What is Naive Bayes Classifier? [Explained With Example]

There are so many instances when you are working on machine learning (ML), deep learning (DL), mining data from a set of data, programming on Python, or doing natural language processing (NLP) in which you are required to differentiate discrete objects based on specific attributes. A classifier is a machine learning model used for the purpose. The Naive Bayes Classifier is the crux of this blog post which we will learn further.

Bayes’ Theorem

The British mathematician Reverend Thomas Bayes, Bayes‘ theorem is a mathematical formula used to determine the conditional probability, which is the likelihood of an outcome occurring based on a previous outcome.


Using this formula, we can find the probability of A when B has occurred.


A is the proposition;

B is the evidence;

P(A) is the prior probability of proposition;

P(B) is the prior probability of evidence;

P(A/B) is called the posterior and

P(B/A) is called the likelihood.


Posterior = (Likelihood)(Proposition in prior probability)


                      Evidence Prior probability

This formula assumes that the predictors or features are independent, and one’s presence does not affect another’s feature. Hence, it is called ‘naïve.’

Example Displaying Naïve Bayes Classifier

We are taking an example of a better understanding of the topic.

Problem Statement:

We are creating a classifier that depicts if a text is about sports or not.

The training data has five sentences:

Sentence Label
“A great game” Sports
“The election was over” Not sports
“Very clean match” Sports
“It was a close election” Not sports
“A clean but forgettable game” Sports

Here, you need to find the sentence ‘A very close game’ is of which label?

Naive Bayes, as a classifier, calculates the probability of the sentence “A very close game” is Sports with the probability ‘Not Sports.’

Mathematically, we want to know P (Sports | a very close game), probability of the label Sports in the sentence “A very close game.”

Now, the next step is calculating the probabilities.

But before that, let’s take a look at some concepts.

Feature Engineering

We need to first determine the features to use while a machine learning model creation. Features are the chunks of information from the text given to the algorithm.

In the above example, we have data as text. So, we need to convert the text into numbers in which we will perform calculations.

Hence, instead of text, we will use the frequencies of the words occurring in the text. The features will be the number of these words.

Applying Bayes’ Theorem

We will convert the probability to be calculated using the count of the frequency of words. For this, we will use Bayes’ Theorem and some basic concepts of probability.

P(A/B) = P(B/A) x P(A)



We have P (Sports | a very close game), and by using Bayes theorem, we will countermand the conditional probability:

P (sports/ a very close game) = P(a very close game/ sports) x P(sports)


                                                        P (a very close game)

We will abandon the divisor same for both the labels and compare

P(a very close game/ Sports) x P(Sports)


P(a very close game/ Not Sports) x P(Not Sports)

We can calculate the probabilities by calculating the counts the sentence “A very close game” emerges in the label ‘Sports’.  To determine P (a very close game | Sports), divide it by the total.

But, in the training data, ‘A very close game’ doesn’t seem anywhere so this probability is zero.

The model won’t be of much use without every sentence we want to classify is present in the training data.

Naïve Bayes Classifier

Now comes the core part here, ‘Naïve.’ Every word in a sentence is independent of the other, we’re not looking at the entire sentences, but at single words. Learn more about naive bayes classifier.

P(a very close game) = P(a) x P(very) x P(close) x P(game)

This presumption is powerful and useful too. The subsequent step is to apply:

P(a very close game/Sports) = P(a/Sports) x P(very/Sports) x P(close/Sports) x P(game/Sports)

These individual words appear many times in the training data that we can compute.

Computing Probability

The finishing step is to calculate the probabilities and look at which one is larger.
First, we calculate the a priori probability of the labels: for the sentences in the given training data. The probability of it being Sports P (Sports) will be ⅗, and P (Not Sports) will be ⅖.

While calculating P (game/ Sports), we count the times the word “game” appears in Sports text (here 2) divided by the words in sports (11).

P(game/Sports) = 2/11

But, the word “close” isn’t present in any Sports text!

This means P (close | Sports) = 0 and is inconvenient as we will multiply it with other probabilities, 

P(a/Sports) x P(very/Sports) x 0 x P(game/Sports)

The end result will be 0, and the entire calculation will be nullified. But this is not what we want, so we seek some other way around.

Laplace Smoothing

We can eliminate the above issue with Laplace smoothing, where we will sum up 1 to every count; so that it is never zero.

We will add the possible number words to the divisor, and the division will not be more than 1.

In this case, the set of possible words are 

[‘a’, ‘great’, ‘very’, ‘over’, ‘it’, ‘but’, ‘game’, ‘match’, ‘clean’, ‘election’, ‘close’, ‘the’, ‘was’, ‘forgettable’].

The possible number of words is 14; by applying Laplace smoothing, 

P(game/Sports) = 2+1


                     11 + 14

Final Outcome:

Word P (word | Sports) P (word | Not Sports)
a (2 + 1) ÷ (11 + 14) (1 + 1) ÷ (9 + 14)
very (1 + 1) ÷ (11 + 14) (0 + 1) ÷ (9 + 14)
close (0 + 1) ÷ (11 + 14) (1 + 1) ÷ (9 + 14)
game (2 + 1) ÷ (11 + 14) (0 + 1) ÷ (9 + 14)

Now, multiplying all the probabilities to find which is bigger:

P(a/Sports) x P(very/Sports) x P(game/Sports)x P(game/Sports)x P(Sports)

= 2.76 x 10 ^-5

= 0.0000276

P(a/Non Sports) x P(very/ Non Sports) x P(game/ Non Sports)x P(game/ Non Sports)x P(Non Sports)

= 0.572 x 10 ^-5

= 0.00000572

Hence, we have finally got our classifier that gives “A very close game” the label Sports as its probability is high and we infer that the sentence belongs to the Sports category.

Checkout: Machine Learning Models Explained

Types of Naive Bayes Classifier

Now that we have understood what a Naïve Bayes Classifier is and have seen an example too, let’s see the types of it:

1. Multinomial Naive Bayes Classifier

This is used mostly for document classification problems, whether a document belongs to the categories such as politics, sports, technology, etc. The predictor used by this classifier is the frequency of the words in the document.

2. Bernoulli Naive Bayes Classifier

This is similar to the multinomial Naive Bayes Classifier, but its predictors are boolean variables. The parameters we use to predict the class variable take up the values yes or no only. For instance, whether a word occurs in a text or not.

3. Gaussian Naive Bayes Classifier

When the predictors take a constant value, we assume that these values are sampled from a Gaussian distribution.


Since the values present in the dataset changes, the conditional probability formula changes to,



We hope we could guide you on what Naive Bayes Classifier is and how it is used to classify text. This simple method works wonders in classification problems. Whether you’re a Machine Learning expert or not, you can build your own Naive Bayes Classifier without spending hours on coding.

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