What is Bayes Theorem?
Bayes Theorem is a mathematical representation for determining conditional probability. It is the probability of an event occurring based on the previous occurring outcome. Bayes theorem is also called Bayes Rule or Bayes Law and is the foundation of Bayesian statistics. The mathematical formula for Bayes Theorem is:
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P(A/B) the probability of event A occurring, given event B has already occurred.
P(B/A) the probability of event B occurring, given event A has already occurred.
P(A) the probability of event A occurring
P(B) the possibility of event B occurring
What are we trying to calculate?
So basically, we are trying to find the probability of event A happening, given that event B is true. The probability of event B occurring is termed as evidence here. A's probability, i.e., P(A), is the probability/ priori of event A happening before the evidence is seen. Evidence is an attribute of an unknown instance. Here, P(AB) is the probability of event A occurring after the evidence is seen, i.e., posteriori.
Therefore, the Bayes Theorem equation can be rewritten as:
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Where,
P(c/x) is the posterior probability, i.e., the probability of event A occurring, given event B has already occurred.
P(x/c) is the likelihood probability, i.e., the probability of event B occurring, given event A has already occurred.
P(c) is the prior probability that is the probability of event A occurring
P(x) is the marginal probability that is the probability of event B occurring
Further, the above formula can be rewritten as:
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How does the Naive Bayes algorithm work?
Let us understand the functioning of the Naive Bayes algorithm using an example. Below is a training dataset of weather and corresponding target variable Play. Now we need to classify whether we should play or not based on the weather conditions.
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The first step is to construct a frequency table of the given data. Next, we will create a likelihood table by finding the probabilities of sunny, rainy, etc. The final step is to use the Naive Bayes equation and find the probability of each category.
Problem statement: Players will play if the weather is sunny. Is this the correct statement?
Weâ€™ll solve the question by using the Naive Bayes formula.
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P(Yes / Sunny) = P( Sunny / Yes) * P(Yes) / P (Sunny)
Here we have,
P (Sunny /Yes) = 3/9 = 0.33,
P(Sunny) = 5/14 = 0.36,
P( Yes)= 9/14 = 0.64
Now, P (Yes  Sunny) = 0.33 * 0.64 / 0.36 = 0.60
which has a higher probability.
Types of Naive Bayes Models
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There are three categories of Naive Bayes models.
Gaussian Naive Bayes Model
This model assumes that normal distribution is followed by attributes. This means if the predictor takes a continuous value instead of a discrete one, the model assumes that these values are a sample of the Gaussian distribution; this distribution is also known as Normal distribution. When plotted, it gives a bellshaped curve, which means that all the values are distributed across the mean value. The graph is represented like:
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This type of model is helpful for continuous values, and the assumption for this model justifies as constant values associated with each class are distributed normally. The following formula assumes the likelihood of the features:
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Where,
y is the variance of y
Creating a simple Gaussian Naive Bayes assumes that a Gaussian distribution describes the data with no covariance (independent) between dimensions. This model can be fit simply by finding the mean and standard deviation of the points within each label, all needed to define such a distribution.
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This illustration indicates how a Gaussian Naive Bayes classifier works. At every data point, the zscore distance between that point and each classmean is calculated, namely the distance from the class mean divided by the standard deviation of that class.
Let us have a look at a working example of Gaussian Naive Bayes code:
#importing the necessary libraries
import numpy as np
from sklearn.naive_bayes import GaussianNB
#Creating a sample dataset
X = np.array([[1, 1], [2, 1], [3, 2], [1, 1], [2, 1], [3, 2]])
Y = np.array([1, 1, 1, 2, 2, 2])
print("Values in Array X:",X, "Values in Array Y:",Y, sep='\n')
#Creating the Gaussian Naive Bayes Classifier model with the inbuilt function.
model = GaussianNB()
model.fit(X, Y)
print("Predicted value for the input (4,7) is:",model.predict([[4, 7]]))
print("Predicted value for the input (3,7) is:",model.predict([[3, 7]]))
You can also try this code with Online Python Compiler
Run Code
This code will generate the following output:
Values in Array X:
[[1 1]
[2 1]
[3 2]
[ 1 1]
[ 2 1]
[ 3 2]]
Values in Array Y:
[1 1 1 2 2 2]
Predicted value for the input (4,7) is: [1]
Predicted value for the input (3,7) is: [2]
Multinomial Naive Bayes Model
This type of classifier is used for document classification questions. It uses the frequency of words to anticipate. It predicts that a particular document would fit into which category, for example, Politics, novel, fiction, horror, etc.
