`
``
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});
`

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: