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Naive Bayes (and text classification)[ Edit ]

Naive Bayes is a widely used classification algorithm. It is a supervised learning algorithm based on Bayes’ Theorem. The word naive comes from the assumption of independence among features. That is, if our input vector is (x₁, x₂,...,x_n), then x_i's are conditionally independent given y.

Deriving the algorithm

Let's start with Bayes' theorem (for naive bayes, x is the input and y is the output):

P( y | x ) = \frac{P(y)P(x | y)}{P(x)}

When we have more than one feature, we can rewrite Bayes' theorem as:

P( y | x_1,...,x_n) = \frac{P(y)P(x_1,...,x_n | y)}{P(x_1,x_2,...,x_n)}

Since we are making the assumption that x_i's are conditionally independent given y, we can rewrite the above as

P(y|x_1,...,x_n) = \frac{P(y) \prod_{i=1}^{n} P(x_i|y)}{P(x_1,x_2,...,x_n)}

but we also know that P(x₁, x₂, .., x_n) is a constant given the input, i.e.

P(y|x_1,...,x_n) \propto P(y) \prod_{i=1}^{n} P(x_i|y) \quad \quad (1)

Notice that

the left hand side is the term we are interested in, probability distribution of the output y given input x
P(y) can be estimated by counting the number of times each class y appears in our training data (this is called Maximum a Posteriori estimation)
P(x_i|y) can be estimated by counting the number of times each value of x_i appears for each class y in our training data

Pseudocode

Training:

Estimate P(y): P(y=t) = number of times class t appears in the dataset / size of dataset
Estimate P(x_i|y): P(x_i=k|y=t) = number of times x_i has value k and y has value t / number of data points of class t

Predicting:

Estimate P(y|x₁,...,x_n): Use above estimated values of P(y) and P(x_i|y) and equation (1). Thereafter, normalize the values.

Variants

There are several variants of naive bayes which use different distributions for P(x_i|y) such as gaussian distribution (gaussian naive bayes), multinomial distribution (multinomial naive bayes) and bernoulli distribution (bernoulli naive bayes).

Scikit-learn implementation

# we will use iris dataset
from sklearn.datasets import load_iris
from sklearn.naive_bayes import GaussianNB
import numpy as np
 
# load the dataset
data = load_iris()
 
model = GaussianNB()
model.fit(data.data, data.target)
 
# evalaute
print(model.score(data.data, data.target))
# output = 0.96
 
# predict
model.predict([[4.2, 3, 0.9, 2.1]])
# 0 = setosa, 1 = versicolor, and 2 = virginica

Applications

Naive bayes is one of the simplest yet effective algorithms for

Text classification: For example, we have a number of news articles, and we want to learn to classify if the article is about politics, health, technology, sports or lifestyle.
Spam filtering: We have a number of emails, and we want to learn to classify if the email is spam or not.
Gender classification: Given features such as height, weight, etc, predict whether the person is male or female.

Mark as completed

Quiz (and Assignment): Support Vector Machines

Hands-on Assignment: Sentiment Classification with Naive Bayes (from Scratch)

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