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<title>Write your own Bayesian Classifier!</title>

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<div class='slide'>
<h1>Write your own Bayesian Classifier!</h1>
<p>John Melesky
(Open Source Bridge, June 2009)</p>
</div>


<div class='slide'>
<h1>What's a Bayesian Classifier?</h1>
</div>


<div class='slide'>
<h1>What's a Bayesian Classifier?</h1>
<p>Something which classifies based on:</p>
<ol>
<li>Information about past categorizations</li>
<li>Bayesian statistics (Bayes' Theorem)</li>
</ol>
</div>


<div class='slide'>
<h1>What's Bayes' Theorem?</h1>
<p>Let's check <a href="http://phaedrusdeinus.org/Bayes&apos;_theorem.html">Wikipedia</a>.</p>
</div>


<div class='slide'>
<h1>Derrr....</h1>
</div>


<div class='slide'>
<h1>An example: random drug testing</h1>
<p>3% of the population are using Zopadrine.</p>
<p>We have a drug test with a 98% accuracy rate.</p>
</div>


<div class='slide'>
<h1>An example: random drug testing</h1>
<p>3% of the population are using Zopadrine.</p>
<p>We have a drug test with a 98% accuracy rate.</p>
<p>Bob is tested, and the result is positive. How likely is it that Bob uses Zopadrine?</p>
</div>


<div class='slide'>
<h1>Break it down</h1>
<p>Let's assume a population of 10000 people.</p>
</div>


<div class='slide'>
<h1>Break it down</h1>
<p>3% are users.</p>
<table border=1>
<tr><td></td><td>Population</td></tr>
<tr><td>Clean</td><td>9700</td></tr>
<tr><td>Users</td><td>300</td></tr>
<tr><td>Total</td><td>10000</td></tr>
</table>
</div>


<div class='slide'>
<h1>Break it down</h1>
<p>The test is 98% accurate.</p>
<table border=1>
<tr><td></td><td>Population</td><td>Test negative</td><td>Test positive</td></tr>
<tr><td>Clean</td><td>9700</td><td>9506</td><td>194</td></tr>
<tr><td>Users</td><td>300</td><td>6</td><td>294</td></tr>
<tr><td>Total</td><td>10000</td><td>9512</td><td>488</td></tr>
</table>
</div>


<div class='slide'>
<h1>Break it down</h1>
<p>Bob is tested, and the result is positive. How likely is it that Bob uses Zopadrine?</p>
<table border=1>
<tr><td></td><td>Population</td><td>Test negative</td><td>Test positive</td></tr>
<tr><td>Clean</td><td>9700</td><td>9506</td><td>194</td></tr>
<tr><td>Users</td><td>300</td><td>6</td><td bgcolor="#ff6666">294</td></tr>
<tr><td>Total</td><td>10000</td><td>9512</td><td bgcolor="#ff6666">488</td></tr>
</table>
</div>


<div class='slide'>
<h1>Break it down</h1>
<p>294 / 488 = 60.24%</p>
</div>


<div class='slide'>
<h1>Back to Bayes' Theorem</h1>
<p><img alt="Bayes&apos; Theorem" src="img/bayes.png" /></p>
</div>


<div class='slide'>
<h1>Back to Bayes' Theorem</h1>
<table>
<tr><td>P = probability</td><td rowspan="4"><img src="img/bayes.png"></td></tr>
<tr><td>A = "is a user"</td></tr>
<tr><td>B = "tests positive"</td></tr>
<tr><td>x|y = x, given y</td></tr>
</table>
</div>


<div class='slide'>
<h1>Back to Bayes' Theorem</h1>
<table>
<tr><td>P(A) = probability of being a user</td><td rowspan="4"><img src="img/bayes.png"></td></tr>
<tr><td>P(B|A) = probability of testing positive, given being a user</td></tr>
<tr><td>P(B) = probability of testing positive</td></tr>
<tr><td>P(A|B) = probability Bob's a user</td></tr>
</table>
</div>


<div class='slide'>
<h1>Back to Bayes' Theorem</h1>
<table>
<tr><td>P(A) = 3%</td><td rowspan="4"><img src="img/bayes.png"></td></tr>
<tr><td>P(B|A) = probability of testing positive, given being a user</td></tr>
<tr><td>P(B) = probability of testing positive</td></tr>
<tr><td>P(A|B) = probability Bob's a user</td></tr>
</table>
</div>


<div class='slide'>
<h1>Back to Bayes' Theorem</h1>
<table>
<tr><td>P(A) = 3%</td><td rowspan="4"><img src="img/bayes.png"></td></tr>
<tr><td>P(B|A) = 98%</td></tr>
<tr><td>P(B) = probability of testing positive</td></tr>
<tr><td>P(A|B) = probability Bob's a user</td></tr>
</table>
</div>


