Understanding Probability Normalization in Naive Bayes

In the context of the Naive Bayes classifier, probability normalization plays a vital role, especially when we want our probabilities to reflect the true likelihood of an event occurring in comparison to other events.

When predicting class labels using the Naive Bayes formula, we compute the product of feature probabilities for each class. However, these products do not sum up to 1 across classes, making them hard to interpret as probabilities. This is where normalization comes in.

Let's understand this with a basic example:

Suppose we have two classes, A and B, and we compute the product of feature probabilities for both:

P(features|A) = 1/2 and P(features|B) = 1/4.

To get the probability of a data point belonging to class A or B, we need to normalize:

Normalized P(A|features) 
= P(features|A) / [P(features|A) + P(features|B)] 
= 0.5 / (0.5 + 0.25) 
= 2/3 or 0.67.

Similarly, for class B:

Normalized P(B|features) 
= P(features|B) / [P(features|A) + P(features|B)] 
= 0.25 / 0.75 
= 1/3 or 0.33.

After normalization, the probabilities for the classes A and B will sum up to 1. This way, Naive Bayes ensures that our class predictions are probabilistically coherent.