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<section id="bayesian-inference">
<h1>Bayesian inference<a class="headerlink" href="#bayesian-inference" title="Link to this heading">¶</a></h1>
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<p>With Bayesian inference we ask the question ‘how does our understanding of the inputs change given some observation of the outputs of the model?’, i.e. we perform an updating step of the prior distributions to posterior, based on some observations.
Bayesvalidrox provides a dedicated class to perform this task, <a class="reference internal" href="_autosummary/bayesvalidrox.bayes_inference.bayes_inference.BayesInference.html#bayesvalidrox.bayes_inference.bayes_inference.BayesInference" title="bayesvalidrox.bayes_inference.bayes_inference.BayesInference"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.bayes_inference.bayes_inference.BayesInference</span></code></a>, which infers the posterior via <code class="docutils literal notranslate"><span class="pre">rejection-sampling</span></code> or <code class="docutils literal notranslate"><span class="pre">MCMC</span></code>.
The likelihood in rejection sampling is estimated with the help of <code class="docutils literal notranslate"><span class="pre">bootstrapping</span></code>.
MCMC-specific parameters are to be given as a dictionary called <code class="docutils literal notranslate"><span class="pre">mcmc_params</span></code> and can include</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">init_samples</span></code>: initial samples</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">n_steps</span></code>: number of steps</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">n_walkers</span></code>: number of walkers</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">n_burn</span></code>: length of the burn-in</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">moves</span></code>: function to use for the moves, e.g. taken from <code class="docutils literal notranslate"><span class="pre">emcee</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">multiprocessing</span></code>: setting for multiprocessing</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">verbose</span></code>: verbosity</p></li>
</ul>
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<a class="reference internal image-reference" href="_images/bayesian_validation.png"><img alt="UML diagram for classes related to Bayesian inference." src="_images/bayesian_validation.png" style="width: 300px;" /></a>
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</div>
<p>The observation should be set as <code class="docutils literal notranslate"><span class="pre">Model.observations</span></code> in the <code class="docutils literal notranslate"><span class="pre">Engine</span></code>, and an estimation of its uncertainty can be provided as a <a class="reference internal" href="_autosummary/bayesvalidrox.bayes_inference.discrepancy.Discrepancy.html#bayesvalidrox.bayes_inference.discrepancy.Discrepancy" title="bayesvalidrox.bayes_inference.discrepancy.Discrepancy"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.bayes_inference.discrepancy.Discrepancy</span></code></a> object.</p>
<section id="example">
<h2>Example<a class="headerlink" href="#example" title="Link to this heading">¶</a></h2>
<p>For this example we need to add the following imports.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">bayesvalidrox</span> <span class="kn">import</span> <span class="n">Discrepancy</span><span class="p">,</span> <span class="n">BayesInference</span>
</pre></div>
</div>
<p>In order to run Bayesian inference we first need to provide an observation.
For this example we take an evaluation of the model on some chosen sample and add the resulting values as <code class="docutils literal notranslate"><span class="pre">Model.observations</span></code>.
As this expects a 1D-array for each output key, we need to change the format slightly.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">true_sample</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">2</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">observation</span> <span class="o">=</span> <span class="n">Model</span><span class="o">.</span><span class="n">run_model_parallel</span><span class="p">(</span><span class="n">true_sample</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">Model</span><span class="o">.</span><span class="n">observations</span> <span class="o">=</span> <span class="p">{}</span>
<span class="gp">>>> </span><span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">observation</span><span class="p">:</span>
<span class="gp">>>> </span> <span class="k">if</span> <span class="n">key</span> <span class="o">==</span> <span class="s1">'x_values'</span><span class="p">:</span>
<span class="gp">>>> </span> <span class="n">Model</span><span class="o">.</span><span class="n">observations</span><span class="p">[</span><span class="n">key</span><span class="p">]</span><span class="o">=</span><span class="n">observation</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
<span class="gp">>>> </span> <span class="k">else</span><span class="p">:</span>
<span class="gp">>>> </span> <span class="n">Model</span><span class="o">.</span><span class="n">observations</span><span class="p">[</span><span class="n">key</span><span class="p">]</span><span class="o">=</span><span class="n">observation</span><span class="p">[</span><span class="n">key</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
</pre></div>
</div>
<p>Next we define the uncertainty on the observation with the class <a class="reference internal" href="_autosummary/bayesvalidrox.bayes_inference.discrepancy.Discrepancy.html#bayesvalidrox.bayes_inference.discrepancy.Discrepancy" title="bayesvalidrox.bayes_inference.discrepancy.Discrepancy"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.bayes_inference.discrepancy.Discrepancy</span></code></a>.
For this example we set the uncertainty to be zero-mean gaussian and dependent on the values in the observation, i.e. larger values have a larger uncertainty associated with them.
