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<li class="toctree-l2"><a class="reference internal" href="#mcmc">4.1. MCMC</a></li>
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<section id="examples">
<h1><span class="section-number">4. </span>Examples<a class="headerlink" href="#examples" title="Permalink to this heading"></a></h1>
<p>The following examples are available in a checkout of the repository
in the <code class="docutils literal notranslate"><span class="pre">examples/</span></code> directory.</p>
<section id="mcmc">
<h2><span class="section-number">4.1. </span>MCMC<a class="headerlink" href="#mcmc" title="Permalink to this heading"></a></h2>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="sd">'''This example illustrates how to run a Markov Chain using pypmc'''</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pypmc</span>
<span class="c1"># define a proposal</span>
<span class="n">prop_dof</span> <span class="o">=</span> <span class="mf">1.</span>
<span class="n">prop_sigma</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.1</span> <span class="p">,</span> <span class="mf">0.</span> <span class="p">]</span>
<span class="p">,[</span><span class="mf">0.</span> <span class="p">,</span> <span class="mf">0.02</span><span class="p">]])</span>
<span class="n">prop</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">student_t</span><span class="o">.</span><span class="n">LocalStudentT</span><span class="p">(</span><span class="n">prop_sigma</span><span class="p">,</span> <span class="n">prop_dof</span><span class="p">)</span>
<span class="c1"># define the target; i.e., the function you want to sample from.</span>
<span class="c1"># In this case, it is a Gaussian with mean "target_mean" and</span>
<span class="c1"># covariance "target_sigma".</span>
<span class="c1">#</span>
<span class="c1"># Note that the target function "log_target" returns the log of the</span>
<span class="c1"># unnormalized gaussian density.</span>
<span class="n">target_sigma</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.01</span> <span class="p">,</span> <span class="mf">0.003</span> <span class="p">]</span>
<span class="p">,[</span><span class="mf">0.003</span><span class="p">,</span> <span class="mf">0.0025</span><span class="p">]])</span>
<span class="n">inv_target_sigma</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">target_sigma</span><span class="p">)</span>
<span class="n">target_mean</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">4.3</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">])</span>
<span class="k">def</span> <span class="nf">unnormalized_log_pdf_gauss</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">inv_sigma</span><span class="p">):</span>
<span class="n">diff</span> <span class="o">=</span> <span class="n">x</span> <span class="o">-</span> <span class="n">mu</span>
<span class="k">return</span> <span class="o">-</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">diff</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">inv_sigma</span><span class="p">)</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">diff</span><span class="p">)</span>
<span class="n">log_target</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">unnormalized_log_pdf_gauss</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">target_mean</span><span class="p">,</span> <span class="n">inv_target_sigma</span><span class="p">)</span>
<span class="c1"># choose a bad initialization</span>
<span class="n">start</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">-</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">10.</span><span class="p">])</span>
<span class="c1"># define the markov chain object</span>
<span class="n">mc</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">sampler</span><span class="o">.</span><span class="n">markov_chain</span><span class="o">.</span><span class="n">AdaptiveMarkovChain</span><span class="p">(</span><span class="n">log_target</span><span class="p">,</span> <span class="n">prop</span><span class="p">,</span> <span class="n">start</span><span class="p">)</span>
<span class="c1"># run burn-in</span>
<span class="n">mc</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">4</span><span class="p">)</span>
<span class="c1"># delete burn-in from samples</span>
<span class="n">mc</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
<span class="c1"># run 100,000 steps adapting the proposal every 500 steps</span>
<span class="c1"># hereby save the accept count which is returned by mc.run</span>
<span class="n">accept_count</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">200</span><span class="p">):</span>
<span class="n">accept_count</span> <span class="o">+=</span> <span class="n">mc</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mi">500</span><span class="p">)</span>
<span class="n">mc</span><span class="o">.</span><span class="n">adapt</span><span class="p">()</span>
<span class="c1"># extract a reference to the history of all visited points</span>
<span class="n">values</span> <span class="o">=</span> <span class="n">mc</span><span class="o">.</span><span class="n">samples</span><span class="p">[:]</span>
<span class="n">accept_rate</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">accept_count</span><span class="p">)</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">values</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"The chain accepted </span><span class="si">%4.2f%%</span><span class="s2"> of the proposed points"</span> <span class="o">%</span> <span class="p">(</span><span class="n">accept_rate</span> <span class="o">*</span> <span class="mi">100</span><span class="p">)</span> <span class="p">)</span>
<span class="c1"># plot the result</span>
<span class="k">try</span><span class="p">:</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'For plotting "matplotlib" needs to be installed'</span><span class="p">)</span>
<span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">hexbin</span><span class="p">(</span><span class="n">values</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">values</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">gridsize</span> <span class="o">=</span> <span class="mi">40</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'gray_r'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p>(<a class="reference download internal" download="" href="_downloads/07517a7d7ca8c2300bd6a1fd546f1bc5/markov_chain.py"><code class="xref download docutils literal notranslate"><span class="pre">Source</span> <span class="pre">code</span></code></a>, <a class="reference download internal" download="" href="_downloads/d5d0d3aa599f5627ef87b45b3887ca8c/markov_chain.png"><code class="xref download docutils literal notranslate"><span class="pre">png</span></code></a>, <a class="reference download internal" download="" href="_downloads/5c6b6e4219dedecfc502b679827e20d2/markov_chain.hires.png"><code class="xref download docutils literal notranslate"><span class="pre">hires.png</span></code></a>, <a class="reference download internal" download="" href="_downloads/545efd64ee52b97db872938e1980591c/markov_chain.pdf"><code class="xref download docutils literal notranslate"><span class="pre">pdf</span></code></a>)</p>
<figure class="align-default">
<img alt="_images/markov_chain.png" class="plot-directive" src="_images/markov_chain.png" />
</figure>
</section>
<section id="pmc">
<h2><span class="section-number">4.2. </span>PMC<a class="headerlink" href="#pmc" title="Permalink to this heading"></a></h2>
<section id="serial">
<h3><span class="section-number">4.2.1. </span>Serial<a class="headerlink" href="#serial" title="Permalink to this heading"></a></h3>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="sd">'''This example shows how to use importance sampling and how to</span>
<span class="sd">adapt the proposal density using the pmc algorithm.</span>
<span class="sd">'''</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pypmc</span>
<span class="c1"># define the target; i.e., the function you want to sample from.</span>
<span class="c1"># In this case, it is a bimodal Gaussian</span>
<span class="c1">#</span>
<span class="c1"># Note that the target function "log_target" returns the log of the</span>
<span class="c1"># target function.</span>
<span class="n">component_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">])</span>
<span class="n">mean0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span> <span class="p">([</span> <span class="mf">5.0</span> <span class="p">,</span> <span class="mf">0.01</span> <span class="p">])</span>
<span class="n">covariance0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">0.01</span> <span class="p">,</span> <span class="mf">0.003</span> <span class="p">],</span>
<span class="p">[</span> <span class="mf">0.003</span><span class="p">,</span> <span class="mf">0.0025</span><span class="p">]])</span>
<span class="n">inv_covariance0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">covariance0</span><span class="p">)</span>
<span class="n">mean1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span> <span class="p">([</span><span class="o">-</span><span class="mf">4.0</span> <span class="p">,</span> <span class="mf">1.0</span> <span class="p">])</span>
<span class="n">covariance1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">0.1</span> <span class="p">,</span> <span class="mf">0.</span> <span class="p">],</span>
<span class="p">[</span> <span class="mf">0.</span> <span class="p">,</span> <span class="mf">0.02</span> <span class="p">]])</span>
<span class="n">inv_covariance1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">covariance1</span><span class="p">)</span>
<span class="n">component_means</span> <span class="o">=</span> <span class="p">[</span><span class="n">mean0</span><span class="p">,</span> <span class="n">mean1</span><span class="p">]</span>
<span class="n">component_covariances</span> <span class="o">=</span> <span class="p">[</span><span class="n">covariance0</span><span class="p">,</span> <span class="n">covariance1</span><span class="p">]</span>
<span class="n">target_mixture</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">mixture</span><span class="o">.</span><span class="n">create_gaussian_mixture</span><span class="p">(</span><span class="n">component_means</span><span class="p">,</span> <span class="n">component_covariances</span><span class="p">,</span> <span class="n">component_weights</span><span class="p">)</span>
<span class="n">log_target</span> <span class="o">=</span> <span class="n">target_mixture</span><span class="o">.</span><span class="n">evaluate</span>
<span class="c1"># define the initial proposal density</span>
<span class="c1"># In this case a three-modal gaussian used</span>
<span class="c1"># the initial covariances are set to the unit-matrix</span>
<span class="c1"># the initial component weights are set equal</span>
