Mahout currently has two implementations of Bayesian classifiers.  One is the traditional Naive Bayes approach, and the other is called Complementary Naive Bayes.
ImplementationsNaiveBayes (MAHOUT-9)
Complementary Naive Bayes (MAHOUT-60)
The Naive Bayes implementations in Mahout follow the paperhttp://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:

• j = 0 to N features
• k = 0 to L labels
  

Then:

• 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
• 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.
• 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  ) ]

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

• Wikipedia Bayes Example - Classify Wikipedia data.
  

• Twenty Newsgroups - Classify the classic Twenty Newsgroups data.