<div class='slide'>
<h1>Back to the numbers</h1>
<table border=1>
<tr><td></td><td>Population</td><td>Test negative</td><td>Test positive</td></tr>
<tr><td>Clean</td><td>9700</td><td>9506</td><td>194</td></tr>
<tr><td>Users</td><td>300</td><td>6</td><td>294</td></tr>
<tr><td>Total</td><td bgcolor="#ff6666">10000</td><td>9512</td><td bgcolor="#ff6666">488</td></tr>
</table>
</div>


<div class='slide'>
<h1>Back to Bayes' Theorem</h1>
<table>
<tr><td>P(A) = 3%</td><td rowspan="4"><img src="img/bayes.png"></td></tr>
<tr><td>P(B|A) = 98%</td></tr>
<tr><td>P(B) = 4.88%</td></tr>
<tr><td>P(A|B) = probability Bob's a user</td></tr>
</table>
</div>


<div class='slide'>
<h1>Back to Bayes' Theorem</h1>
<table>
<tr><td>P(A) = 3%</td><td rowspan="4"><img src="img/bayes.png"></td></tr>
<tr><td>P(B|A) = 98%</td></tr>
<tr><td>P(B) = 4.88%</td></tr>
<tr><td>P(A|B) = (98% * 3%)/4.88% = 60.24%</td></tr>
</table>
</div>


<div class='slide'>
<h1>This works with population numbers, too</h1>
<pre><code>P(A)   = 300
P(B|A) = 9800
P(B)   = 488
P(A|B) = 6024
</code></pre>
<p>Which is useful for reasons we'll see later.</p>
</div>


<div class='slide'>
<h1>Bayes' Theorem, in code</h1>
<p>My examples are going to be in perl.</p>
<pre><code>sub bayes {
  my ($p_a, $p_b, $p_b_a) = @_;

  my $p_a_b = ($p_b_a * $p_a) / $p_b;

  return $p_a_b;
}
</code></pre>
</div>


<div class='slide'>
<h1>Bayes' Theorem, in code</h1>
<p>But you could just as easily work in Python.</p>
<pre><code>def bayes(p_a, p_b, p_b_a):
  return (p_b_a * p_a) / p_b
</code></pre>
</div>


<div class='slide'>
<h1>Bayes' Theorem, in code</h1>
<p>Or Java</p>
<pre><code>public static Double bayes(Double p_a, Double p_b, Double p_b_a) {
  Double p_a_b = (p_b_a * p_a) / p_b;

  return p_a_b;
}
</code></pre>
</div>


<div class='slide'>
<h1>Bayes' Theorem, in code</h1>
<p>Or SML</p>
<pre><code>let bayes(p_a, p_b, p_b_a) = (p_b_a * p_a) / p_b
</code></pre>
</div>


<div class='slide'>
<h1>Bayes' Theorem, in code</h1>
<p>Or Erlang</p>
<pre><code>bayes(p_a, p_b, p_b_a) -&gt;
  (p_b_a * p_a) / p_b.
</code></pre>
</div>


<div class='slide'>
<h1>Bayes' Theorem, in code</h1>
<p>Or Haskell</p>
<pre><code>bayes p_a p_b p_b_a = (p_b_a * p_a) / p_b
</code></pre>
</div>


<div class='slide'>
<h1>Bayes' Theorem, in code</h1>
<p>Or Scheme</p>
<pre><code>(define (bayes p_a p_b p_b_a)
  (/ (* p_b_a p_a) p_b))
</code></pre>
</div>


<div class='slide'>
<h1>Bayes' Theorem, in code</h1>
<p>LOLCODE, anyone? Befunge? Unlambda?</p>
<p>If it supports floating point operations, you're set.</p>
</div>


<div class='slide'>
<h1>How does that make a classifier?</h1>
<pre><code>A = "is spam"
B = "contains the string 'viagra'"
</code></pre>
<p>What's P(A|B)?</p>
</div>


<div class='slide'>
<h1>What do we need for a classifier?</h1>
<ol>
<li>We need to tokenize our training set</li>
<li>Then build a model</li>
<li>Then test that model</li>
<li>Then apply that model to new data</li>
</ol>
</div>


<div class='slide'>
<h1>What do we need for a classifier?</h1>
<ol>
<li><strong>We need to tokenize our training set</strong></li>
<li>Then build a model</li>
<li>Then test that model</li>
<li>Then apply that model to new data</li>
</ol>
</div>


