Spam Filtering with Naive Bayes Classifier

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1 Spam Filtering with Naive Bayes Classifier Yuriy Arabskyy June 6, 2017

2 Table of contents What is spam? Different spam types Anti-Spam Techniques Probability theory basics Conditional probability Bayes Theorem Naive Bayes Theorem Spam filtering with Naive Bayes Classifier (NBC) Definition of terms Feature representation Evaluation Comparison to Logistic Classifier (LC)

3 What is spam? Spam mass-mailing of a message over the internet, for the purposes of advertising.

4 What is spam? Spam mass-mailing of a message over the internet, for the purposes of advertising. HOT sing1e women seeking your attention near this AREA!!!%%###!!! Just follow this link...

5 What is spam? Spam mass-mailing of a message over the internet, for the purposes of advertising. HOT sing1e women seeking your attention near this AREA!!!%%###!!! Just follow this link... I am Wumi Abdul; the only Daughter of late Mr and Mrs George Abdul. My father was a very wealthy cocoa merchant in Abidjan, he was poisoned to death by his business associates... I seek for a foreign partner. Please provide a Bank account where this money would be transferred to.

6 Different spam types Figure: Spam chart

7 Anti-Spam Techniques End-user techniques Discretion Address munging Ham passwords

8 Anti-Spam Techniques End-user techniques Discretion Address munging Ham passwords Mail server level filtering Realtime Blackhole Lists Spamtrapping SMTP callback verification Statistical spam filtering

9 Probability theory basics Conditional Probability: Pr[X Y ] = Pr[Y X ] Pr[X ] (1)

10 Probability theory basics Conditional Probability: Pr[X Y ] = Pr[Y X ] Pr[X ] (1) Figure: Weather - conditional probability

11 Probability theory basics Pr[X Y ] = Pr[Y X ] Pr[X ] = Pr[X Y ] Pr[Y ] (2)

12 Probability theory basics Pr[X Y ] = Pr[Y X ] Pr[X ] = Pr[X Y ] Pr[Y ] (2) Bayes Theorem: Pr[Y X ] = Pr[X Y ] Pr[Y ] Pr[X ] (3)

13 Probability theory basics Pr[X Y ] = Pr[Y X ] Pr[X ] = Pr[X Y ] Pr[Y ] (2) Bayes Theorem: Pr[Y X ] = Pr[X Y ] Pr[Y ] Pr[X ] (3) Bayes Theorem is a way of updating of what we think about the world, based on what we know about it.

14 Probability theory basics Multiple variables Pr[x 1, x 2,..., x n ] = Pr[x 1 x 2, x 3,..., x n ] Pr[x 2, x 3,..., x n ] (4) Pr[x 2, x 3,..., x n ] = Pr[x 2 x 3, x 4,..., x n ] Pr[x 3, x 4,..., x n ] (5)

15 Probability theory basics Multiple variables Pr[x 1, x 2,..., x n ] = Pr[x 1 x 2, x 3,..., x n ] Pr[x 2, x 3,..., x n ] (4) Pr[x 2, x 3,..., x n ] = Pr[x 2 x 3, x 4,..., x n ] Pr[x 3, x 4,..., x n ] (5) Assuming x i and x j are independent: Pr[x i x j ] = Pr[x i ] (6)

16 Probability theory basics Multiple variables Pr[x 1, x 2,..., x n ] = Pr[x 1 x 2, x 3,..., x n ] Pr[x 2, x 3,..., x n ] (4) Pr[x 2, x 3,..., x n ] = Pr[x 2 x 3, x 4,..., x n ] Pr[x 3, x 4,..., x n ] (5) Assuming x i and x j are independent: Pr[x i x j ] = Pr[x i ] (6) Previous formula may be simplified to the following one: Pr[x 1, x 2,..., x n ] = Pr[x 1 ] Pr[x 2 ]... Pr[x n ] (7)

17 Spam filtering with NBC Bayes theorem rewritten using the naive assumption: Pr(c x 1, x 2,..., x n ) = Pr(c)Pr(x 1 c)pr(x 2 c) Pr(x n c) Pr(x 1, x 2,..., x n ) (8)

18 Spam filtering with NBC Bayes theorem rewritten using the naive assumption: Pr(c x 1, x 2,..., x n ) = Pr(c)Pr(x 1 c)pr(x 2 c) Pr(x n c) Pr(x 1, x 2,..., x n ) (8) Class of d i = argmax c Pr(c d i ) (9)

19 Spam filtering with NBC Defintion of terms: Vocabulary (V) is an ordered collection of words i.e., V = (v 1, v 2, v 3,..., v n ) used to classify an . Document (D) is an ordered collection of words used in a message D = (w 1, w 2, w 3,..., w n ). The classifier is a machine that, when given a document D and a collection of parameters θ, deterministically returns the class of the document.

20 Spam filtering with NBC Document representation Binary vector of length V is used to represent a document. x i means the absence of the word v i in the specified document.

21 Spam filtering with NBC Document representation Binary vector of length V is used to represent a document. x i means the absence of the word v i in the specified document. Bernoulli event model Pr[x i c k ] = p x i ki (1 p ki) 1 x i (10) p ki is the probability of class c k generating the word v i and can be calculated as follows: d c p ki = k ispresent(v i, d) (11) # of documents in c k

22 Evaluation Legitimate Spam Classifier accepted a b Classifier rejected c d b accepted even though it was spam c legitimate mail is classified as spam (very bad!) Recall = a a + c Precision = a a + b (12)

23 Comparison to Logistic Classifier Advantage NBC requires less training data to be able to function properly. Disadvantage Logistic Classifier can reach a lower error rate when given enough data.

24 Comparison to Logistic Classifier Figure: Dashed LC; Solid NBC; Y-axis error; X-axis - m (1000 random train splits

25 Thank you for listening!

26 Thank you for listening! And remember, what do we say to nigerian princes who want to make business with you? :)

27 Thank you for listening! And remember, what do we say to nigerian princes who want to make business with you? :) If you re interested, listen to James Veitch s talk about answering spam: happens_when_you_reply_to_spam_

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