<span class="n">initial_prop_means</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">initial_prop_means</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="mf">4.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span> <span class="p">)</span>
<span class="n">initial_prop_means</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">-</span><span class="mf">5.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span> <span class="p">)</span>
<span class="n">initial_prop_means</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span> <span class="p">)</span>
<span class="n">initial_prop_covariance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="n">initial_prop_components</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span>
<span class="n">initial_prop_components</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">gauss</span><span class="o">.</span><span class="n">Gauss</span><span class="p">(</span><span class="n">initial_prop_means</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">initial_prop_covariance</span><span class="p">))</span>
<span class="n">initial_proposal</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">mixture</span><span class="o">.</span><span class="n">MixtureDensity</span><span class="p">(</span><span class="n">initial_prop_components</span><span class="p">)</span>
<span class="c1"># define an ImportanceSampler object</span>
<span class="n">sampler</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">sampler</span><span class="o">.</span><span class="n">importance_sampling</span><span class="o">.</span><span class="n">ImportanceSampler</span><span class="p">(</span><span class="n">log_target</span><span class="p">,</span> <span class="n">initial_proposal</span><span class="p">)</span>
<span class="c1"># draw 10,000 samples adapting the proposal every 1,000 samples</span>
<span class="c1"># hereby save the generating proposal component for each sample which is</span>
<span class="c1"># returned by sampler.run</span>
<span class="c1"># Note: With too few samples components may die out, and one mode might be lost.</span>
<span class="n">generating_components</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="se">\r</span><span class="s2">step"</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="s2">"...</span><span class="se">\n\t</span><span class="s2">"</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s1">''</span><span class="p">)</span>
<span class="c1"># draw 1,000 samples and save the generating component</span>
<span class="n">generating_components</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sampler</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="n">trace_sort</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span>
<span class="c1"># get a reference to the weights and samples that have just been generated</span>
<span class="n">samples</span> <span class="o">=</span> <span class="n">sampler</span><span class="o">.</span><span class="n">samples</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">weights</span> <span class="o">=</span> <span class="n">sampler</span><span class="o">.</span><span class="n">weights</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][:,</span><span class="mi">0</span><span class="p">]</span>
<span class="c1"># update the proposal using the pmc algorithm in the non Rao-Blackwellized form</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">mix_adapt</span><span class="o">.</span><span class="n">pmc</span><span class="o">.</span><span class="n">gaussian_pmc</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span><span class="p">,</span> <span class="n">weights</span><span class="p">,</span> <span class="n">generating_components</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span>
<span class="n">mincount</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">rb</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">copy</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="se">\r</span><span class="s2">sampling finished"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span> <span class="s1">'-----------------'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="se">\n</span><span class="s1">'</span><span class="p">)</span>
<span class="c1"># print information about the adapted proposal</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'initial component weights:'</span><span class="p">,</span> <span class="n">initial_proposal</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'final component weights:'</span><span class="p">,</span> <span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'target component weights:'</span><span class="p">,</span> <span class="n">component_weights</span><span class="p">)</span>
<span class="nb">print</span><span class="p">()</span>
<span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">([</span><span class="n">mean0</span><span class="p">,</span> <span class="n">mean1</span><span class="p">,</span> <span class="kc">None</span><span class="p">]):</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'initial mean of component </span><span class="si">%i</span><span class="s1">:'</span> <span class="o">%</span><span class="n">k</span><span class="p">,</span> <span class="n">initial_proposal</span><span class="o">.</span><span class="n">components</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">mu</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'final mean of component </span><span class="si">%i</span><span class="s1">:'</span> <span class="o">%</span><span class="n">k</span><span class="p">,</span> <span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span><span class="o">.</span><span class="n">components</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">mu</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'target mean of component </span><span class="si">%i</span><span class="s1">:'</span> <span class="o">%</span><span class="n">k</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
<span class="nb">print</span><span class="p">()</span>
<span class="nb">print</span><span class="p">()</span>
<span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">([</span><span class="n">covariance0</span><span class="p">,</span> <span class="n">covariance1</span><span class="p">,</span> <span class="kc">None</span><span class="p">]):</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'initial covariance of component </span><span class="si">%i</span><span class="s1">:</span><span class="se">\n</span><span class="s1">'</span> <span class="o">%</span><span class="n">k</span><span class="p">,</span> <span class="n">initial_proposal</span><span class="o">.</span><span class="n">components</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">sigma</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s1">''</span><span class="p">)</span>
<span class="nb">print</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'final covariance of component </span><span class="si">%i</span><span class="s1">:</span><span class="se">\n</span><span class="s1">'</span> <span class="o">%</span><span class="n">k</span><span class="p">,</span> <span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span><span class="o">.</span><span class="n">components</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">sigma</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s1">''</span><span class="p">)</span>
<span class="nb">print</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'target covariance of component </span><span class="si">%i</span><span class="s1">:</span><span class="se">\n</span><span class="s1">'</span> <span class="o">%</span><span class="n">k</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s1">''</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="se">\n</span><span class="s1">'</span><span class="p">)</span>
<span class="c1"># plot results</span>
<span class="k">try</span><span class="p">:</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'For plotting "matplotlib" needs to be installed'</span><span class="p">)</span>
<span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">set_axlimits</span><span class="p">():</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mf">6.0</span><span class="p">,</span> <span class="o">+</span><span class="mf">6.000</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="o">+</span><span class="mf">1.401</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">221</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="s1">'target mixture'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">target_mixture</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'jet'</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">222</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="s1">'pmc fit'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'nipy_spectral'</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">223</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="s1">'target mixture and pmc fit'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">target_mixture</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'jet'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'nipy_spectral'</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">224</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="s1">'weighted samples'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span><span class="n">sampler</span><span class="o">.</span><span class="n">samples</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">sampler</span><span class="o">.</span><span class="n">samples</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">weights</span><span class="o">=</span><span class="n">sampler</span><span class="o">.</span><span class="n">weights</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'gray_r'</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">200</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p>(<a class="reference download internal" download="" href="_downloads/cd2788d8d48584761eb5872d6d059628/pmc.py"><code class="xref download docutils literal notranslate"><span class="pre">Source</span> <span class="pre">code</span></code></a>, <a class="reference download internal" download="" href="_downloads/6b617f89fa123d5c67a4c78da9753e60/pmc.png"><code class="xref download docutils literal notranslate"><span class="pre">png</span></code></a>, <a class="reference download internal" download="" href="_downloads/de6220cd981351b2766e1242da7253a7/pmc.hires.png"><code class="xref download docutils literal notranslate"><span class="pre">hires.png</span></code></a>, <a class="reference download internal" download="" href="_downloads/c22502c1068309cffd6e631cff5b10c4/pmc.pdf"><code class="xref download docutils literal notranslate"><span class="pre">pdf</span></code></a>)</p>
<figure class="align-default">
<img alt="_images/pmc.png" class="plot-directive" src="_images/pmc.png" />