The <code class="docutils literal notranslate"><span class="pre">parameters</span></code> contain the variance for each point in the observation.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">obsData</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">Model</span><span class="o">.</span><span class="n">observations</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">Model</span><span class="o">.</span><span class="n">Output</span><span class="o">.</span><span class="n">names</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">DiscrepancyOpts</span> <span class="o">=</span> <span class="n">Discrepancy</span><span class="p">(</span><span class="s1">''</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">DiscrepancyOpts</span><span class="o">.</span><span class="n">type</span> <span class="o">=</span> <span class="s1">'Gaussian'</span>
<span class="gp">>>> </span><span class="n">DiscrepancyOpts</span><span class="o">.</span><span class="n">parameters</span> <span class="o">=</span> <span class="n">obsData</span><span class="o">**</span><span class="mi">2</span>
</pre></div>
</div>
<p>Now we can initialize an object of class <a class="reference internal" href="_autosummary/bayesvalidrox.bayes_inference.bayes_inference.BayesInference.html#bayesvalidrox.bayes_inference.bayes_inference.BayesInference" title="bayesvalidrox.bayes_inference.bayes_inference.BayesInference"><code class="xref any py py-class docutils literal notranslate"><span class="pre">bayesvalidrox.bayes_inference.bayes_inference.BayesInference</span></code></a> with all the wanted properties.
This object has to be given our <code class="docutils literal notranslate"><span class="pre">Engine</span></code>.
If it should use the surrogate during inference, set <code class="docutils literal notranslate"><span class="pre">emulator</span></code> to <code class="docutils literal notranslate"><span class="pre">True</span></code>, otherwise the model will be evaluated directly.
We also set the defined <code class="docutils literal notranslate"><span class="pre">Discrepancy</span></code>. and set <code class="docutils literal notranslate"><span class="pre">post_plot_pred</span></code> if posterior predictions should be visualized.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">BayesObj</span> <span class="o">=</span> <span class="n">BayesInference</span><span class="p">(</span><span class="n">Engine_</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">BayesObj</span><span class="o">.</span><span class="n">emulator</span> <span class="o">=</span> <span class="kc">True</span>
<span class="gp">>>> </span><span class="n">BayesObj</span><span class="o">.</span><span class="n">Discrepancy</span> <span class="o">=</span> <span class="n">DiscrepancyOpts</span>
<span class="gp">>>> </span><span class="n">BayesObj</span><span class="o">.</span><span class="n">plot_post_pred</span> <span class="o">=</span> <span class="kc">True</span>
</pre></div>
</div>
<p>In order to run with rejection sampling, we set the <code class="docutils literal notranslate"><span class="pre">inference_method</span></code> accordingly and add properties for <code class="docutils literal notranslate"><span class="pre">bootstrap</span></code>.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">BayesObj</span><span class="o">.</span><span class="n">inference_method</span> <span class="o">=</span> <span class="s1">'rejection'</span>
<span class="gp">>>> </span><span class="n">BayesObj</span><span class="o">.</span><span class="n">bootstrap</span> <span class="o">=</span> <span class="kc">True</span>
<span class="gp">>>> </span><span class="n">BayesObj</span><span class="o">.</span><span class="n">n_bootstrap_itrs</span> <span class="o">=</span> <span class="mi">500</span>
<span class="gp">>>> </span><span class="n">BayesObj</span><span class="o">.</span><span class="n">bootstrap_noise</span> <span class="o">=</span> <span class="mi">2</span>
</pre></div>
</div>
<p>If the sampling should be done with MCMC, then this is set as the <code class="docutils literal notranslate"><span class="pre">inference_method</span></code> and additional properties are given in <code class="docutils literal notranslate"><span class="pre">mcmc_params</span></code>.
For this example we use the python package <code class="docutils literal notranslate"><span class="pre">emcee</span></code> to define the MCMC moves.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">BayesObj</span><span class="o">.</span><span class="n">inference_method</span> <span class="o">=</span> <span class="s1">'MCMC'</span>
<span class="gp">>>> </span><span class="kn">import</span> <span class="nn">emcee</span>
<span class="gp">>>> </span><span class="n">BayesObj</span><span class="o">.</span><span class="n">mcmc_params</span> <span class="o">=</span> <span class="p">{</span>
<span class="gp">>>> </span> <span class="s1">'n_steps'</span><span class="p">:</span> <span class="mf">1e4</span><span class="p">,</span><span class="c1">#5,</span>
<span class="gp">>>> </span> <span class="s1">'n_walkers'</span><span class="p">:</span> <span class="mi">30</span><span class="p">,</span>
<span class="gp">>>> </span> <span class="s1">'moves'</span><span class="p">:</span> <span class="n">emcee</span><span class="o">.</span><span class="n">moves</span><span class="o">.</span><span class="n">KDEMove</span><span class="p">(),</span>
<span class="gp">>>> </span> <span class="s1">'multiprocessing'</span><span class="p">:</span> <span class="kc">False</span><span class="p">,</span>
<span class="gp">>>> </span> <span class="s1">'verbose'</span><span class="p">:</span> <span class="kc">False</span>
<span class="gp">>>> </span> <span class="p">}</span>
</pre></div>
</div>
<p>Then we run the inference.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">BayesObj</span><span class="o">.</span><span class="n">create_inference</span><span class="p">()</span>
</pre></div>
</div>
<p>If the output directory <code class="docutils literal notranslate"><span class="pre">BayesObj.out_dir</span></code> is not set otherwise, the outputs are written into the folder ``</p>
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