frequency counts required for estimating a single weight of evidence. and thus: E(Ui ) = 0 E(Ui2 ) = θi (1 − θi ) E(Ui3 ) = Ni

2 E(Ui4 ) = 3θi (1 − θi )2 + Ni−1 θi (1 − θi )(1 − 6θi (1 − θi )).

θi (1 − θi )(1 − 2θi )

We consider: W (S : T ) − W (S : T ) R1 + a R2 + c θ1 = log − log − log N1 + b N +d θ2 √ 2 √ N1 θ1 + U1 N 1 + a N2 θ2 + U2 N 2 + c θ1 = log − log − log . N1 + b N2 + d θ2 U1 a d dU1 ad √ + = log 1 + √ + + + θ1 N1 N1 θ1 N2 θ1 N2 N1 θ1 N1 N2 U2 c b bU2 bc √ − log 1 + √ + + + + θ2 N2 N2 θ2 N1 θ2 N1 N2 θ2 N1 N2 Expanding this expression, taking expectations, and ignoring terms of order N −2 and lower, where N = min(N1 , N2 ) we have: E W (S : T ) − W (S : T ) =

a − 1/2 c − 1/2 b − 1/2 d − 1/2 − − + + O(N −2 ). θ1 N1 θ2 N2 N1 N2

The absolute value of this expression is clearly minimized when a = b = c = d = 1/2. With these values, the bias is given by: −
2 2 1 − θ1 1 − θ2 + + O(N −3 ). 2 2 2 2 24θ1 N1 24θ2 N2


It is straightforward but tedious to show that setting a = b = c = d = 1/2 also minimizes the bias for the Poisson and multinomial sampling schemes. Curiously, for a binomial sampling that conditions on the row totals of Table 6 instead of the column totals, simple expressions for a, b, c, and d do not seem to exist.

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the 1998 Workshop. AAAI Technical Report WS-98-05. Sebastiani, F. (2002). Machine learning in automated text categorization. ACM Computing Surveys, 34, 1–47. Spiegelhalter, D.J. and Knill-Jones, R.P. (1984). Statistical and knowledge based approaches to clinical decision support systems, with an application to gastroenterology (with discussion). Journal of the Royal Statistical Society (Series A), 147, 35-77.


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