Title: Bayesian
<a name="Bayesian-Intro"></a>
# Intro

Mahout currently has two implementations of Bayesian classifiers.  One is
the traditional Naive Bayes approach, and the other is called Complementary
Naive Bayes.

<a name="Bayesian-Implementations"></a>
# Implementations

[NaiveBayes](naivebayes.html)
 ([MAHOUT-9|http://issues.apache.org/jira/browse/MAHOUT-9])

[Complementary Naive Bayes](complementary-naive-bayes.html)
 ([MAHOUT-60|http://issues.apache.org/jira/browse/MAHOUT-60])

The Naive Bayes implementations in Mahout follow the paper [http://people.csail.mit.edu/jrennie/papers/icml03-nb.pdf](http://people.csail.mit.edu/jrennie/papers/icml03-nb.pdf)
 Before we get to the actual algorithm lets discuss the terminology

Given, in an input set of classified documents: 
1. j = 0 to N features 
1. k = 0 to L labels

Then:

1. Normalized Frequency for a term(feature) in a document is calculated by
dividing the term frequency by the root mean square of terms frequencies in
that document
1. Weight Normalized Tf for a given feature in a given label = sum of
Normalized Frequency of the feature across all the documents in the label. 
1. Weight Normalized Tf-Idf for a given feature in a label is the Tf-idf
calculated using standard idf multiplied by the Weight Normalized Tf

Once Weight Normalized Tf-idf(W-N-Tf-idf) is calculated, the final weight
matrix for Bayes and Cbayes are calculated as follows

We calculate the sum of W-N-Tf-idf for all the features in a label called
as Sigma_k or sumLabelWeight

For Bayes

    Weight = Log [ ( W-N-Tf-Idf + alpha_i ) / ( Sigma_k + N  ) ]

For CBayes

We calculate the Sum of W-N-Tf-Idf across all labels for a given feature.
We call this sumFeatureWeight of Sigma_j
Also we sum the entire W-N-Tf-Idf weights for all feature,label pair in the
train set. Call this Sigma_jSigma_k

Final Weight is calculated as

    Weight = Log [ ( Sigma_j - W-N-Tf-Idf + alpha_i ) / ( Sigma_jSigma_k - Sigma_k + N  ) ]


<a name="Bayesian-Examples"></a>
# Examples

In Mahout's example code, there are two samples that can be used:

1. [Wikipedia Bayes Example](wikipedia-bayes-example.html)
 - Classify Wikipedia data.

1. [Twenty Newsgroups](twenty-newsgroups.html)
 - Classify the classic Twenty Newsgroups data.