NAME

       Algorithm::NaiveBayes - Bayesian prediction of categories

SYNOPSIS

         use Algorithm::NaiveBayes;
         my $nb = Algorithm::NaiveBayes->new;

         $nb->add_instance
           (attributes => {foo => 1, bar => 1, baz => 3},
            label => 'sports');

         $nb->add_instance
           (attributes => {foo => 2, blurp => 1},
            label => ['sports', 'finance']);

         ... repeat for several more instances, then:
         $nb->train;

         # Find results for unseen instances
         my $result = $nb->predict
           (attributes => {bar => 3, blurp => 2});

DESCRIPTION

       This module implements the classic "Naive Bayes" machine learning algorithm.  It is a
       well-studied probabilistic algorithm often used in automatic text categorization.
       Compared to other algorithms (kNN, SVM, Decision Trees), it's pretty fast and reasonably
       competitive in the quality of its results.

       A paper by Fabrizio Sebastiani provides a really good introduction to text categorization:
       <http://faure.iei.pi.cnr.it/~fabrizio/Publications/ACMCS02.pdf>

METHODS

       new()
           Creates a new "Algorithm::NaiveBayes" object and returns it.  The following parameters
           are accepted:

           purge
               If set to a true value, the "do_purge()" method will be invoked during "train()".
               The default is true.  Set this to a false value if you'd like to be able to add
               additional instances after training and then call "train()" again.

       add_instance( attributes => HASH, label => STRING|ARRAY )
           Adds a training instance to the categorizer.  The "attributes" parameter contains a
           hash reference whose keys are string attributes and whose values are the weights of
           those attributes.  For instance, if you're categorizing text documents, the attributes
           might be the words of the document, and the weights might be the number of times each
           word occurs in the document.

           The "label" parameter can contain a single string or an array of strings, with each
           string representing a label for this instance.  The labels can be any arbitrary
           strings.  To indicate that a document has no applicable labels, pass an empty array
           reference.

       train()
           Calculates the probabilities that will be necessary for categorization using the
           "predict()" method.

       predict( attributes => HASH )
           Use this method to predict the label of an unknown instance.  The attributes should be
           of the same format as you passed to "add_instance()".  "predict()" returns a hash
           reference whose keys are the names of labels, and whose values are the score for each
           label.  Scores are between 0 and 1, where 0 means the label doesn't seem to apply to
           this instance, and 1 means it does.

           In practice, scores using Naive Bayes tend to be very close to 0 or 1 because of the
           way normalization is performed.  I might try to alleviate this in future versions of
           the code.

       labels()
           Returns a list of all the labels the object knows about (in no particular order), or
           the number of labels if called in a scalar context.

       do_purge()
           Purges training instances and their associated information from the NaiveBayes object.
           This can save memory after training.

       purge()
           Returns true or false depending on the value of the object's "purge" property.  An
           optional boolean argument sets the property.

       save_state($path)
           This object method saves the object to disk for later use.  The $path argument
           indicates the place on disk where the object should be saved:

             $nb->save_state($path);

       restore_state($path)
           This class method reads the file specified by $path and returns the object that was
           previously stored there using "save_state()":

             $nb = Algorithm::NaiveBayes->restore_state($path);

THEORY

       Bayes' Theorem is a way of inverting a conditional probability. It states:

                       P(y|x) P(x)
             P(x|y) = -------------
                          P(y)

       The notation "P(x|y)" means "the probability of "x" given "y"."  See also
       "/mathforum.org/dr.math/problems/battisfore.03.22.99.html"" in "http: for a simple but
       complete example of Bayes' Theorem.

       In this case, we want to know the probability of a given category given a certain string
       of words in a document, so we have:

                           P(words | cat) P(cat)
         P(cat | words) = --------------------
                                  P(words)

       We have applied Bayes' Theorem because "P(cat | words)" is a difficult quantity to compute
       directly, but "P(words | cat)" and "P(cat)" are accessible (see below).

       The greater the expression above, the greater the probability that the given document
       belongs to the given category.  So we want to find the maximum value.  We write this as

                                        P(words | cat) P(cat)
         Best category =   ArgMax      -----------------------
                          cat in cats          P(words)

       Since "P(words)" doesn't change over the range of categories, we can get rid of it.
       That's good, because we didn't want to have to compute these values anyway.  So our new
       formula is:

         Best category =   ArgMax      P(words | cat) P(cat)
                          cat in cats

       Finally, we note that if "w1, w2, ... wn" are the words in the document, then this
       expression is equivalent to:

         Best category =   ArgMax      P(w1|cat)*P(w2|cat)*...*P(wn|cat)*P(cat)
                          cat in cats

       That's the formula I use in my document categorization code.  The last step is the only
       non-rigorous one in the derivation, and this is the "naive" part of the Naive Bayes
       technique.  It assumes that the probability of each word appearing in a document is
       unaffected by the presence or absence of each other word in the document.  We assume this
       even though we know this isn't true: for example, the word "iodized" is far more likely to
       appear in a document that contains the word "salt" than it is to appear in a document that
       contains the word "subroutine".  Luckily, as it turns out, making this assumption even
       when it isn't true may have little effect on our results, as the following paper by Pedro
       Domingos argues: "/www.cs.washington.edu/homes/pedrod/mlj97.ps.gz"" in "http:

HISTORY

       My first implementation of a Naive Bayes algorithm was in the now-obsolete AI::Categorize
       module, first released in May 2001.  I replaced it with the Naive Bayes implementation in
       AI::Categorizer (note the extra 'r'), first released in July 2002.  I then extracted that
       implementation into its own module that could be used outside the framework, and that's
       what you see here.

AUTHOR

       Ken Williams, ken@mathforum.org

COPYRIGHT

       Copyright 2003-2004 Ken Williams.  All rights reserved.

       This library is free software; you can redistribute it and/or modify it under the same
       terms as Perl itself.

SEE ALSO

       AI::Categorizer(3), perl.