# 1.3. Naive Bayes¶

Naive Bayes methods are a set of supervised learning algorithms based on applying Bayes’ theorem with the “naive” assumption of conditional independence between every pair of features given the value of the class variable. Bayes’ theorem states the following relationship, given class variable \(y\) and dependent feature vector \(x_1\) through \(x_n\), :

Using the naive conditional independence assumption that

for all \(i\), this relationship is simplified to

Since \(P(x_1, \dots, x_n)\) is constant given the input, we can use the following classification rule:

and we can use Maximum A Posteriori (MAP) estimation to estimate \(P(y)\) and \(P(x_i \mid y)\); the former is then the relative frequency of class \(y\) in the training set.

The different naive Bayes classifiers differ mainly by the assumptions they make regarding the distribution of \(P(x_i \mid y)\).

In spite of their apparently over-simplified assumptions, naive Bayes classifiers have worked quite well in many real-world situations, famously document classification and spam filtering. They require a small amount of training data to estimate the necessary parameters. (For theoretical reasons why naive Bayes works well, and on which types of data it does, see the references below.)

Naive Bayes learners and classifiers can be extremely fast compared to more sophisticated methods. The decoupling of the class conditional feature distributions means that each distribution can be independently estimated as a one dimensional distribution. This in turn helps to alleviate problems stemming from the curse of dimensionality.

On the flip side, although naive Bayes is known as a decent classifier,
it is known to be a bad estimator, so the probability outputs from
`predict_proba`

are not to be taken too seriously.

References:

- H. Zhang (2004). The optimality of Naive Bayes. Proc. FLAIRS.

## 1.3.1. Gaussian Naive Bayes¶

`GaussianNB`

implements the Gaussian Naive Bayes algorithm for
classification. The likelihood of the features is assumed to be Gaussian:

The parameters \(\sigma_y\) and \(\mu_y\) are estimated using maximum likelihood.

```
>>> from sklearn import datasets
>>> iris = datasets.load_iris()
>>> from pmlearn.naive_bayes import GaussianNB
>>> gnb = GaussianNB()
>>> y_pred = gnb.fit(iris.data, iris.target).predict(iris.data)
>>> print("Number of mislabeled points out of a total %d points : %d"
... % (iris.data.shape[0],(iris.target != y_pred).sum()))
Number of mislabeled points out of a total 150 points : 6
```