</figure>
</section>
<section id="parallel">
<h3><span class="section-number">4.2.2. </span>Parallel<a class="headerlink" href="#parallel" title="Permalink to this heading"></a></h3>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="sd">'''This example shows how to use importance sampling and how to</span>
<span class="sd">adapt the proposal density using the pmc algorithm in an MPI</span>
<span class="sd">parallel environment.</span>
<span class="sd">In order to have a multiprocessing enviroment invoke this script with</span>
<span class="sd">"mpirun -n 10 python pmc_mpi.py".</span>
<span class="sd">'''</span>
<span class="kn">from</span> <span class="nn">mpi4py.MPI</span> <span class="kn">import</span> <span class="n">COMM_WORLD</span> <span class="k">as</span> <span class="n">comm</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pypmc</span>
<span class="kn">import</span> <span class="nn">pypmc.tools.parallel_sampler</span> <span class="c1"># this submodule is NOT imported by ``import pypmc``</span>
<span class="c1"># This script is a parallelized version of the PMC example ``pmc.py``.</span>
<span class="c1"># The following lines just define a target density and an initial proposal.</span>
<span class="c1"># These steps are exactly the same as in ``pmc.py``:</span>
<span class="c1"># define the target; i.e., the function you want to sample from.</span>
<span class="c1"># In this case, it is a bimodal Gaussian</span>
<span class="c1">#</span>
<span class="c1"># Note that the target function "log_target" returns the log of the</span>
<span class="c1"># target function.</span>
<span class="n">component_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">])</span>
<span class="n">mean0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span> <span class="p">([</span> <span class="mf">5.0</span> <span class="p">,</span> <span class="mf">0.01</span> <span class="p">])</span>
<span class="n">covariance0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">0.01</span> <span class="p">,</span> <span class="mf">0.003</span> <span class="p">],</span>
<span class="p">[</span> <span class="mf">0.003</span><span class="p">,</span> <span class="mf">0.0025</span><span class="p">]])</span>
<span class="n">inv_covariance0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">covariance0</span><span class="p">)</span>
<span class="n">mean1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span> <span class="p">([</span><span class="o">-</span><span class="mf">4.0</span> <span class="p">,</span> <span class="mf">1.0</span> <span class="p">])</span>
<span class="n">covariance1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">0.1</span> <span class="p">,</span> <span class="mf">0.</span> <span class="p">],</span>
<span class="p">[</span> <span class="mf">0.</span> <span class="p">,</span> <span class="mf">0.02</span> <span class="p">]])</span>
<span class="n">inv_covariance1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">covariance1</span><span class="p">)</span>
<span class="n">component_means</span> <span class="o">=</span> <span class="p">[</span><span class="n">mean0</span><span class="p">,</span> <span class="n">mean1</span><span class="p">]</span>
<span class="n">component_covariances</span> <span class="o">=</span> <span class="p">[</span><span class="n">covariance0</span><span class="p">,</span> <span class="n">covariance1</span><span class="p">]</span>
<span class="n">target_mixture</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">mixture</span><span class="o">.</span><span class="n">create_gaussian_mixture</span><span class="p">(</span><span class="n">component_means</span><span class="p">,</span> <span class="n">component_covariances</span><span class="p">,</span> <span class="n">component_weights</span><span class="p">)</span>
<span class="n">log_target</span> <span class="o">=</span> <span class="n">target_mixture</span><span class="o">.</span><span class="n">evaluate</span>
<span class="c1"># define the initial proposal density</span>
<span class="c1"># In this case it has three Gaussians:</span>
<span class="c1"># the initial covariances are set to the unit-matrix,</span>
<span class="c1"># the initial component weights are set equal</span>
<span class="n">initial_prop_means</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">initial_prop_means</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="mf">4.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span> <span class="p">)</span>
<span class="n">initial_prop_means</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">-</span><span class="mf">5.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span> <span class="p">)</span>
<span class="n">initial_prop_means</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span> <span class="p">)</span>
<span class="n">initial_prop_covariance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="n">initial_prop_components</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span>
<span class="n">initial_prop_components</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">gauss</span><span class="o">.</span><span class="n">Gauss</span><span class="p">(</span><span class="n">initial_prop_means</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">initial_prop_covariance</span><span class="p">))</span>
<span class="n">initial_proposal</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">mixture</span><span class="o">.</span><span class="n">MixtureDensity</span><span class="p">(</span><span class="n">initial_prop_components</span><span class="p">)</span>
<span class="c1"># -----------------------------------------------------------------------------------------------------------------------</span>
<span class="c1"># In ``pmc.py`` the following line defines the sequential single process sampler:</span>
<span class="c1"># sampler = pypmc.sampler.importance_sampling.ImportanceSampler(log_target, initial_proposal)</span>
<span class="c1">#</span>
<span class="c1"># We now use the parallel MPISampler instead:</span>
<span class="n">SequentialIS</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">sampler</span><span class="o">.</span><span class="n">importance_sampling</span><span class="o">.</span><span class="n">ImportanceSampler</span>
<span class="n">parallel_sampler</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">parallel_sampler</span><span class="o">.</span><span class="n">MPISampler</span><span class="p">(</span><span class="n">SequentialIS</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="n">log_target</span><span class="p">,</span> <span class="n">proposal</span><span class="o">=</span><span class="n">initial_proposal</span><span class="p">)</span>
<span class="c1"># Draw 10,000 samples adapting the proposal every 1,000 samples:</span>
<span class="c1"># make sure that every process has a different random number generator seed</span>
<span class="k">if</span> <span class="n">comm</span><span class="o">.</span><span class="n">Get_rank</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">seed</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mf">1e5</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">seed</span> <span class="o">=</span> <span class="kc">None</span>
<span class="n">seed</span> <span class="o">=</span> <span class="n">comm</span><span class="o">.</span><span class="n">bcast</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="n">seed</span> <span class="o">+</span> <span class="n">comm</span><span class="o">.</span><span class="n">Get_rank</span><span class="p">())</span>
<span class="n">generating_components</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
<span class="c1"># With the invocation "mpirun -n 10 python pmc_mpi.py", there are</span>
<span class="c1"># 10 processes which means in order to draw 1,000 samples</span>
<span class="c1"># ``parallel_sampler.run(1000//comm.Get_size())`` makes each process draw</span>
<span class="c1"># 100 samples.</span>
<span class="c1"># Hereby the generating proposal component for each sample in each process</span>
<span class="c1"># is returned by ``parallel_sampler.run``.</span>
<span class="c1"># In the master process, ``parallel_sampler.run`` is a list containing the</span>
<span class="c1"># return values of the sequential ``run`` method of every process.</span>
<span class="c1"># In all other processes, ``parallel_sampler.run`` returns the generating</span>
<span class="c1"># component for its own samples only.</span>
<span class="n">last_generating_components</span> <span class="o">=</span> <span class="n">parallel_sampler</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mi">1000</span><span class="o">//</span><span class="n">comm</span><span class="o">.</span><span class="n">Get_size</span><span class="p">(),</span> <span class="n">trace_sort</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1"># In addition to the generating components, the ``sampler.run``</span>
<span class="c1"># method automatically sends all samples to the master</span>
<span class="c1"># process i.e. the process which fulfills comm.Get_rank() == 0.</span>
<span class="k">if</span> <span class="n">comm</span><span class="o">.</span><span class="n">Get_rank</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="se">\r</span><span class="s2">step"</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="s2">"...</span><span class="se">\n\t</span><span class="s2">"</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s1">''</span><span class="p">)</span>
<span class="c1"># Now let PMC run only in the master process:</span>
<span class="c1"># ``sampler.samples_list`` and ``sampler.weights_list`` store the weighted samples</span>
<span class="c1"># sorted by the resposible process:</span>
<span class="c1"># The History objects that are held by process i can be accessed via</span>
<span class="c1"># ``sampler.<samples/weights>_list[i]``. The master process (i=0) also produces samples.</span>
<span class="c1"># Combine the weights and samples to two arrays of 1,000 samples</span>
<span class="n">samples</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">history_item</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">history_item</span> <span class="ow">in</span> <span class="n">parallel_sampler</span><span class="o">.</span><span class="n">samples_list</span><span class="p">])</span>
<span class="n">weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">history_item</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">history_item</span> <span class="ow">in</span> <span class="n">parallel_sampler</span><span class="o">.</span><span class="n">weights_list</span><span class="p">])[:,</span><span class="mi">0</span><span class="p">]</span>
<span class="c1"># The latent variables are stored in ``last_generating_components``.</span>
<span class="c1"># ``last_generating_components[i]`` returns an array with the generating</span>
