Building a Spam Filter with Naive Bayes

In this project, we're going to build a spam filter for SMS messages using the multinomial Naive Bayes algorithm. Our goal is to write a program that classifies new messages with an accuracy greater than 80% — so we expect that more than 80% of the new messages will be classified correctly as spam or ham (non-spam).

To train the algorithm, we'll use a dataset of 5,572 SMS messages that are already classified by humans. The dataset was put together by Tiago A. Almeida and José María Gómez Hidalgo, and it can be downloaded from the The UCI Machine Learning Repository. The data collection process is described in more details on this page, where you can also find some of the papers authored by Tiago A. Almeida and José María Gómez Hidalgo.

Exploring the Dataset

We'll now start by reading in the dataset.

Below, we see that about 87% of the messages are ham, and the remaining 13% are spam. This sample looks representative, since in practice most messages that people receive are ham.

Training and Test Set

We're now going to split our dataset into a training and a test set, where the training set accounts for 80% of the data, and the test set for the remaining 20%.

We'll now analyze the percentage of spam and ham messages in the training and test sets. We expect the percentages to be close to what we have in the full dataset, where about 87% of the messages are ham, and the remaining 13% are spam.

The results look good! We'll now move on to cleaning the dataset.

Data Cleaning

To calculate all the probabilities required by the algorithm, we'll first need to perform a bit of data cleaning to bring the data in a format that will allow us to extract easily all the information we need.

Essentially, we want to bring data to this format:

Letter Case and Punctuation

We'll begin with removing all the punctuation and bringing every letter to lower case.

Creating the Vocabulary

Let's now move to creating the vocabulary, which in this context means a list with all the unique words in our training set.

It looks like there are 7,783 unique words in all the messages of our training set.

The Final Training Set

We're now going to use the vocabulary we just created to make the data transformation we want.

Calculating Constants First

We're now done with cleaning the training set, and we can begin creating the spam filter. The Naive Bayes algorithm will need to answer these two probability questions to be able to classify new messages:

\begin{equation} P(Spam | w_1,w_2, ..., w_n) \propto P(Spam) \cdot \prod_{i=1}^{n}P(w_i|Spam) \end{equation}

\begin{equation} P(Ham | w_1,w_2, ..., w_n) \propto P(Ham) \cdot \prod_{i=1}^{n}P(w_i|Ham) \end{equation}

Also, to calculate P(w_i|Spam) and P(w_i|Ham) inside the formulas above, we'll need to use these equations:

\begin{equation} P(w_i|Spam) = \frac{N_{w_i|Spam} + \alpha}{N_{Spam} + \alpha \cdot N_{Vocabulary}} \end{equation}

\begin{equation} P(w_i|Ham) = \frac{N_{w_i|Ham} + \alpha}{N_{Ham} + \alpha \cdot N_{Vocabulary}} \end{equation}

Some of the terms in the four equations above will have the same value for every new message. We can calculate the value of these terms once and avoid doing the computations again when a new messages comes in. Below, we'll use our training set to calculate:

• P(Spam) and P(Ham)
• N_Spam, N_Ham, N_Vocabulary

We'll also use Laplace smoothing and set $\alpha = 1$.

Calculating Parameters

Now that we have the constant terms calculated above, we can move on with calculating the parameters $P(w_i|Spam)$ and $P(w_i|Ham)$. Each parameter will thus be a conditional probability value associated with each word in the vocabulary.

The parameters are calculated using the formulas:

\begin{equation} P(w_i|Spam) = \frac{N_{w_i|Spam} + \alpha}{N_{Spam} + \alpha \cdot N_{Vocabulary}} \end{equation}

\begin{equation} P(w_i|Ham) = \frac{N_{w_i|Ham} + \alpha}{N_{Ham} + \alpha \cdot N_{Vocabulary}} \end{equation}

Classifying A New Message

Now that we have all our parameters calculated, we can start creating the spam filter. The spam filter can be understood as a function that:

• Takes in as input a new message (w₁, w₂, ..., w_n).
• Calculates P(Spam|w₁, w₂, ..., w_n) and P(Ham|w₁, w₂, ..., w_n).
• Compares the values of P(Spam|w₁, w₂, ..., w_n) and P(Ham|w₁, w₂, ..., w_n), and:
  • If P(Ham|w₁, w₂, ..., w_n) > P(Spam|w₁, w₂, ..., w_n), then the message is classified as ham.
  • If P(Ham|w₁, w₂, ..., w_n) < P(Spam|w₁, w₂, ..., w_n), then the message is classified as spam.
  • If P(Ham|w₁, w₂, ..., w_n) = P(Spam|w₁, w₂, ..., w_n), then the algorithm may request human help.

Measuring the Spam Filter's Accuracy

The two results above look promising, but let's see how well the filter does on our test set, which has 1,114 messages.

We'll start by writing a function that returns classification labels instead of printing them.

Now that we have a function that returns labels instead of printing them, we can use it to create a new column in our test set.

Now, we'll write a function to measure the accuracy of our spam filter to find out how well our spam filter does.

The accuracy is close to 98.74%, which is really good. Our spam filter looked at 1,114 messages that it hasn't seen in training, and classified 1,100 correctly.

Next Steps

In this project, we managed to build a spam filter for SMS messages using the multinomial Naive Bayes algorithm. The filter had an accuracy of 98.74% on the test set we used, which is a pretty good result. Our initial goal was an accuracy of over 80%, and we managed to do way better than that.

Next steps include:

• Analyze the 14 messages that were classified incorrectly and try to figure out why the algorithm classified them incorrectly
• Make the filtering process more complex by making the algorithm sensitive to letter case