**Introduction  **

In mathematics and programming, some of the simplest solutions are usually the most powerful ones. The naïve Bayes Algorithm comes as a classic example of this statement. Even with the strong and rapid advancement and development in the field of Machine Learning, this Naïve Bayes Algorithm still stands strong as one of the most widely used and efficient algorithms. The naïve Bayes Algorithm finds its applications in a variety of problems including Classification tasks and Natural Language Processing (NLP) problems.

The mathematical hypothesis of the Bayes Theorem serves as the fundamental concept behind this Naïve Bayes Algorithm. In this article, we shall go through the basics of Bayes Theorem, the Naïve Bayes Algorithm along with its implementation in Python with a real-time example problem. Along with these, we shall also look at some advantages and disadvantages of the Naïve Bayes Algorithm in comparison with its competitors.

**Basics of Probability **

Before we venture out on understanding the Bayes Theorem and Naïve Bayes Algorithm, let us brush up our existing knowledge upon the fundamentals of Probability.

As we all know by definition, given an event A, the probability of that event occurring is given by P(A). In probability, two events A and B are termed as independent events if the occurrence of event A does not alter the probability of occurrence of event B and vice versa. On the other hand, if one’s occurrence changes the probability of the other, then they are termed as Dependent events.

Let us get introduced to a new term called Conditional Probability. In mathematics, Conditional Probability for two events A and B given by P (A| B) is defined as the probability of the occurrence of event A given that event B has already occurred. Depending upon the relationship between the two events A and B as to whether they are dependent or independent, Conditional Probability is calculated in two ways.

• The conditional probability of two dependent events A and B is given by P (A| B) = P (A and B) / P (B)
• The expression for the conditional probability of two independent events A and B is given by, P (A| B) = P (A)

Knowing the math behind Probability and Conditional Probabilities, let us now move on towards the Bayes Theorem.

**Bayes Theorem **

In statistics and probability theory, the Bayes’ Theorem also known as the Bayes’ rule is used to determine the conditional probability of events. In other words, the Bayes’ theorem describes the probability of an event based on prior knowledge of the conditions that might be relevant to the event.

To understand it in a simpler way, consider that we need to know the probability of the price of a house is very high. If we know about the other parameters such as the presence of schools, medical shops and hospitals nearby, then we can make a more accurate assessment of the same. This is exactly what the Bayes Theorem performs.

Such that,

• P(A|B) – the conditional probability of event A occurring, given event B has occurred also known as Posterior Probability.
• P(B|A) – the conditional probability of event B occurring, given event A has occurred also known as Likelihood Probability.
• P(A) – the probability of event A occurring also known as Prior Probability.
• P(B) – the probability of event B occurring also known as Marginal Probability.

Suppose we have a simple Machine Learning problem with ‘n’ independent variables and the dependent variable which is the output is a Boolean value (True or False). Suppose the independent attributes are categorical in nature let us consider 2 categories for this example. Hence, with these data, we need to calculate the value of the Likelihood Probability, P(B|A).

Hence, on observing the above we find that we need to calculate 2*(2^n -1) parameters in order to learn this Machine Learning model. Similarly, if we have 30 Boolean independent attributes, then the total number of parameters to be calculated will be close to 3 billion which is extremely high in computational cost.

This difficulty in building a Machine Learning model with the Bayes Theorem led to the birth and development of the Naïve Bayes Algorithm.

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