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<!DOCTYPE html>
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<div class=" highlight hl-ipython2"><pre><span class="o">%</span><span class="k">pylab</span> inline
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<pre>Populating the interactive namespace from numpy and matplotlib
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<h1 id="Decomposing-Classification-For-Mixture-Models">Decomposing Classification For Mixture Models<a class="anchor-link" href="#Decomposing-Classification-For-Mixture-Models">¶</a></h1><p>By Kyle Cranmer, May 2015
@kylecranmer</p>
<p>A few slides related to Section 5.4 of <em>Approximating Likelihood Ratios with Calibrated Discriminative Classifiers</em>,
<a href="http://arxiv.org/abs/1506.02169">http://arxiv.org/abs/1506.02169</a></p>
<p><a href="https://creativecommons.org/licenses/by/4.0/"><img src="https://i.creativecommons.org/l/by/4.0/88x31.png" alt="CC BY"></a></p>
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<h1 id="Introduction">Introduction<a class="anchor-link" href="#Introduction">¶</a></h1><p>We consider the situation where you want to train a classifier between two different classes of events, and each class of events is generated from a mixture model</p>
\begin{equation}
p(x|\theta)=\sum_c w_c(\theta) p_c(x| \theta) \;,
\end{equation}<p>where $x$ is a high-dimensional feature vector, the index $c$ represents different classes of events with corresponding mixture coefficient $w_c(\theta)$ and probability distribution $p_c(x|\theta)$.</p>
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<p>Our goal is to train a classifier to discriminate between $p(x|\theta_0)$ and $p(x|\theta_1)$.</p>
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<p>We are considering the situation that both the coefficients and the distributions are parametrized by $\theta$, which may have several components.</p>
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<p>Special case: the mixture coefficents are not really functions of $\theta$. This can be accomodated by this formalism by identifying one of the components of $\theta$ with the mixture coefficient, i.e. $w_c(\theta) = \theta_i$</p>
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<p>Special case: the component distributions don't depend on $\theta$, i.e. $p_c(x|\theta) = p_c(x)$</p>
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<h1 id="The-problem">The problem<a class="anchor-link" href="#The-problem">¶</a></h1><p>The simplest thing we can do is:</p>
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<li>Generate training data $\{(x,y=0)\}$ from $p(x|\theta_0)$ and $\{(x,y=1)\}$ from $p(x|\theta_1)$, where $y$ is the class label</li>
<li>Train classifier based on this training data</li>
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<p>This is fine, but if the difference between $p(x|\theta_0)$ and $p(x|\theta_1)$ is due to a component with a small $w_c$, then most of the training examples do not focus on the relevant difference between the two classes.</p>
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<h1 id="An-example">An example<a class="anchor-link" href="#An-example">¶</a></h1><p>In particle physics, often one model is the <em>background-only</em> hypothesis</p>
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<li>$p(x|\theta_0) = p_b(x)$ is the only component</li>
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<p>the other is the <em>signal-plus-background</em> hypothesis,</p>
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<li>$p(x|\theta_1) = w_s p_s(x) + (1-w_s) p_b(x)$</li>
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<p>and $w_s$ is very small.</p>
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<h2 id="Capacity-Focusing-Technique">Capacity Focusing Technique<a class="anchor-link" href="#Capacity-Focusing-Technique">¶</a></h2><p>In order to focus the capacity of the classifier on the relevant regions of $x$, the classifier is not trained to distinguish $p(x|\theta_0)$ vs. $p(x|\theta_1)$.</p>
<p>Instead, a classifier is trained to distinguish $p_b(x)$ vs. $p_s(x)$.</p>
<p>This works because the likleihood ratio $p_s(x)/p_b(x)$ is 1-to-1 with $p(x|\theta_0)/p(x|\theta_1)$.</p>
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<h1 id="Decomposing-the-classification">Decomposing the classification<a class="anchor-link" href="#Decomposing-the-classification">¶</a></h1><p>We can generalize this capacity focusing technique to comparisons of mixture models, by rewriting the desired likelihood ratio</p>
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\begin{eqnarray}
\frac{p(x|\theta_0)}{p(x|\theta_1)} & =& \frac{\sum_c w_c(\theta_0) p_c(x| \theta_0)}{\sum_{c'} w_{c'}(\theta_1) p_{c'}(x| \theta_1)}
\\
%&=& \sum_c \frac{ w_c(\theta_0) p_c(x| \theta_0)}{\sum_{c'} w_{c'}(\theta_1) p_{c'}(x| \theta_1)} \\
%&=& \sum_c \left[ \frac{\sum_{c'} w_{c'}(\theta_1) p_{c'}(x| \theta_1)}{ w_c(\theta_0) p_c(x| \theta_0)} \right]^{-1} \\
&=& \sum_c \left[ \sum_{c'} \frac{ w_{c'}(\theta_1)}{w_c(\theta_0)} \frac{ p_{c'}(x| \theta_1)}{ p_c(x| \theta_0)} \right]^{-1} \\
&=& \sum_c \left[ \sum_{c'} \frac{ w_{c'}(\theta_1)}{w_c(\theta_0)} \frac{ p_{c'}(s_{c,c',\theta_0, \theta_1}| \theta_1)}{ p_c(s_{c,c',\theta_0, \theta_1}| \theta_0)} \right]^{-1}
\end{eqnarray}
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\begin{eqnarray}
\frac{p(x|\theta_0)}{p(x|\theta_1)} & =& \frac{\sum_c w_c(\theta_0) p_c(x| \theta_0)}{\sum_{c'} w_{c'}(\theta_1) p_{c'}(x| \theta_1)}
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&=& \sum_c \left[ \sum_{c'} \frac{ w_{c'}(\theta_1)}{w_c(\theta_0)} \frac{ p_{c'}(x| \theta_1)}{ p_c(x| \theta_0)} \right]^{-1} \\
%&=& \sum_c \left[ \sum_{c'} \frac{ w_{c'}(\theta_1)}{w_c(\theta_0)} \frac{ p_{c'}(s_{c,c',\theta_0, \theta_1}| \theta_1)}{ p_c(s_{c,c',\theta_0, \theta_1}| \theta_0)} \right]^{-1}
\end{eqnarray}
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<p>The second line is a trivial, but useful decomposition into pair-wise classification between $p_{c'}(x|\theta_1)$ and $p_c(x|\theta_0)$.</p>
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<p>In the situation where the only free parameters of the mixture model are the coefficients $w_c$, then the distributions $p_{c}(s_{c,c',\theta_0, \theta_1}| \theta)$ are independent of $\theta$ and can be pre-computed (after training the discriminative classifier, but before performing the generalized likelihood ratio test).</p>
