Naive Bayes Classifier Tutorial

by keshav

Naive Bayes Classifier is a classification technique based on Bayes’ Theorem. It is base on the principle that the predictors are independent of each other. In simple words, we can say that the Naive Bayes classifier assumes that the presence of a particular feature in a class is independent(Unrelated) with the presence of any other feature in the same class. Let's understand this concept by an example, suppose a fruit may be considered to be orange if it is orange in color, approximately round, and about 2.5 inches in diameter. Here we can see that all of these properties independently contribute to the probability that this fruit is orange, even if these features depend on each other. This is the reason, why it is known as ‘Naive’. (Naive meaning: Unaffected).

Naive Bayes algorithm is simple to understand and easy to build. It does not contain any complicated iterative parameter estimation. We can use a Naive Bayes classifier in small data set as well as with a large data set that may be highly sophisticated classification.

The naive Bayes classifier is based on the Bayes theorem of probability. Bayes theorem can be used for calculating posterior probability P(y|X) from P(y), P(X), and P(X|y). The mathematical equation for Bayes Theorem is,

From the equation, we have,

X is the feature vector represented as,

P(y|X) is the posterior probability (A posterior probability, in Bayesian statistics, is the revised or updated probability of an event occurring after taking into consideration new information ~ Investopedia) of class (y, target) given predictor (X, attributes).
P(y) is the prior probability(probability as assessed before making reference to certain relevant observations) class.
P(X|y) is the probability(Likelihood) of the predictor given class.
P(X) is the prior probability of predictor.

Since Naive Bayes classifier assumes the independence of predictors (features), so for independent features, we calculate the output probability using Bayes theorem as,

Which can be represented as,

Since the denominator is constant, we can write,

Now, To create a Naive Bayes classifier model, we find the probability of a given set of inputs for all possible values of the class variable y and pick up the output with maximum probability. This can be expressed mathematically as:

So, finally, we are left with the task of calculating P(y) and P(xi | y).

NOTE: P(y) is also called class probability and P(xi | y) is called conditional probability.

How does the Naive Bayes classifier work?

Let’s understand the working and algorithm of Naive Bayes Classifier using an example. Below is the training data set for playing golf under different circumstances. We have different features as Outlook, Temperature, Humidity, Windy, and we are given a label as play golf under different situations of those features. We need to predict whether to play or not for new test data(that we provide) by using a naive Bayes classification algorithm. Let’s do it step by step and learn this algorithm.

	OUTLOOK	TEMPERATURE	HUMIDITY	WINDY	PLAY GOLF
0	Rainy	Hot	High	False	No
1	Rainy	Hot	High	True	No
2	Overcast	Hot	High	False	Yes
3	Sunny	Mild	High	False	Yes
4	Sunny	Cool	Normal	False	Yes
5	Sunny	Cool	Normal	True	No
6	Overcast	Cool	Normal	True	Yes
7	Rainy	Mild	High	False	No
8	Rainy	Cool	Normal	False	Yes
9	Sunny	Mild	Normal	False	Yes
10	Rainy	Mild	Normal	True	Yes
11	Overcast	Mild	High	True	Yes
12	Overcast	Hot	Normal	False	Yes
13	Sunny	Mild	High	True	No

Here, The attributes are Outlook, Windy, Temperature, and humidity. And the class (or Target) is Play Golf.

Step 1: Convert the given training data set into a frequency table

Step 2: Create a Likelihood table (or you can say a probability table) by finding the probabilities.

probability calculation table

In those tables we have calculated both P(y) (i.e. P(yes) and P(no)) and P(xi | y) (e.g. p(humidity,high|yes)).

Step 3: Now, apply the Naive Bayesian equation to calculate the posterior probability for each class. The class with the highest posterior probability will be the outcome of the prediction.

Let's Suppose our test data be, test = (Sunny, Hot, Normal, False). For this, we need to predict whether it will be okay to play golf or not.

Let's Calculate:

Probability of playing golf:

Probability of not playing golf:

Here, we can see that in both of the probabilities there is a common factor p(test), so we can ignore it. Thus we get the calculation as follows,

and,

To convert these numbers into actual probabilities, we normalize them as follows,

and,

From the above calculations, we see that

Thus, the prediction for golf played is 'Yes'.

What are the Pros and Cons of Naive Bayes Classifier?

Pros:

Naive Bayes Classifier is simple to understand, easy and fast to predict the class of test data set.
It performs quite well in multi-class prediction.
It performs well in the case of categorical input variables compared to a numerical variable(s).

Cons:

The model will assign a 0 (zero) probability and will be unable to make a prediction If a categorical variable has a category (in the test data set), which was not present in the training data set. This type of error is often known as “Zero Frequency”.
Another limitation with the Naive Bayes is the assumption of independence. In real life, it is almost impossible that we get a set of predictors which are completely independent of each other.

Applications of Naive Bayes Classifier

Real-time Prediction: Naive Bayes Classifier is an eager (not a lazy learner) learning classifier and it is sure fast. Therefore, it could be used for making real-time predictions.
Multi-class Prediction: This algorithm is also well known for its multi-class prediction feature. Here we can predict the probability of multiple classes of the target variables.
Text classification/ Spam Filtering/ Sentiment Analysis: Naive Bayes classifiers widely used in text classification due to better results in multi-class problems and independence rules. It has a higher success rate as compared to other algorithms. As a result, it is widely used in Spam filtering (identification between ham and spam e-mail) and Sentiment Analysis (in social media analysis, to identify positive and negative sentiments in comments and reviews)