<span class="c1"># components of the samples produced by process number "i".</span>
<span class="c1"># ``np.hstack(last_generating_components)`` combines the generating components</span>
<span class="c1"># from all processes to one array holding all 1,000 entries.</span>
<span class="n">generating_components</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">(</span><span class="n">last_generating_components</span><span class="p">)</span> <span class="p">)</span>
<span class="c1"># adapt the proposal using the samples from all processes</span>
<span class="n">new_proposal</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">mix_adapt</span><span class="o">.</span><span class="n">pmc</span><span class="o">.</span><span class="n">gaussian_pmc</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">parallel_sampler</span><span class="o">.</span><span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span><span class="p">,</span>
<span class="n">weights</span><span class="p">,</span> <span class="n">generating_components</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span>
<span class="n">mincount</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">rb</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="c1"># In order to broadcast the ``new_proposal``, define a dummy variable in the other processes</span>
<span class="c1"># see "MPI4Py tutorial", section "Collective Communication": http://mpi4py.scipy.org/docs/usrman/tutorial.html</span>
<span class="n">new_proposal</span> <span class="o">=</span> <span class="kc">None</span>
<span class="c1"># broadcast the ``new_proposal``</span>
<span class="n">new_proposal</span> <span class="o">=</span> <span class="n">comm</span><span class="o">.</span><span class="n">bcast</span><span class="p">(</span><span class="n">new_proposal</span><span class="p">)</span>
<span class="c1"># replace the old proposal</span>
<span class="n">parallel_sampler</span><span class="o">.</span><span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span> <span class="o">=</span> <span class="n">new_proposal</span>
<span class="c1"># only the master process shall print out any final information</span>
<span class="k">if</span> <span class="n">comm</span><span class="o">.</span><span class="n">Get_rank</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">all_samples</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">history_item</span><span class="p">[</span> <span class="p">:]</span> <span class="k">for</span> <span class="n">history_item</span> <span class="ow">in</span> <span class="n">parallel_sampler</span><span class="o">.</span><span class="n">samples_list</span><span class="p">])</span>
<span class="n">all_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">history_item</span><span class="p">[</span> <span class="p">:]</span> <span class="k">for</span> <span class="n">history_item</span> <span class="ow">in</span> <span class="n">parallel_sampler</span><span class="o">.</span><span class="n">weights_list</span><span class="p">])</span>
<span class="n">last_samples</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">history_item</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">history_item</span> <span class="ow">in</span> <span class="n">parallel_sampler</span><span class="o">.</span><span class="n">samples_list</span><span class="p">])</span>
<span class="n">last_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">history_item</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">history_item</span> <span class="ow">in</span> <span class="n">parallel_sampler</span><span class="o">.</span><span class="n">weights_list</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="se">\r</span><span class="s2">sampling finished"</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s1">', '</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"collected "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">all_samples</span><span class="p">))</span> <span class="o">+</span> <span class="s2">" samples"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span> <span class="s1">'------------------------------------------'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="se">\n</span><span class="s1">'</span><span class="p">)</span>
<span class="c1"># print information about the adapted proposal</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'initial component weights:'</span><span class="p">,</span> <span class="n">initial_proposal</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'final component weights:'</span><span class="p">,</span> <span class="n">parallel_sampler</span><span class="o">.</span><span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'target component weights:'</span><span class="p">,</span> <span class="n">component_weights</span><span class="p">)</span>
<span class="nb">print</span><span class="p">()</span>
<span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">([</span><span class="n">mean0</span><span class="p">,</span> <span class="n">mean1</span><span class="p">,</span> <span class="kc">None</span><span class="p">]):</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'initial mean of component </span><span class="si">%i</span><span class="s1">:'</span> <span class="o">%</span><span class="n">k</span><span class="p">,</span> <span class="n">initial_proposal</span><span class="o">.</span><span class="n">components</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">mu</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'final mean of component </span><span class="si">%i</span><span class="s1">:'</span> <span class="o">%</span><span class="n">k</span><span class="p">,</span> <span class="n">parallel_sampler</span><span class="o">.</span><span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span><span class="o">.</span><span class="n">components</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">mu</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'target mean of component </span><span class="si">%i</span><span class="s1">:'</span> <span class="o">%</span><span class="n">k</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
<span class="nb">print</span><span class="p">()</span>
<span class="nb">print</span><span class="p">()</span>
<span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">([</span><span class="n">covariance0</span><span class="p">,</span> <span class="n">covariance1</span><span class="p">,</span> <span class="kc">None</span><span class="p">]):</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'initial covariance of component </span><span class="si">%i</span><span class="s1">:</span><span class="se">\n</span><span class="s1">'</span> <span class="o">%</span><span class="n">k</span><span class="p">,</span> <span class="n">initial_proposal</span><span class="o">.</span><span class="n">components</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">sigma</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s1">''</span><span class="p">)</span>
<span class="nb">print</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'final covariance of component </span><span class="si">%i</span><span class="s1">:</span><span class="se">\n</span><span class="s1">'</span> <span class="o">%</span><span class="n">k</span><span class="p">,</span> <span class="n">parallel_sampler</span><span class="o">.</span><span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span><span class="o">.</span><span class="n">components</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">sigma</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s1">''</span><span class="p">)</span>
<span class="nb">print</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'target covariance of component </span><span class="si">%i</span><span class="s1">:</span><span class="se">\n</span><span class="s1">'</span> <span class="o">%</span><span class="n">k</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s1">''</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="se">\n</span><span class="s1">'</span><span class="p">)</span>
<span class="k">if</span> <span class="n">comm</span><span class="o">.</span><span class="n">Get_size</span><span class="p">()</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'******************************************************'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'********** NOTE: There is only one process. **********'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'******** try "mpirun -n 10 python pmc_mpi.py" ********'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'******************************************************'</span><span class="p">)</span>
<span class="c1"># plot results</span>
<span class="k">try</span><span class="p">:</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'For plotting "matplotlib" needs to be installed'</span><span class="p">)</span>
<span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">set_axlimits</span><span class="p">():</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mf">6.0</span><span class="p">,</span> <span class="o">+</span><span class="mf">6.000</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="o">+</span><span class="mf">1.401</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">221</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="s1">'target mixture'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">target_mixture</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'jet'</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">222</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="s1">'pmc fit'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">parallel_sampler</span><span class="o">.</span><span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'nipy_spectral'</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">223</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="s1">'target mixture and pmc fit'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">target_mixture</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'jet'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">parallel_sampler</span><span class="o">.</span><span class="n">sampler</span><span class="o">.</span><span class="n">proposal</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'nipy_spectral'</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">224</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="s1">'weighted samples'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span><span class="n">last_samples</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">last_samples</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">weights</span><span class="o">=</span><span class="n">last_weights</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'gray_r'</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">200</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p>(<a class="reference download internal" download="" href="_downloads/99da80a624f5a7b06ec0d644e52e5375/pmc_mpi.py"><code class="xref download docutils literal notranslate"><span class="pre">Source</span> <span class="pre">code</span></code></a>, <a class="reference download internal" download="" href="_downloads/22ab20da30b87770407b52bac2d18407/pmc_mpi.png"><code class="xref download docutils literal notranslate"><span class="pre">png</span></code></a>, <a class="reference download internal" download="" href="_downloads/0bc31b71c9a71b3690e24f121701b3e7/pmc_mpi.hires.png"><code class="xref download docutils literal notranslate"><span class="pre">hires.png</span></code></a>, <a class="reference download internal" download="" href="_downloads/67d2e871da8466232c20d593f5290532/pmc_mpi.pdf"><code class="xref download docutils literal notranslate"><span class="pre">pdf</span></code></a>)</p>