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<h1 id="Demonstration">Demonstration<a class="anchor-link" href="#Demonstration">¶</a></h1><p>Consider these three distributions for the components</p>
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<div class=" highlight hl-ipython2"><pre><span class="k">def</span> <span class="nf">f0</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="o">*-</span><span class="mi">1</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">f1</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">+.</span><span class="mo">01</span>
<span class="k">def</span> <span class="nf">f2</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
<span class="n">fs</span> <span class="o">=</span> <span class="p">[</span><span class="n">f0</span><span class="p">,</span> <span class="n">f1</span><span class="p">,</span> <span class="n">f2</span><span class="p">]</span>
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<p>with these mixture coefficients</p>
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<div class=" highlight hl-ipython2"><pre><span class="n">w0</span> <span class="o">=</span> <span class="p">[</span><span class="o">.</span><span class="mi">1</span><span class="p">,</span><span class="o">.</span><span class="mi">2</span><span class="p">,</span><span class="o">.</span><span class="mi">7</span><span class="p">]</span>
<span class="n">w1</span> <span class="o">=</span> <span class="p">[</span><span class="o">.</span><span class="mi">2</span><span class="p">,</span><span class="o">.</span><span class="mi">5</span><span class="p">,</span><span class="o">.</span><span class="mi">3</span><span class="p">]</span>
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<p>Corresponding to the two mixture models</p>
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<div class=" highlight hl-ipython2"><pre><span class="k">def</span> <span class="nf">F0</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="nb">sum</span> <span class="o">=</span> <span class="mf">0.</span>
<span class="k">for</span> <span class="n">w</span><span class="p">,</span> <span class="n">f</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">w0</span><span class="p">,</span> <span class="n">fs</span><span class="p">):</span>
<span class="nb">sum</span> <span class="o">+=</span> <span class="n">w</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">return</span> <span class="nb">sum</span>
<span class="k">def</span> <span class="nf">F1</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="nb">sum</span> <span class="o">=</span> <span class="mf">0.</span>
<span class="k">for</span> <span class="n">w</span><span class="p">,</span> <span class="n">f</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">w1</span><span class="p">,</span> <span class="n">fs</span><span class="p">):</span>
<span class="nb">sum</span><span class="o">+=</span> <span class="n">w</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">return</span> <span class="nb">sum</span>
</pre></div>
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<h1 id="Visualization-of-the-models">Visualization of the models<a class="anchor-link" href="#Visualization-of-the-models">¶</a></h1>
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<div class="prompt input_prompt">In [52]:</div>
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<div class=" highlight hl-ipython2"><pre><span class="n">xarray</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
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<div class="prompt input_prompt">In [53]:</div>
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<div class=" highlight hl-ipython2"><pre><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xarray</span><span class="p">,</span><span class="n">f0</span><span class="p">(</span><span class="n">xarray</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s">'r'</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s">'--'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xarray</span><span class="p">,</span><span class="n">f1</span><span class="p">(</span><span class="n">xarray</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s">'y'</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s">'--'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xarray</span><span class="p">,</span><span class="n">f2</span><span class="p">(</span><span class="n">xarray</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s">'orange'</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s">'--'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xarray</span><span class="p">,</span><span class="n">F0</span><span class="p">(</span><span class="n">xarray</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s">'black'</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s">'-'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xarray</span><span class="p">,</span><span class="n">F1</span><span class="p">(</span><span class="n">xarray</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s">'b'</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s">'-'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
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<div class="output_area"><div class="prompt output_prompt">Out[53]:</div>
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<pre>[<matplotlib.lines.Line2D at 0x10727ee50>]</pre>
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<h1 id="Visualization-of-the-target-likleihood-ratio">Visualization of the target likleihood ratio<a class="anchor-link" href="#Visualization-of-the-target-likleihood-ratio">¶</a></h1>
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<div class="prompt input_prompt">In [54]:</div>
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<div class=" highlight hl-ipython2"><pre><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xarray</span><span class="p">,</span><span class="n">F0</span><span class="p">(</span><span class="n">xarray</span><span class="p">)</span><span class="o">/</span><span class="n">F1</span><span class="p">(</span><span class="n">xarray</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">'x'</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">'$F_0(x)/F_1(x)$'</span><span class="p">)</span>
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<div class="output_area"><div class="prompt output_prompt">Out[54]:</div>
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<pre><matplotlib.text.Text at 0x10701ae90></pre>
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