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:

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.