<figure class="align-default">
<img alt="_images/pmc_mpi.png" class="plot-directive" src="_images/pmc_mpi.png" />
</figure>
</section>
</section>
<section id="grouping-by-gelman-rubin-r-value">
<h2><span class="section-number">4.3. </span>Grouping by Gelman-Rubin R value<a class="headerlink" href="#grouping-by-gelman-rubin-r-value" title="Permalink to this heading"></a></h2>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="sd">'''This example illustrates how to group Markov Chains according to the</span>
<span class="sd">Gelman-Rubin R value (see [GR92]_).</span>
<span class="sd">'''</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pypmc</span>
<span class="c1"># A Markov Chain can only explore a local mode of the target function.</span>
<span class="c1"># The Gelman-Rubin R value can be used to determine whether N chains</span>
<span class="c1"># explored the same mode. Pypmc offers a function which groups chains</span>
<span class="c1"># with a common R value less than some ``critical_r``.</span>
<span class="c1">#</span>
<span class="c1"># In this example, we run five Markov Chains initialized in different</span>
<span class="c1"># modes and then group those chains together that explored same mode.</span>
<span class="c1"># define a proposal</span>
<span class="c1"># this defines the same initial proposal for all chains</span>
<span class="n">prop_dof</span> <span class="o">=</span> <span class="mf">50.</span>
<span class="n">prop_sigma</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.1</span> <span class="p">,</span> <span class="mf">0.</span> <span class="p">]</span>
<span class="p">,[</span><span class="mf">0.</span> <span class="p">,</span> <span class="mf">0.02</span><span class="p">]])</span>
<span class="n">prop</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">student_t</span><span class="o">.</span><span class="n">LocalStudentT</span><span class="p">(</span><span class="n">prop_sigma</span><span class="p">,</span> <span class="n">prop_dof</span><span class="p">)</span>
<span class="c1"># define the target; i.e., the function you want to sample from.</span>
<span class="c1"># In this case, it is a bimodal Gaussian with well separated modes.</span>
<span class="c1">#</span>
<span class="c1"># Note that the target function "log_target" returns the log of the</span>
<span class="c1"># target function.</span>
<span class="n">component_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">])</span>
<span class="n">mean0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span> <span class="p">([</span> <span class="mf">5.0</span> <span class="p">,</span> <span class="mf">0.01</span> <span class="p">])</span>
<span class="n">covariance0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">0.01</span> <span class="p">,</span> <span class="mf">0.003</span> <span class="p">],</span>
<span class="p">[</span> <span class="mf">0.003</span><span class="p">,</span> <span class="mf">0.0025</span><span class="p">]])</span>
<span class="n">inv_covariance0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">covariance0</span><span class="p">)</span>
<span class="n">mean1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span> <span class="p">([</span><span class="o">-</span><span class="mf">4.0</span> <span class="p">,</span> <span class="mf">1.0</span> <span class="p">])</span>
<span class="n">covariance1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">0.1</span> <span class="p">,</span> <span class="mf">0.</span> <span class="p">],</span>
<span class="p">[</span> <span class="mf">0.</span> <span class="p">,</span> <span class="mf">0.02</span> <span class="p">]])</span>
<span class="n">inv_covariance1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">covariance1</span><span class="p">)</span>
<span class="n">component_means</span> <span class="o">=</span> <span class="p">[</span><span class="n">mean0</span><span class="p">,</span> <span class="n">mean1</span><span class="p">]</span>
<span class="n">component_covariances</span> <span class="o">=</span> <span class="p">[</span><span class="n">covariance0</span><span class="p">,</span> <span class="n">covariance1</span><span class="p">]</span>
<span class="n">target_mixture</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">mixture</span><span class="o">.</span><span class="n">create_gaussian_mixture</span><span class="p">(</span><span class="n">component_means</span><span class="p">,</span> <span class="n">component_covariances</span><span class="p">,</span> <span class="n">component_weights</span><span class="p">)</span>
<span class="n">log_target</span> <span class="o">=</span> <span class="n">target_mixture</span><span class="o">.</span><span class="n">evaluate</span>
<span class="c1"># choose initializations for the chains</span>
<span class="c1"># Here we place two chains into the mode at [5, 0.01] and three into the mode at [-4,1].</span>
<span class="c1"># In such a setup, the chains will only explore the mode where they are initialized.</span>
<span class="c1"># Different random numbers are used in each chain.</span>
<span class="n">starts</span> <span class="o">=</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">4.999</span><span class="p">,</span> <span class="mf">0.</span><span class="p">])]</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">-</span><span class="mf">4.0001</span><span class="p">,</span> <span class="mf">0.999</span><span class="p">])]</span> <span class="o">*</span> <span class="mi">3</span>
<span class="c1"># define the markov chain objects</span>
<span class="n">mcs</span> <span class="o">=</span> <span class="p">[</span><span class="n">pypmc</span><span class="o">.</span><span class="n">sampler</span><span class="o">.</span><span class="n">markov_chain</span><span class="o">.</span><span class="n">AdaptiveMarkovChain</span><span class="p">(</span><span class="n">log_target</span><span class="p">,</span> <span class="n">prop</span><span class="p">,</span> <span class="n">start</span><span class="p">)</span> <span class="k">for</span> <span class="n">start</span> <span class="ow">in</span> <span class="n">starts</span><span class="p">]</span>
<span class="c1"># run and discard burn-in</span>
<span class="k">for</span> <span class="n">mc</span> <span class="ow">in</span> <span class="n">mcs</span><span class="p">:</span>
<span class="n">mc</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="n">mc</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
<span class="c1"># run 10,000 steps adapting the proposal every 500 steps</span>
<span class="k">for</span> <span class="n">mc</span> <span class="ow">in</span> <span class="n">mcs</span><span class="p">:</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">20</span><span class="p">):</span>
<span class="n">mc</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mi">500</span><span class="p">)</span>
<span class="n">mc</span><span class="o">.</span><span class="n">adapt</span><span class="p">()</span>
<span class="c1"># extract a reference to the samples from all chains</span>
<span class="n">stacked_values</span> <span class="o">=</span> <span class="p">[</span><span class="n">mc</span><span class="o">.</span><span class="n">samples</span><span class="p">[:]</span> <span class="k">for</span> <span class="n">mc</span> <span class="ow">in</span> <span class="n">mcs</span><span class="p">]</span>
<span class="c1"># find the chain groups</span>
<span class="c1"># chains 0 and 1 are initialized in the same mode (at [5, 0.01])</span>
<span class="c1"># chains 2, 3 and 4 are initialized in the same mode (at [-4, 0])</span>
<span class="c1"># expect chain groups:</span>
<span class="n">expected_groups</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]]</span>
<span class="c1"># R value calculation only needs the means, variances (diagonal</span>
<span class="c1"># elements of covariance matrix) and number of samples,</span>
<span class="c1"># axis=0 ensures that we get variances separately for each parameter.</span>
<span class="c1"># critical_r can be set manually, here the default value is used</span>
<span class="n">found_groups</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">mix_adapt</span><span class="o">.</span><span class="n">r_value</span><span class="o">.</span><span class="n">r_group</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">chain</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">chain</span> <span class="ow">in</span> <span class="n">stacked_values</span><span class="p">],</span>
<span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">var</span> <span class="p">(</span><span class="n">chain</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">chain</span> <span class="ow">in</span> <span class="n">stacked_values</span><span class="p">],</span>
<span class="nb">len</span><span class="p">(</span><span class="n">stacked_values</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
<span class="c1"># print the result</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Expect </span><span class="si">%s</span><span class="s2">"</span> <span class="o">%</span> <span class="n">expected_groups</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Have </span><span class="si">%s</span><span class="s2">"</span> <span class="o">%</span> <span class="n">found_groups</span><span class="p">)</span>
<span class="c1"># Hint: ``stacked_values`` is an example of what `pypmc.mix_adapt.r_value.make_r_gaussmix()` expects as ``data``</span>
<span class="n">result</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">mix_adapt</span><span class="o">.</span><span class="n">r_value</span><span class="o">.</span><span class="n">make_r_gaussmix</span><span class="p">(</span><span class="n">stacked_values</span><span class="p">)</span>
<span class="k">try</span><span class="p">:</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'For plotting "matplotlib" needs to be installed'</span><span class="p">)</span>
