A Generative supervised learning algorithm used when the input space is $\{0,1\}^n$, so an input point consists of $x_i \in \{0,1\}$, $i=1,2,...,n$.

The particular variant below is called Multivariate Bernoulli event model, as it is part of a family of models called event models (another example of which is below).

Even though the assumptions in naive Bayes are too simplifying, the model often gives good results, because it isn't prone to Overfiting due to the relatively small number of parameters. It is however, not very good as a Generative model, but it does well as a classifier.

See Wiki

Intro lec vid

Definition of naive Bayes assumption

Problem with Naive Bayes. Probabilities $P(X|Y)=0$ for $X$s that have never been observed, and this causes problems ($P(Y|X)$ undefined). How to fix it actually here... (the method is known as Laplace smoothing).

Naive Bayes is a linear classifier, just like Gaussian discriminant analysis, it creates a Hyperplane Classification boundary, and a sigmoid posterior, like Logistic regression

Generalization to when $x_i$ can take any of $k$ values

here. Can arrise when discretizing a real valued $x_i$.

Variant for sequences (Multinomial event model)

here. description of model.

This is the log likelihood

Maximum likelihood estimates Laplace-smoothed estimate