<div class='slide'>
<h1>Tokenizing your training set</h1>
<p><em>Fancy</em> perl</p>
<pre><code>sub tokenize {
  my $contents = shift;

  my %tokens = map { $_ =&gt; 1 } split(/\s+/, $contents);
  return %tokens;
}
</code></pre>
</div>


<div class='slide'>
<h1>Tokenizing your training set</h1>
<pre><code>sub tokenize_file {
  my $filename = shift;

  my $contents = '';
  open(FILE, $filename);
  read(FILE, $contents, -s FILE);
  close(FILE);

  return tokenize($contents);
}
</code></pre>
</div>


<div class='slide'>
<h1>Tokenizing your training set</h1>
<p>This is the "bag of words" model.</p>
<p>For each category (spam, not spam), we need to know how many documents in the training set contain a given word.</p>
</div>


<div class='slide'>
<h1>Tokenizing your training set</h1>
<pre><code>my %spam_tokens = ();
my %notspam_tokens = ();

foreach my $file (@spam_files) {
  my %tokens = tokenize_file($file);
  %spam_tokens = combine_hash(\%spam_tokens, \%tokens);
}

foreach my $file (@notspam_files) {
  my %tokens = tokenize_file($file);
  %notspam_tokens = combine_hash(\%notspam_tokens, \%tokens);
}
</code></pre>
</div>


<div class='slide'>
<h1>Tokenizing your training set</h1>
<pre><code>sub combine_hash {
  my ($hash1, $hash2) = @_;

  my %resulthash = %{ $hash1 };

  foreach my $key (keys(%{ $hash2 })) {
    if ($resulthash{$key}) {
      $resulthash{$key} += $hash2-&gt;{$key};
    } else {
      $resulthash{$key} = $hash2-&gt;{$key};
    }
  }

  return %resulthash;
}
</code></pre>
</div>


<div class='slide'>
<h1>What do we need for a classifier?</h1>
<ol>
<li>We need to tokenize our training set</li>
<li><strong>Then build a model</strong></li>
<li>Then test that model</li>
<li>Then apply that model to new data</li>
</ol>
</div>


<div class='slide'>
<h1>Build a model</h1>
<pre><code>my %total_tokens = combine_hash(\%spam_tokens, \%notspam_tokens);

my $total_spam_files = scalar(@spam_files);
my $total_notspam_files = scalar(@notspam_files);
my $total_files = $total_spam_files + $total_notspam_files;
my $probability_spam = $total_spam_files / $total_files;
my $probability_notspam = $total_notspam_files / $total_files;
</code></pre>
</div>


<div class='slide'>
<h1>Build a model</h1>
<p>In this case, our model is just a bunch of numbers. </p>
</div>


<div class='slide'>
<h1>Build a model</h1>
<p>In this case, our model is just a bunch of numbers. </p>
<p>(a little secret: it's <em>all</em> a bunch of numbers)</p>
</div>


<div class='slide'>
<h1>What do we need for a classifier?</h1>
<ol>
<li>We need to tokenize our training set</li>
<li>Then build a model</li>
<li><strong>Then test that model</strong></li>
<li>Then apply that model to new data</li>
</ol>
</div>


<div class='slide'>
<h1>*cough* *cough*</h1>
</div>


<div class='slide'>
<h1>What do we need for a classifier?</h1>
<ol>
<li>We need to tokenize our training set</li>
<li>Then build a model</li>
<li>Then test that model</li>
<li><strong>Then apply that model to new data</strong></li>
</ol>
</div>


<div class='slide'>
<h1>Apply that model to new data</h1>
<pre><code>my %test_tokens = tokenize_file($test_file);

foreach my $token (keys(%test_tokens)) {
  if (exists($total_tokens{$token})) {
    my $p_t_s = (($spam_tokens{$token} || 0) + 1) /
                ($total_spam_files + $total_tokens);
    $spam_accumulator = $spam_accumulator * $p_t_s;

    my $p_t_ns = (($notspam_tokens{$token} || 0) + 1) /
                 ($total_notspam_files + $total_tokens);

    $notspam_accumulator = $notspam_accumulator * $p_t_ns;
  }
}
</code></pre>
</div>


<div class='slide'>
<h1>Apply that model to new data</h1>
<pre><code>my $score_spam = bayes( $probability_spam,
                        $total_tokens,
                        $spam_accumulator );

my $score_notspam = bayes( $probability_notspam,
                           $total_tokens,
                           $notspam_accumulator );

my $likelihood_spam = $score_spam / ($score_spam + $score_notspam);
my $likelihood_notspam = $score_notspam / ($score_spam + $score_notspam);

printf("likelihood of spam email: %0.2f %%\n", ($likelihood_spam * 100));
</code></pre>
</div>