<span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'jet'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p>(<a class="reference download internal" download="" href="_downloads/a2839dca3b9ecf9dd48b2dd2dd18cc48/r_group.py"><code class="xref download docutils literal notranslate"><span class="pre">Source</span> <span class="pre">code</span></code></a>, <a class="reference download internal" download="" href="_downloads/19d2810114c5b354ac92a455a985ac24/r_group.png"><code class="xref download docutils literal notranslate"><span class="pre">png</span></code></a>, <a class="reference download internal" download="" href="_downloads/2bf633f86dee2ab678bbf0ea9ded3a25/r_group.hires.png"><code class="xref download docutils literal notranslate"><span class="pre">hires.png</span></code></a>, <a class="reference download internal" download="" href="_downloads/f15ec6eadf505b7a4d1760e3f0668837/r_group.pdf"><code class="xref download docutils literal notranslate"><span class="pre">pdf</span></code></a>)</p>
<figure class="align-default">
<img alt="_images/r_group.png" class="plot-directive" src="_images/r_group.png" />
</figure>
</section>
<section id="variational-bayes">
<h2><span class="section-number">4.4. </span>Variational Bayes<a class="headerlink" href="#variational-bayes" title="Permalink to this heading"></a></h2>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="sd">'''This example shows how to generate a "best fit" Gaussian mixture density</span>
<span class="sd">from data using variational Bayes.</span>
<span class="sd">'''</span>
<span class="c1">## in this example, we will:</span>
<span class="c1">## 1. Define a Gaussian mixture</span>
<span class="c1">## 2. Generate demo data from that Gaussian mixture</span>
<span class="c1">## 3. Generate a Gaussian mixture out of the data</span>
<span class="c1">## 4. Plot the original and the generated mixture</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pypmc</span>
<span class="c1"># -------------------- 1. Define a Gaussian mixture --------------------</span>
<span class="n">component_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">])</span>
<span class="n">mean0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span> <span class="p">([</span> <span class="mf">5.0</span> <span class="p">,</span> <span class="mf">0.01</span> <span class="p">])</span>
<span class="n">covariance0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">0.01</span> <span class="p">,</span> <span class="mf">0.003</span> <span class="p">],</span>
<span class="p">[</span> <span class="mf">0.003</span><span class="p">,</span> <span class="mf">0.0025</span><span class="p">]])</span>
<span class="n">mean1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span> <span class="p">([</span><span class="o">-</span><span class="mf">4.0</span> <span class="p">,</span> <span class="mf">1.0</span> <span class="p">])</span>
<span class="n">covariance1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">0.1</span> <span class="p">,</span> <span class="mf">0.</span> <span class="p">],</span>
<span class="p">[</span> <span class="mf">0.</span> <span class="p">,</span> <span class="mf">0.02</span> <span class="p">]])</span>
<span class="n">component_means</span> <span class="o">=</span> <span class="p">[</span><span class="n">mean0</span><span class="p">,</span> <span class="n">mean1</span><span class="p">]</span>
<span class="n">component_covariances</span> <span class="o">=</span> <span class="p">[</span><span class="n">covariance0</span><span class="p">,</span> <span class="n">covariance1</span><span class="p">]</span>
<span class="n">target_mix</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">mixture</span><span class="o">.</span><span class="n">create_gaussian_mixture</span><span class="p">(</span><span class="n">component_means</span><span class="p">,</span> <span class="n">component_covariances</span><span class="p">,</span> <span class="n">component_weights</span><span class="p">)</span>
<span class="c1"># -------------------- 2. Generate demo data ---------------------------</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">target_mix</span><span class="o">.</span><span class="n">propose</span><span class="p">(</span><span class="mi">500</span><span class="p">)</span>
<span class="c1"># -------------------- 3. Adapt a Gaussian mixture ---------------------</span>
<span class="c1"># maximum number of components</span>
<span class="n">K</span> <span class="o">=</span> <span class="mi">20</span>
<span class="c1"># Create a "GaussianInference" object.</span>
<span class="c1"># The following command passes just the two essential arguments to "GaussianInference":</span>
<span class="c1"># The ``data`` and a maximum number of ``components``.</span>
<span class="c1"># For reasonable results in more complicated settings, a careful choice for ``W0``</span>
<span class="c1"># is crucial. As a rule of thumb, choose ``inv(W0)`` much smaller than the expected</span>
<span class="c1"># covariance. In this case, however, the default (``W0`` = unit matrix) is good enough.</span>
<span class="n">vb</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">mix_adapt</span><span class="o">.</span><span class="n">variational</span><span class="o">.</span><span class="n">GaussianInference</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
<span class="c1"># adapt the variational parameters</span>
<span class="n">converged</span> <span class="o">=</span> <span class="n">vb</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'-----------------------------'</span><span class="p">)</span>
<span class="c1"># generate a Gaussian mixture with the most probable parameters</span>
<span class="n">fit_mixture</span> <span class="o">=</span> <span class="n">vb</span><span class="o">.</span><span class="n">make_mixture</span><span class="p">()</span>
<span class="c1"># -------------------- 4. Plot/print results ---------------------------</span>
<span class="k">if</span> <span class="n">converged</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="se">\n</span><span class="s1">The adaptation did not converge.</span><span class="se">\n</span><span class="s1">'</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="se">\n</span><span class="s1">Converged after </span><span class="si">%i</span><span class="s1"> iterations.</span><span class="se">\n</span><span class="s1">'</span> <span class="o">%</span><span class="n">converged</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"final component weights: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">fit_mixture</span><span class="o">.</span><span class="n">weights</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"target component weights: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span> <span class="n">target_mix</span><span class="o">.</span><span class="n">weights</span><span class="p">))</span>
<span class="k">try</span><span class="p">:</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'For plotting "matplotlib" needs to be installed'</span><span class="p">)</span>
<span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">set_axlimits</span><span class="p">():</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mf">6.0</span><span class="p">,</span> <span class="o">+</span><span class="mf">6.000</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="o">+</span><span class="mf">1.401</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">221</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="s1">'target mixture'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">target_mix</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'winter'</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">222</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="s1">'"best fit"'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">fit_mixture</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'nipy_spectral'</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">223</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="s1">'target mixture and "best fit"'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">target_mix</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'winter'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">fit_mixture</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'nipy_spectral'</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">224</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="s1">'data'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">hexbin</span><span class="p">(</span><span class="n">data</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">data</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'gray_r'</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p>(<a class="reference download internal" download="" href="_downloads/1482810eeba281459c9a382b21b20428/variational.py"><code class="xref download docutils literal notranslate"><span class="pre">Source</span> <span class="pre">code</span></code></a>, <a class="reference download internal" download="" href="_downloads/ce0689ed982932dfc4b40ab8f2724883/variational.png"><code class="xref download docutils literal notranslate"><span class="pre">png</span></code></a>, <a class="reference download internal" download="" href="_downloads/d23739137788ce36d409f22a24edbe7c/variational.hires.png"><code class="xref download docutils literal notranslate"><span class="pre">hires.png</span></code></a>, <a class="reference download internal" download="" href="_downloads/d3935fa45c55be1940f33f670d9b2573/variational.pdf"><code class="xref download docutils literal notranslate"><span class="pre">pdf</span></code></a>)</p>
<figure class="align-default">
<img alt="_images/variational.png" class="plot-directive" src="_images/variational.png" />
</figure>
</section>
<section id="mixture-reduction">
<span id="ex-mix-red"></span><h2><span class="section-number">4.5. </span>Mixture reduction<a class="headerlink" href="#mixture-reduction" title="Permalink to this heading"></a></h2>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="sd">'''Demonstrate the usage of hierarchical clustering and variational</span>
<span class="sd">Bayes (VBMerge) to reduce a given Gaussian mixture to a Gaussian</span>
<span class="sd">mixture with a reduced number of components.</span>
<span class="sd">'''</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="kn">import</span> <span class="n">chi2</span>
<span class="kn">import</span> <span class="nn">pypmc</span>
<span class="c1"># dimension</span>
<span class="n">D</span> <span class="o">=</span> <span class="mi">2</span>
<span class="c1"># number of components</span>
<span class="n">K</span> <span class="o">=</span> <span class="mi">400</span>
<span class="c1"># Wishart parameters: mean W, degree of freedom nu</span>
<span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
<span class="n">nu</span> <span class="o">=</span> <span class="mi">5</span>
<span class="c1"># "draw" covariance matrices from Wishart distribution</span>
<span class="k">def</span> <span class="nf">wishart</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">W</span><span class="p">):</span>
