# 3.7. Naive Bayes¶

Naive Bayes methods are a set of supervised learning algorithms based on applying Bayes’ theorem with the “naive” assumption of independence between every pair of features. Given a class variable and a dependent feature vector through , Bayes’ theorem states the following relationship:

Using the naive independence assumption that

for all , this relationship is simplified to

Since is constant given the input, we can use the following classification rule:

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

The different Naive Bayes classifiers differ mainly by the assumptions they make regarding the distribution of .

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 requires 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.

## 3.7.1. Gaussian Naive Bayes¶

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

The parameters and are estimated using maximum likelihood.

```
>>> from sklearn import datasets
>>> iris = datasets.load_iris()
>>> from sklearn.naive_bayes import GaussianNB
>>> gnb = GaussianNB()
>>> y_pred = gnb.fit(iris.data, iris.target).predict(iris.data)
>>> print "Number of mislabeled points : %d" % (iris.target != y_pred).sum()
Number of mislabeled points : 6
```

## 3.7.2. Multinomial Naive Bayes¶

`MultinomialNB` implements the Naive Bayes algorithm for multinomially
distributed data, and is one of the two classic Naive Bayes variants used in
text classification (where the data are typically represented as word vector
counts, although tf-idf vectors are also known to work well in practice).
The distribution is parametrized by vectors
for each class , where is the number of features
(in text classification, the size of the vocabulary)
and is the probability
of feature appearing in a sample belonging to class .

The parameters is estimated by a smoothed version of maximum likelihood, i.e. relative frequency counting:

where is the number of times feature appears in a sample of class in the training set , and is the total count of all features for class .

The smoothing priors accounts for features not present in the learning samples and prevents zero probabilities in further computations. Setting is called Laplace smoothing, while is called Lidstone smoothing.

## 3.7.3. Bernoulli Naive Bayes¶

`BernoulliNB` implements the Naive Bayes training and classification
algorithms for data that is distributed according to multivariate Bernoulli
distributions; i.e., there may be multiple features but each one is assumed
to be a binary-valued (Bernoulli, boolean) variable.
Therefore, this class requires samples to be represented as binary-valued
feature vectors; if handed any other kind of data, a `BernoulliNB` instance
may binarizes its input (depending on the `binarize` parameter).

The decision rule for Bernoulli Naive Bayes is based on

which differs from multinomial NB’s rule in that it explicitly penalizes the non-occurrence of a feature that is an indicator for class , where the multinomial variant would simply ignore a non-occurring feature.

In the case of text classification, word occurrence vectors (rather than word
count vectors) may be used to train and use this classifier. `BernoulliNB`
might perform better on some datasets, especially those with shorter documents.
It is advisable to evaluate both models, if time permits.

References:

- C.D. Manning, P. Raghavan and H. Schütze (2008). Introduction to Information Retrieval. Cambridge University Press, pp. 234-265.
- A. McCallum and K. Nigam (1998). A comparison of event models for Naive Bayes text classification. Proc. AAAI/ICML-98 Workshop on Learning for Text Categorization, pp. 41-48.
- V. Metsis, I. Androutsopoulos and G. Paliouras (2006). Spam filtering with Naive Bayes – Which Naive Bayes? 3rd Conf. on Email and Anti-Spam (CEAS).