### Summary

In the world of artificial intelligence and machine learning, the Naive Bayes Classifier is a popular algorithm used for binary and multiclass classification tasks. This simple yet effective model is based on applying Bayes’ theorem and makes the ’naive’ assumption that all features are independent of each other.

### ELI5

Imagine you are a detective. You have a big box of clues related to many different investigations. Each clue has different properties, like color, size, or shape. You want to use these properties to sort out which investigation the clue belongs to, but you’re not sure how. However, you do know that, based on your previous experiences, certain properties tend to show up more often for some investigations. The Naive Bayes Classifier is like your clever assistant that helps you sort these clues, assuming each property is independent and doesn’t affect the others. It calculates the probabilities, and gives you the most likely investigation for each clue.

### In-depth explanation

The Naive Bayes Classifier is a probabilistic learning algorithm, widely used for classification problems. The algorithm builds on Bayes’ theorem, a fundamental theorem in probability theory and statistics that describes the relationship of conditional probabilities of statistical quantities.

In particular, the Naive Bayes Classifier computes the posterior probability P(c|x) of a class (target) given predictor (attribute) values of a particular instance. The class with the highest posterior probability is considered as the prediction.

The ’naive’ in Naive Bayes comes from the assumption that the probabilities of each attribute are independent of each other. In reality, this might not be the case, and attributes can be dependent. However, despite its simple architecture and the somewhat ’naive’ assumption, Naive Bayes Classifier often performs surprisingly well and is particularly useful when dimensionality of the input is high.

The simple architecture of Naive Bayes Classifier lends itself well to a vast array of applications from spam filtering, to text categorization, to medical diagnosis.

Despite its advantages, the Naive Bayes classifier is not suitable for all tasks. It tends to perform poorly if correlation between features exists, as it’s incapable of modeling such relationships. Also, it requires a reasonable estimation of probabilities and large reduction in computational complexity to work effectively. That being said, if the feature independence assumption holds, or the correlation among features is negligible, then Naive Bayes can outperform even the most sophisticated classification methods.

Bayes’ theorem, Classification, Feature Independence, Multinomial Naive Bayes, Bernoulli Naive Bayes, Gaussian Naive Bayes, Prior Probability, Posterior Probability, Maximum A Posteriori (MAP) Decision Rule, Likelihood Function