<span class="n">dim</span> <span class="o">=</span> <span class="n">W</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="n">chol</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cholesky</span><span class="p">(</span><span class="n">W</span><span class="p">)</span>
<span class="n">tmp</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">dim</span><span class="p">,</span><span class="n">dim</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">dim</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
<span class="n">tmp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">chi2</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">nu</span><span class="o">-</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">tmp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">chol</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">tmp</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">tmp</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">chol</span><span class="o">.</span><span class="n">T</span><span class="p">)))</span>
<span class="n">covariances</span> <span class="o">=</span> <span class="p">[</span><span class="n">wishart</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">W</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">K</span><span class="p">)]</span>
<span class="c1"># put components at positions drawn from a Gaussian around mu</span>
<span class="n">mu</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">D</span><span class="p">)</span>
<span class="n">means</span> <span class="o">=</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">(</span><span class="n">mu</span><span class="p">,</span> <span class="n">sigma</span><span class="p">)</span> <span class="k">for</span> <span class="n">sigma</span> <span class="ow">in</span> <span class="n">covariances</span><span class="p">]</span>
<span class="c1"># equal weights for every component</span>
<span class="n">weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">K</span><span class="p">)</span>
<span class="c1"># weights are automatically normalized</span>
<span class="n">input_mixture</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">mixture</span><span class="o">.</span><span class="n">create_gaussian_mixture</span><span class="p">(</span><span class="n">means</span><span class="p">,</span> <span class="n">covariances</span><span class="p">,</span> <span class="n">weights</span><span class="p">)</span>
<span class="c1"># create initial guess from first K_out components</span>
<span class="n">K_out</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">initial_guess</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">mixture</span><span class="o">.</span><span class="n">create_gaussian_mixture</span><span class="p">(</span><span class="n">means</span><span class="p">[:</span><span class="n">K_out</span><span class="p">],</span> <span class="n">covariances</span><span class="p">[:</span><span class="n">K_out</span><span class="p">],</span> <span class="n">weights</span><span class="p">[:</span><span class="n">K_out</span><span class="p">])</span>
<span class="c1">###</span>
<span class="c1"># hierarchical clustering</span>
<span class="c1">#</span>
<span class="c1"># - the output closely resembles the initial guess</span>
<span class="c1"># - components laid out spherically symmetric</span>
<span class="c1"># - every component is preserved</span>
<span class="c1">###</span>
<span class="n">h</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">mix_adapt</span><span class="o">.</span><span class="n">hierarchical</span><span class="o">.</span><span class="n">Hierarchical</span><span class="p">(</span><span class="n">input_mixture</span><span class="p">,</span> <span class="n">initial_guess</span><span class="p">)</span>
<span class="n">h</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1">###</span>
<span class="c1"># VBMerge</span>
<span class="c1">#</span>
<span class="c1"># - N is the number of samples that gave rise to the input mixture. It</span>
<span class="c1"># is arbitrary, so play around with it. You might have to adjust the</span>
<span class="c1"># ``prune`` parameter in the ``run()`` method</span>
<span class="c1"># - only one component survives, again it is spherically symmetric</span>
<span class="c1">###</span>
<span class="n">vb</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">mix_adapt</span><span class="o">.</span><span class="n">variational</span><span class="o">.</span><span class="n">VBMerge</span><span class="p">(</span><span class="n">input_mixture</span><span class="p">,</span> <span class="n">N</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span>
<span class="n">initial_guess</span><span class="o">=</span><span class="n">initial_guess</span><span class="p">)</span>
<span class="nb">print</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Start variational Bayes:"</span><span class="p">)</span>
<span class="n">vb</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1"># plot results</span>
<span class="k">try</span><span class="p">:</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'For plotting "matplotlib" needs to be installed'</span><span class="p">)</span>
<span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">set_axlimits</span><span class="p">():</span>
<span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="s1">'equal'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mf">12.0</span><span class="p">,</span> <span class="o">+</span><span class="mf">12.0</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mf">12.0</span><span class="p">,</span> <span class="o">+</span><span class="mf">12.0</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">221</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="s1">'input mixture'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">input_mixture</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">222</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="s1">'initial guess'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">initial_guess</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">223</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="s1">'variational Bayes'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">vb</span><span class="o">.</span><span class="n">make_mixture</span><span class="p">(),</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'autumn'</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">224</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="s1">'hierarchical output'</span><span class="p">)</span>
<span class="n">pypmc</span><span class="o">.</span><span class="n">tools</span><span class="o">.</span><span class="n">plot_mixture</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">g</span><span class="p">)</span>
<span class="n">set_axlimits</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p>(<a class="reference download internal" download="" href="_downloads/12310153aa4d9cbac087860ccb9a1acb/mixture_reduction.py"><code class="xref download docutils literal notranslate"><span class="pre">Source</span> <span class="pre">code</span></code></a>, <a class="reference download internal" download="" href="_downloads/7de791b7b93fd7c82652c26e0734ab5a/mixture_reduction.png"><code class="xref download docutils literal notranslate"><span class="pre">png</span></code></a>, <a class="reference download internal" download="" href="_downloads/c0e0c73281e182f970899012a7d4fa19/mixture_reduction.hires.png"><code class="xref download docutils literal notranslate"><span class="pre">hires.png</span></code></a>, <a class="reference download internal" download="" href="_downloads/4e825e52376c1352e0e49785fb4ecdd8/mixture_reduction.pdf"><code class="xref download docutils literal notranslate"><span class="pre">pdf</span></code></a>)</p>
<figure class="align-default">
<img alt="_images/mixture_reduction.png" class="plot-directive" src="_images/mixture_reduction.png" />
</figure>
</section>
<section id="mcmc-variational-bayes">
<span id="ex-mcmc-vb"></span><h2><span class="section-number">4.6. </span>MCMC + variational Bayes<a class="headerlink" href="#mcmc-variational-bayes" title="Permalink to this heading"></a></h2>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="sd">'''This example illustrates how pypmc can be used to integrate a</span>
<span class="sd">non-negative function. The presented algorithm needs very little</span>
<span class="sd">analytical knowledge about the function.</span>
<span class="sd">'''</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pypmc</span>
<span class="c1"># The idea is to find a good proposal function for importance sampling</span>
<span class="c1"># with as little information about the target function as possible.</span>
<span class="c1">#</span>
<span class="c1"># In this example we will first map out regions of interest using Markov</span>
<span class="c1"># chains, then we use the variational Bayes to approximate the target</span>
<span class="c1"># with a Gaussian mixture.</span>
<span class="c1"># *************************** Important: ***************************</span>
<span class="c1"># * The target function must be defined such that it returns the *</span>
<span class="c1"># * log of the function of interest. The methods we use imply that *</span>
<span class="c1"># * the function is interpreted as an unnormalized probability *</span>
<span class="c1"># * density. *</span>
<span class="c1"># ******************************************************************</span>
<span class="c1"># Define the target; i.e., the function you want to sample from. In</span>
<span class="c1"># this case, it is a Student's t mixture of three components with</span>
<span class="c1"># different degrees of freedom. They are located close to each other.</span>
<span class="c1"># If you want a multimodal target, adjust the means.</span>
<span class="n">dim</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">mean0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span> <span class="p">([</span><span class="o">-</span><span class="mf">6.0</span><span class="p">,</span> <span class="mf">7.3</span> <span class="p">])</span>
<span class="n">covariance0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">0.8</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.3</span> <span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">1.25</span><span class="p">]])</span>
<span class="n">mean1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span> <span class="p">([</span><span class="o">-</span><span class="mf">7.0</span><span class="p">,</span> <span class="mf">8.0</span> <span class="p">])</span>
<span class="n">covariance1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.</span> <span class="p">],</span>
<span class="p">[</span> <span class="mf">0.</span> <span class="p">,</span> <span class="mf">0.2</span> <span class="p">]])</span>