<div class='slide'>
<h1>Boom</h1>
</div>


<div class='slide'>
<h1>What sucks?</h1>
</div>


<div class='slide'>
<h1>What sucks?</h1>
<ul>
<li>Our tokenization</li>
</ul>
</div>


<div class='slide'>
<h1>What sucks?</h1>
<ul>
<li>Our tokenization</li>
<li>Our memory limitations</li>
</ul>
</div>


<div class='slide'>
<h1>What sucks?</h1>
<ul>
<li>Our tokenization</li>
<li>Our memory limitations</li>
<li>Saving/loading models</li>
</ul>
</div>


<div class='slide'>
<h1>Improve memory use</h1>
<h3>Limit the number of tokens</h3>
<p>We want to use the tokens with the highest information values. That means tokens that are predominantly in one category but not the other.</p>
</div>


<div class='slide'>
<h1>Improve memory use</h1>
<h3>Limit the number of tokens</h3>
<p>We want to use the tokens with the highest information values. That means tokens that are predominantly in one category but not the other.</p>
<p>There are a bunch of ways to calculate this, though the big one is Information Gain.</p>
</div>


<div class='slide'>
<h1>Improve tokenization, simple stuff</h1>
<ul>
<li>Weed out punctuation</li>
<li>Weed out stopwords</li>
<li>normaLize CASE</li>
<li>Strip out markup</li>
</ul>
</div>


<div class='slide'>
<h1>Improve tokenization, advanced stuff</h1>
<h3>Stemming</h3>
<p>"wrestling", "wrestler", "wrestled", and "wrestle" are all the same word concept.</p>
<p>Pros: fewer tokens, related tokens match</p>
<p>Cons: some words are hard to stem correctly (e.g. "cactus")</p>
</div>


<div class='slide'>
<h1>Improve tokenization, advanced stuff</h1>
<h3>Include bigrams</h3>
<p>Bigrams are token pairs. For example, "open source", "ron paul", "twitter addict".</p>
<p>Pros: we start distinguishing between Star Wars and astronomy wars</p>
<p>Cons: our memory use balloons</p>
</div>


<div class='slide'>
<h1>Improve tokenization, advanced stuff</h1>
<h3>Use numbers</h3>
<p>Instead of binary (word x is in doc y), we store frequencies (word x appears z times in doc y).</p>
<p>Pros: damage from weak associations is reduced; easier to find the important words in a document</p>
<p>Cons: the math becomes more complex; in many cases, accuracy doesn't actually increase</p>
</div>


<div class='slide'>
<h1>Improve tokenization, advanced stuff</h1>
<h3>Use non-token features</h3>
<p>Sometimes we want to use non-textual attributes of documents. For example, length of document, percent of capital letters.</p>
</div>


<div class='slide'>
<h1>Improve tokenization, advanced stuff</h1>
<h3>Use non-token features</h3>
<p>Sometimes we want to use non-textual attributes of documents. For example, length of document, percent of capital letters.</p>
<p>We can also grab structural information, like the sender, or subject line, and treat them differently. Or whether the word appears early or late in the document.</p>
</div>


<div class='slide'>
<h1>Improve tokenization, advanced stuff</h1>
<h3>Use non-token features</h3>
<p>Sometimes we want to use non-textual attributes of documents. For example, length of document, percent of capital letters.</p>
<p>We can also grab structural information, like the sender, or subject line, and treat them differently. Or whether the word appears early or late in the document.</p>
<p>Pros: a little can go a long way</p>
<p>Cons: selecting these can be a dark art. or an incredible memory burden.</p>
</div>


<div class='slide'>
<h1>Which leads us to</h1>
</div>


<div class='slide'>
<h1>Which leads us to</h1>
<p>Tokenization == Vectorization</p>
</div>


<div class='slide'>
<h1>In other words</h1>
<p>Our documents are all just vectors of numbers.</p>
</div>


<div class='slide'>
<h1>Or even</h1>
<p>Our documents are all just points in a high-dimensional Cartesian space.</p>
</div>


<div class='slide'>
<h1>Vectors of numbers</h1>
<p>This concept opens up a whole world of statistical methods for categorization, including decision trees, linear separations, and support vector machines.</p>
</div>


<div class='slide'>
<h1>Points in space</h1>
<p>And this opens up a whole different world of geometric methods for categorization and information manipulation, including k-nearest-neighbor classification and various clustering algorithms.</p>
</div>


<div class='slide'>
<h1>Alright</h1>
<p>It's been a long trip. Any questions?</p>
</div>


<div class='slide'>
<h1>Thanks</h1>
<p>Thanks for coming. Thanks to OS Bridge for having me.</p>
</div>


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