It is helpful to model feature vectors where each value represents the number of occurrences of a term or its relative frequency. If the feature vectors have n elements and each of them can assume k different values with probability pk, then:
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Let us have a look at a working sample code:
#importing the necessary libraries
import numpy as np
from sklearn.naive_bayes import MultinomialNB
#Creating a sample dataset
rng = np.random.RandomState(1)
X = rng.randint(5, size=(6,6))
y = np.array([1, 2, 3, 4, 5, 6])
print("X:",X,"y:",y, sep='\n')
#Creating the Multinomial Naive Bayes Classifier model with the inbuilt function.
model = MultinomialNB()
model.fit(X, y)
print("Predicted value for the input value of (2,3) is:" ,model.predict(X[2:3]))
You can also try this code with Online Python Compiler
Run Code
This code will generate the following output:
X:
[[3 4 0 1 3 0]
[0 1 4 4 1 2]
[4 2 4 3 4 2]
[4 2 4 1 1 0]
[1 1 1 1 0 4]
[1 0 0 3 2 1]]
y:
[1 2 3 4 5 6]
Predicted value for the input value of (2,3) is: [3]
Bernoulli Naive Bayes Model
This type of classifier is used for discrete data, and it works on Bernoulli distribution. The primary feature of the Bernoulli Naive Bayes Classifier is that it accepts features only as binary values like, yes or no, success or failure, true or false, 0 or 1, and so on. Therefore, when the feature values are binary, we must use Bernoulli Naive Bayes classifier. The Bernoulli distribution can be represented in a mathematical formula as:
Source: Link
Let us have a look at a working sample code:
#importing the necessary libraries
import numpy as np
from sklearn.naive_bayes import BernoulliNB
#Creating a sample dataset
rng = np.random.RandomState(1)
X = rng.randint(5, size=(6,6))
Y = np.array([1, 2, 3, 4, 5, 6])
print("X:",X,"y:",y, sep='\n')
#Creating the Bernoulli Naive Bayes Classifier model with the inbuilt function.
model = BernoulliNB()
model.fit(X, Y)
print("Predicted value for the input value of (5,6) is:",model.predict(X[5:6]))
You can also try this code with Online Python Compiler
Run Code
The output of the code is as follows:
X:
[[3 4 0 1 3 0]
[0 1 4 4 1 2]
[4 2 4 3 4 2]
[4 2 4 1 1 0]
[1 1 1 1 0 4]
[1 0 0 3 2 1]]
y:
[1 2 3 4 5 6]
Predicted value for the input value of (5,6) is: [6]
Advantages and Disadvantages of Naive Bayes Classifier
Advantages of Naive Bayes Classifier
 Naive Bayes is one of the most fastmoving and effortless Machine Learning algorithms to predict the class of a dataset.
 It can manage to perform Binary as well as Multiclass Classifications.
 It is best suited for multiclass predictions as compared to any other algorithm.
 The most popular practical implementation of this classifier is for text classification problems.
Disadvantages of Naive Bayes Classifier
There are various disadvantages of Naive Bayesian classification,
 The data may not always be independent of each other.
 This algorithm can not be used for an imbalanced dataset.

When we encounter a response vector in the test data for a particular class absent in the training data, we might end up with zero class probabilities; this is known as the Zero probability problem.
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Applications of Naive Bayes Classifier
Naive Bayes Classifier has multiple reallife applications, to name a few are:
 Spam filtration
 Text classification
 Sentiment analysis
 Recommendation System
 Multiclass prediction
Frequently Asked Questions
1. Does Naive Bayes work with missing values?
Ans: Missing values can be handled by the Naive Bayes algorithm. The features are handled separately at the time of model construction and prediction.
2. How do outliers affect naive Bayes?
Ans: The shape of Gaussian distribution will be affected by the outliers and would even affect the mean, standard deviation, and variance.
3. Can the Naive Bayes algorithm be used for regression?
Ans: Originally, Naive Bayes was built for classification problems only, but with some modifications, it can be used for regressions.
Key takeaways
In this article, we learned about the theory of the Naive Bayes Algorithm, various types of Naive Bayes Algorithm, and the advantages and disadvantages. To build a career in Data Science? Check out our industryoriented machine learning course curated by our faculty from Stanford University and Industry experts.