<span class="n">mean2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span> <span class="p">([</span><span class="o">-</span><span class="mf">8.5</span><span class="p">,</span> <span class="mf">7.5</span> <span class="p">])</span>
<span class="n">covariance2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.2</span> <span class="p">],</span>
<span class="p">[</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.2</span> <span class="p">]])</span>
<span class="n">component_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">])</span>
<span class="n">component_means</span> <span class="o">=</span> <span class="p">[</span><span class="n">mean0</span><span class="p">,</span> <span class="n">mean1</span><span class="p">,</span> <span class="n">mean2</span><span class="p">]</span>
<span class="n">component_covariances</span> <span class="o">=</span> <span class="p">[</span><span class="n">covariance0</span><span class="p">,</span> <span class="n">covariance1</span><span class="p">,</span> <span class="n">covariance2</span><span class="p">]</span>
<span class="n">dofs</span> <span class="o">=</span> <span class="p">[</span><span class="mi">13</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>
<span class="n">target_mixture</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">mixture</span><span class="o">.</span><span class="n">create_t_mixture</span><span class="p">(</span><span class="n">component_means</span><span class="p">,</span> <span class="n">component_covariances</span><span class="p">,</span> <span class="n">dofs</span><span class="p">,</span> <span class="n">component_weights</span><span class="p">)</span>
<span class="n">log_target</span> <span class="o">=</span> <span class="n">target_mixture</span><span class="o">.</span><span class="n">evaluate</span>
<span class="c1"># Now we suppose that we only have the following knowledge about the</span>
<span class="c1"># target function: its regions of interest are at a distance of no more</span>
<span class="c1"># than order ten from zero.</span>
<span class="c1"># Now we try to find these with Markov chains. We have to deal with</span>
<span class="c1"># the fact that there may be modes separated by regions of very low</span>
<span class="c1"># probability. It is thus unlikely that a single chain explores more</span>
<span class="c1"># than one mode in such a case. To deal with this multimodality, we</span>
<span class="c1"># start several chains and hope that they find all modes. We will</span>
<span class="c1"># start ten Markov chains at random positions in the square</span>
<span class="c1"># [(-10,-10), (+10,+10)].</span>
<span class="n">starts</span> <span class="o">=</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">dim</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)]</span>
<span class="c1"># For a local-random-walk Markov chain, we also need an initial</span>
<span class="c1"># proposal. Here, we take a gaussian with initial covariance</span>
<span class="c1"># diag(1e-3). The initial covariance should be chosen such that it is</span>
<span class="c1"># of the same order as the real covariance of the mode to be mapped</span>
<span class="c1"># out. For a Gaussian target, the overall scale should</span>
<span class="c1"># decrease as 2.38^2/d as the dimension d increases to achieve an</span>
<span class="c1"># acceptance rate around 20%.</span>
<span class="n">mc_prop</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">density</span><span class="o">.</span><span class="n">gauss</span><span class="o">.</span><span class="n">LocalGauss</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">dim</span><span class="p">)</span> <span class="o">*</span> <span class="mf">2.38</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="n">dim</span><span class="p">)</span>
<span class="n">mcs</span> <span class="o">=</span> <span class="p">[</span><span class="n">pypmc</span><span class="o">.</span><span class="n">sampler</span><span class="o">.</span><span class="n">markov_chain</span><span class="o">.</span><span class="n">AdaptiveMarkovChain</span><span class="p">(</span><span class="n">log_target</span><span class="p">,</span> <span class="n">mc_prop</span><span class="p">,</span> <span class="n">start</span><span class="p">)</span> <span class="k">for</span> <span class="n">start</span> <span class="ow">in</span> <span class="n">starts</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'running Markov chains ...'</span><span class="p">)</span>
<span class="c1"># In general we need to let the chain move to regions of high</span>
<span class="c1"># probability, these samples are not representative, so we discard them</span>
<span class="c1"># as burn-in. Then we let the Markov chains map out the regions of</span>
<span class="c1"># interest. The samples are used to adapt the proposal covariance to</span>
<span class="c1"># yield a satisfactory acceptance rate.</span>
<span class="k">for</span> <span class="n">mc</span> <span class="ow">in</span> <span class="n">mcs</span><span class="p">:</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">20</span><span class="p">):</span>
<span class="n">mc</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mi">500</span><span class="p">)</span>
<span class="n">mc</span><span class="o">.</span><span class="n">adapt</span><span class="p">()</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">mc</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
<span class="n">mc_samples_sorted_by_chain</span> <span class="o">=</span> <span class="p">[</span><span class="n">mc</span><span class="o">.</span><span class="n">samples</span><span class="p">[:]</span> <span class="k">for</span> <span class="n">mc</span> <span class="ow">in</span> <span class="n">mcs</span><span class="p">]</span>
<span class="n">mc_samples</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">(</span><span class="n">mc_samples_sorted_by_chain</span><span class="p">)</span>
<span class="n">means</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="nb">len</span><span class="p">(</span><span class="n">mcs</span><span class="p">),</span> <span class="n">dim</span><span class="p">))</span>
<span class="n">variances</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">means</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">mc</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">mc_samples_sorted_by_chain</span><span class="p">):</span>
<span class="n">means</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">mc</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">variances</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">mc</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># Now we use the Markov chain samples to generate a mixture proposal</span>
<span class="c1"># function for importance sampling. For this purpose, we choose the</span>
<span class="c1"># variational Bayes algorithm that takes samples and an initial guess</span>
<span class="c1"># of the mixture as input. To create the initial guess, we group all</span>
<span class="c1"># chains that mixed, and create 10 components per group. For a</span>
<span class="c1"># unimodal target, all chains should mix. For more information about</span>
<span class="c1"># the following call, check the example "Grouping by Gelman-Rubin R</span>
<span class="c1"># value"(r-group.py) or the reference documentation.</span>
<span class="n">long_patches</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">mix_adapt</span><span class="o">.</span><span class="n">r_value</span><span class="o">.</span><span class="n">make_r_gaussmix</span><span class="p">(</span><span class="n">mc_samples_sorted_by_chain</span><span class="p">,</span> <span class="n">K_g</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
<span class="c1"># Comments on arguments:</span>
<span class="c1"># o mc_samples[::100] - Samples in the Markov chains are strongly correlated</span>
<span class="c1"># => thin the samples to get approx. independent samples</span>
<span class="c1"># o W0=np.eye(dim)*1e10 - The resulting covariance matrices can be very</span>
<span class="c1"># sensitive to W0. Its inverse should be chosen much</span>
<span class="c1"># smaller than the actual covariance. If it is too small,</span>
<span class="c1"># W0 will dominate the resulting covariances and</span>
<span class="c1"># usually lead to very bad results.</span>
<span class="n">vb</span> <span class="o">=</span> <span class="n">pypmc</span><span class="o">.</span><span class="n">mix_adapt</span><span class="o">.</span><span class="n">variational</span><span class="o">.</span><span class="n">GaussianInference</span><span class="p">(</span><span class="n">mc_samples</span><span class="p">[::</span><span class="mi">100</span><span class="p">],</span> <span class="n">initial_guess</span><span class="o">=</span><span class="n">long_patches</span><span class="p">,</span> <span class="n">W0</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">dim</span><span class="p">)</span><span class="o">*</span><span class="mf">1e10</span><span class="p">)</span>
<span class="c1"># When we run variational Bayes, we want unneccessary components to be</span>
<span class="c1"># automatically pruned. The prune parameter sets how many samples a</span>
<span class="c1"># component must effectively have to be considered important. The rule</span>
<span class="c1"># of thumb employed here proved good in our experiments.</span>
<span class="n">vb_prune</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">vb</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="o">/</span> <span class="n">vb</span><span class="o">.</span><span class="n">K</span>
<span class="c1"># Run the variational Bayes for at most 1,000 iterations. But if the</span>
<span class="c1"># lower bound of the model evidence changes by less than `rel_tol`,</span>
<span class="c1"># convergence is declared before. If we increase `rel_tol` to 1e-4, it</span>
<span class="c1"># takes less iterations but potentially more (useless) components</span>
<span class="c1"># survive the pruning. The trade-off depends on the complexity of the</span>
<span class="c1"># problem.</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'running variational Bayes ...'</span><span class="p">)</span>
<span class="n">vb</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="n">rel_tol</span><span class="o">=</span><span class="mf">1e-8</span><span class="p">,</span> <span class="n">abs_tol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span> <span class="n">prune</span><span class="o">=</span><span class="n">vb_prune</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1"># extract the most probable Gaussian mixture given the samples</span>
<span class="n">vbmix</span> <span class="o">=</span> <span class="n">vb</span><span class="o">.</span><span class="n">make_mixture</span><span class="p">()</span>