The Naive Bayes METHOD is a supervised learning algorithm, it is naive since it makes assumptions by applying Bayes’ theorem that all attributes are independent of each other.

Bayes’ theorem states the following relationship, given class VARIABLE y and dependent vector x1  through xn:

P(yi | x1,..., xn) =P(yi)P(x1,..., xn | yi)(P(x1,..., xn)

Using the naive CONDITIONAL independence assumption that each xiis independent: for all I this relationship is simplified to:

P(XI | yi, x1, ..., xi-1, xi+1, ...., xn) = P(xi | yi)

Since, P(x1,..., xn) is a constant given the input, we can use the following classification rule:

P(yi | x1, ..., xn) = P(y) ni=1P(xi | yi)P(x1,...,xn) and we can also use Maximum A Posteriori (MAP) estimation to estimate P(yi)and P(yi | xi) the former is then the relative frequency of class yin the training set.

P(yi | x1,..., xn)  P(yi) ni=1P(xi | yi)

y = arg max P(yi)ni=1P(xi | yi)

The different naive Bayes classifiers mainly differ by the assumptions they make regarding the DISTRIBUTION of P(yi | xi): can be Bernoulli, binomial, Gaussian, and so on.