Bayesian Spam Filter

Thomas Bayes

And God said let their be spam. And ever since, there was spam. It seems like spam these days never ends. It is a constant battle with no winners or losers. Sort of like the war on terrorism. But that’s a topic for another day. However, we are not all doomed. At least for now, thanks to Thomas Bayes and his bayesian spam filtering. So how does this bayes theorem work anyways? Let’s say a given spam has the word ‘free’ in it. We all know nothing is free these days. Bayes theorem states the following:

P(A|B) = [ P(B|A) * P(A)  ] /  P(B)
Read as: The probability of A given B is equal to the probability of B given A times the probability of A divided by the probability of B.

In our case lets specialize this formula to spam:
P(S|W) = [ P(W|S) * P(S) ] / [P(W|S) * P(S)] + [P(W|H) * P(H)]

Looks complicated I know. But it really isn’t.

  • P(S|W)  is the probability that a message is a spam, knowing that the word “free” is in it.
  • P(S) is the overall probability that any given message is spam.
  • P(W|S) is the probability that the word “free” appears in spam messages.
  • P(H) is the overall probability that any given message is not spam (aka ham).
  • P(W|H) is the probability that the word “free” appears in ham messages.

Thomas Bayes

But it does get complicated because we are dealing with multiple words. For multiple words the formula gets a little complicated.

P(S) = p(w1) * p(w2) * … / (p(w1) * p(w2)* … ) + (q(w1) * q(w2) * …)

Now that we have the details taken care of lets look at the code to a simple bayesian spam filter written in c++.

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/***************************************************************************
* Program:
*    Bayesian Spam Filter
*
* Author:
*    Don Page
*
* Summary:
*    This program reads in a hypothetical list of words and frequencies
*    in both spam and ham e-mail messages and then uses Bayesian
*    inference to determine whether hypothetical e-mail messages are spam
*    or not.  The program's judgment may be wrong in some cases, but such
*    is the case with industrial-strength spam filters as well.
******************************************************************************/


#include <iostream>
#include <fstream>
#include <string>
#include <set>
#include <ctype.h>

using namespace std;
class Words
{
private:
int numWords;
string words[200];
float spam[200];
float ham[200];
public:
Words();
int getIndex(string pWord);
float getSpam(string pWord);
float getHam(string pWord);
void insertWord(string pWord, float pSpam, float pHam);

};

/******************************************************************************
* Default Constructor
******************************************************************************/

Words::Words()
{
numWords = 0;
}

/******************************************************************************
* INPUT: pWord - Word to insert
*        pSpam - Probability of spam
*        pHam  - Probability of ham
******************************************************************************/

void Words::insertWord(string pWord, float pSpam, float pHam)
{
words[numWords] = pWord;
spam[numWords]  = pSpam;
ham[numWords]   = pHam;
numWords++;
}

/******************************************************************************
* Read words and their probabilities of being ham and spam from file
******************************************************************************/

Words* readWords(ifstream& words)
{
string word;
float spam;
float ham;
Words* myWords = new Words();

while(!words.eof())
{
words >> word;
words >> spam;
words >> ham;
myWords->insertWord(word, spam, ham);
}

return myWords;
}

/******************************************************************************
* Checks if word is inside spam list. Returns prob of spam or -1 if !found
******************************************************************************/

float Words::getSpam(string pWord)
{

int index = getIndex(pWord);
if (index != -1) // found
return spam[index];
else
return -1.0; // not found
}

/******************************************************************************
* Checks if word is inside ham list. Returns prob of ham or -1 if !found
******************************************************************************/

float Words::getHam(string pWord)
{

int index = getIndex(pWord);
if (index != -1) // found
return ham[index];
else
return -1.0; // not found
}

/******************************************************************************
* Finds a given word inside list of words. Returns -1 if not found
******************************************************************************/

int Words::getIndex(string pWord)
{
//Convert string to lowercase
for (int i = 0; i < pWord.length(); i++)
pWord[i] = tolower(pWord[i]);

int i = 0;
for (; words[i] != pWord && i < numWords; i++);

// Check to see if word was found
if (words[i] == pWord)
return i;
else
return -1;
}

/******************************************************************************
* Process each message using Bayes theorem to calculate if message is Spam
* or Ham. See below for Bayes thereom on multiple words.
* P(S) = p(w1) * p(w2) * ... / (p(w1) * p(w2)* ... ) + (q(w1) * q(w2) * ...)
******************************************************************************/

void processMessages(Words* words, ifstream& messages)
{
int message = 1;
string word;

float probSpam;
float probHam;
string cleanWord = "";
messages >> word;
while(!messages.eof())
{

messages >> word; // First word is message
probSpam = 0.0;
probHam = 0.0;
while (word != "MESSAGE" && !messages.eof()) // New message
{

//Parse word
for (int i = 0; i < word.length(); i++)
{
if (isalpha(word[i]))
{
cleanWord += word[i];
}
if (!isalpha(word[i]) || i == word.length() - 1) // Check cleanWord if it's spam
{
if (cleanWord.length() > 0)
{

float temp = words->getSpam(cleanWord);
if (temp > -1.0) // If Spam Calculate new probSpam
probSpam != 0.0 ? probSpam *= temp : probSpam = temp;

temp = words->getHam(cleanWord);
if (temp > -1.0) // If Ham Calculate new probHam
probHam != 0.0 ? probHam *= temp : probHam = temp;
cleanWord = "";

}
}

}// for
messages >> word;
}// while
// Should we reject message?
float threshold = probSpam / (probSpam + probHam);
cout << "Message " << message << " has a " << threshold << " probability of being spam, ";
if (threshold < .9) // Accept
cout << "so let it through.\n\n";
else // Reject
cout << "so reject it.\n\n";
message++;
}
return;
}

/******************************************************************************
* Opens two input files, words.txt and messages.txt, reads in the words and
* their spam/ham frequencies, and with that data processes the messages
* to determine if they (the messages) are spam or not.
*
* DO NOT MODIFY ANYTHING BELOW THIS LINE!
******************************************************************************/

int main(int argc, const char* argv[])
{
// Set decimal precision for output.
cout.setf(ios::fixed);
cout.setf(ios::showpoint);
cout.precision(2);

if (argc < 3)
{
cerr << "Usage: " << argv[0] << " [words file] [messages file]\n";
exit(1);
}

ifstream words(argv[1]);

ifstream messages(argv[2]);

if (words.fail() || messages.fail())
{
cerr << "File opening failed.\n";
exit(1);
}

processMessages(readWords(words), messages);

return 0;
}