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<title>Binary Data with the Beta Bernouli Distribution</title>
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<div class="section" id="binary-data-with-the-beta-bernouli-distribution">
<h1>Binary Data with the Beta Bernouli Distribution<a class="headerlink" href="#binary-data-with-the-beta-bernouli-distribution" title="Permalink to this headline">¶</a></h1>
<hr class="docutils" />
<p>Let’s consider one of the most basic types of data - binary data</p>
<div class="code python highlight-python"><div class="highlight"><pre>import pandas as pd
import seaborn as sns
import math
import cPickle as pickle
import itertools as it
import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import fetch_mldata
import itertools as it
%matplotlib inline
sns.set_context('talk')
</pre></div>
</div>
<p>Binary data can take various forms:</p>
<p>Image data is often represented as binary images. For example, the MNIST
dataset contains images of handwritten digits.</p>
<p>Let’s convert the MNIST digits into binary images</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">mnist_dataset</span> <span class="o">=</span> <span class="n">fetch_mldata</span><span class="p">(</span><span class="s">'MNIST original'</span><span class="p">)</span>
<span class="n">_</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="n">mnist_dataset</span><span class="p">[</span><span class="s">'data'</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">mnist_dataset</span><span class="p">[</span><span class="s">'data'</span><span class="p">]</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">bool</span><span class="p">)</span>
<span class="n">W</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">D</span><span class="p">))</span>
<span class="k">assert</span> <span class="n">W</span> <span class="o">*</span> <span class="n">W</span> <span class="o">==</span> <span class="n">D</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">Y</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">(</span><span class="n">W</span><span class="p">,</span> <span class="n">W</span><span class="p">)),</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">xticklabels</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">yticklabels</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">cbar</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'Example MNIST Digit'</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><matplotlib.text.Text at 0x1127f1910>
</pre></div>
</div>
<img alt="_images/beta-bernoulli_3_1.png" src="_images/beta-bernoulli_3_1.png" />
<p>Graphs can be represented as binary matrices</p>
<p>In this email communication network from the enron dataset,
<span class="math">\(X_{i,j} = 1\)</span> if and only if person<span class="math">\(_{i}\)</span> sent an email to
person<span class="math">\(_{j}\)</span></p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">enron_utils</span>
<span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s">'results.p'</span><span class="p">)</span> <span class="k">as</span> <span class="n">fp</span><span class="p">:</span>
<span class="n">communications</span> <span class="o">=</span> <span class="n">pickle</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">fp</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">allnames</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
<span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">o</span><span class="p">:</span>
<span class="k">yield</span> <span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="n">names</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">it</span><span class="o">.</span><span class="n">chain</span><span class="o">.</span><span class="n">from_iterable</span><span class="p">(</span><span class="n">allnames</span><span class="p">(</span><span class="n">communications</span><span class="p">)))</span>
<span class="n">names</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">names</span><span class="p">))</span>
<span class="n">namemap</span> <span class="o">=</span> <span class="p">{</span> <span class="n">name</span> <span class="p">:</span> <span class="n">idx</span> <span class="k">for</span> <span class="n">idx</span><span class="p">,</span> <span class="n">name</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">names</span><span class="p">)</span> <span class="p">}</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">names</span><span class="p">)</span>
<span class="n">communications_relation</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="p">,</span> <span class="n">N</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">bool</span><span class="p">)</span>
<span class="k">for</span> <span class="n">sender</span><span class="p">,</span> <span class="n">receivers</span> <span class="ow">in</span> <span class="n">communications</span><span class="p">:</span>
<span class="n">sender_id</span> <span class="o">=</span> <span class="n">namemap</span><span class="p">[</span><span class="n">sender</span><span class="p">]</span>
<span class="k">for</span> <span class="n">receiver</span> <span class="ow">in</span> <span class="n">receivers</span><span class="p">:</span>
<span class="n">receiver_id</span> <span class="o">=</span> <span class="n">namemap</span><span class="p">[</span><span class="n">receiver</span><span class="p">]</span>
<span class="n">communications_relation</span><span class="p">[</span><span class="n">sender_id</span><span class="p">,</span> <span class="n">receiver_id</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
<span class="n">labels</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">if</span> <span class="n">i</span><span class="o">%</span><span class="mi">20</span> <span class="o">==</span> <span class="mi">0</span> <span class="k">else</span> <span class="s">''</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">N</span><span class="p">)]</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">communications_relation</span><span class="p">,</span> <span class="n">linewidths</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">cbar</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">xticklabels</span><span class="o">=</span><span class="n">labels</span><span class="p">,</span> <span class="n">yticklabels</span><span class="o">=</span><span class="n">labels</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">'person number'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">'person number'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'Email Communication Matrix'</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><matplotlib.text.Text at 0x11bca0390>
</pre></div>
</div>
<img alt="_images/beta-bernoulli_5_1.png" src="_images/beta-bernoulli_5_1.png" />
<p>In a Bayesian context, one often models binary data with a beta
Bernoulli distribution</p>
<p>The beta Bernoulli distribution is the posterior of the Bernoulli
distribution and its conjugate prior the beta distribution</p>
<p>Recall that the Bernouli distribution is the likelihood of <span class="math">\(x\)</span>
given some probability <span class="math">\(\theta\)</span></p>
<div class="math">
\[P(x=1)=\theta\]</div>
<div class="math">
\[P(x=0)=1-\theta\]</div>
<div class="math">
\[P(x|\theta)=\theta^x(1-\theta)^{1-x}\]</div>
<p>If we wanted to learn the underlying probability <span class="math">\(\theta\)</span>, we
would use the beta distribution, which is the conjugate prior of the
Bernouli distribution.</p>
<hr class="docutils" />
<p>To import our desired distribution we’d call</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="kn">from</span> <span class="nn">microscopes.models</span> <span class="kn">import</span> <span class="n">bb</span> <span class="k">as</span> <span class="n">beta_bernoulli</span>
</pre></div>
</div>
<p>Then given the specific model we’d want we’d import</p>
<p><tt class="docutils literal"><span class="pre">from</span> <span class="pre">microscopes.model_name.definition</span> <span class="pre">import</span> <span class="pre">model_definition</span></tt></p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="kn">from</span> <span class="nn">microscopes.irm.definition</span> <span class="kn">import</span> <span class="n">model_definition</span> <span class="k">as</span> <span class="n">irm_definition</span>
<span class="kn">from</span> <span class="nn">microscopes.mixture.definition</span> <span class="kn">import</span> <span class="n">model_definition</span> <span class="k">as</span> <span class="n">mm_definition</span>
</pre></div>
</div>
<p>We would then define the model as follows</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">defn_mixture</span> <span class="o">=</span> <span class="n">mm_definition</span><span class="p">(</span><span class="n">Y</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">beta_bernoulli</span><span class="p">]</span><span class="o">*</span><span class="n">D</span><span class="p">)</span>
<span class="n">defn_irm</span> <span class="o">=</span> <span class="n">irm_definition</span><span class="p">([</span><span class="n">N</span><span class="p">],</span> <span class="p">[((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">beta_bernoulli</span><span class="p">)])</span>
</pre></div>
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