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<li><a href="#example-model-neals-funnel" id="toc-example-model-neals-funnel" class="nav-link active" data-scroll-target="#example-model-neals-funnel">Example Model: Neal’s Funnel</a></li>
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<li><a href="#basic-hmc-statistics" id="toc-basic-hmc-statistics" class="nav-link" data-scroll-target="#basic-hmc-statistics">Basic HMC Statistics</a></li>
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<h1 class="title">Understanding Sampler Statistics in Nutpie</h1>
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<p>This guide explains the various statistics that nutpie collects during sampling. We’ll use Neal’s funnel distribution as an example, as it’s a challenging model that demonstrates many important sampling concepts.</p>
<section id="example-model-neals-funnel" class="level2">
<h2 class="anchored" data-anchor-id="example-model-neals-funnel">Example Model: Neal’s Funnel</h2>
<p>Let’s start by implementing Neal’s funnel in PyMC:</p>
<div id="9ef0aa6e" class="cell" data-execution_count="1">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> pymc <span class="im">as</span> pm</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> nutpie</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> seaborn <span class="im">as</span> sns</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> pandas <span class="im">as</span> pd</span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> arviz <span class="im">as</span> az</span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a><span class="co"># Create the funnel model</span></span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a><span class="cf">with</span> pm.Model() <span class="im">as</span> model:</span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a> log_sigma <span class="op">=</span> pm.Normal(<span class="st">'log_sigma'</span>)</span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a> pm.Normal(<span class="st">'x'</span>, sigma<span class="op">=</span>pm.math.exp(log_sigma), shape<span class="op">=</span><span class="dv">5</span>)</span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a><span class="co"># Sample with detailed statistics</span></span>
<span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a>compiled <span class="op">=</span> nutpie.compile_pymc_model(model)</span>
<span id="cb1-16"><a href="#cb1-16" aria-hidden="true" tabindex="-1"></a>trace <span class="op">=</span> nutpie.sample(</span>
<span id="cb1-17"><a href="#cb1-17" aria-hidden="true" tabindex="-1"></a> compiled,</span>
<span id="cb1-18"><a href="#cb1-18" aria-hidden="true" tabindex="-1"></a> tune<span class="op">=</span><span class="dv">1000</span>,</span>
<span id="cb1-19"><a href="#cb1-19" aria-hidden="true" tabindex="-1"></a> store_mass_matrix<span class="op">=</span><span class="va">True</span>,</span>
<span id="cb1-20"><a href="#cb1-20" aria-hidden="true" tabindex="-1"></a> store_gradient<span class="op">=</span><span class="va">True</span>,</span>
<span id="cb1-21"><a href="#cb1-21" aria-hidden="true" tabindex="-1"></a> store_unconstrained<span class="op">=</span><span class="va">True</span>,</span>
<span id="cb1-22"><a href="#cb1-22" aria-hidden="true" tabindex="-1"></a> store_divergences<span class="op">=</span><span class="va">True</span>,</span>
<span id="cb1-23"><a href="#cb1-23" aria-hidden="true" tabindex="-1"></a> seed<span class="op">=</span><span class="dv">42</span>,</span>
<span id="cb1-24"><a href="#cb1-24" aria-hidden="true" tabindex="-1"></a>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display">
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.nutpie th:nth-child(3) { width: var(--column-width-3); }
.nutpie th:nth-child(4) { width: var(--column-width-4); }
.nutpie th:nth-child(5) { width: var(--column-width-5); }
.nutpie progress {
width: 100%;
height: 15px; /* Smaller progress bars */
border-radius: 5px;
}
progress::-webkit-progress-bar {
background-color: #eee;
border-radius: 5px;
}
progress::-webkit-progress-value {
background-color: #5cb85c;
border-radius: 5px;
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progress::-moz-progress-bar {
background-color: #5cb85c;
border-radius: 5px;
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.nutpie .progress-cell {
width: 100%;
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.nutpie p strong { font-size: 16px; font-weight: bold; }
@media (prefers-color-scheme: dark) {
.nutpie {
//color: #ddd;
//background-color: #1e1e1e;
box-shadow: 0 4px 6px rgba(0,0,0,0.2);
}
.nutpie table, .nutpie th, .nutpie td {
border-color: #555;
color: #ccc;
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.nutpie th {
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.nutpie progress::-moz-progress-bar {
background-color: #3178c6;
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}
</style>
</div>
<div class="cell-output cell-output-display">
<div class="nutpie">
<p><strong>Sampler Progress</strong></p>
<p>Total Chains: <span id="total-chains">6</span></p>
<p>Active Chains: <span id="active-chains">0</span></p>
<p>
Finished Chains:
<span id="active-chains">6</span>
</p>
<p>Sampling for now</p>
<p>
Estimated Time to Completion:
<span id="eta">now</span>
</p>
<progress id="total-progress-bar" max="12000" value="12000">
</progress>
<table>
<thead>
<tr>
<th>Progress</th>
<th>Draws</th>
<th>Divergences</th>
<th>Step Size</th>
<th>Gradients/Draw</th>
</tr>
</thead>
<tbody id="chain-details">
<tr>
<td class="progress-cell">
<progress max="2000" value="2000">
</progress>
</td>
<td>2000</td>
<td>21</td>
<td>0.41</td>
<td>7</td>
</tr>
<tr>
<td class="progress-cell">
<progress max="2000" value="2000">
</progress>
</td>
<td>2000</td>
<td>24</td>
<td>0.35</td>
<td>7</td>
</tr>
<tr>
<td class="progress-cell">
<progress max="2000" value="2000">
</progress>
</td>
<td>2000</td>
<td>0</td>
<td>0.35</td>
<td>7</td>
</tr>
<tr>
<td class="progress-cell">
<progress max="2000" value="2000">
</progress>
</td>
<td>2000</td>
<td>13</td>
<td>0.44</td>
<td>7</td>
</tr>
<tr>
<td class="progress-cell">
<progress max="2000" value="2000">
</progress>
</td>
<td>2000</td>
<td>0</td>
<td>0.34</td>
<td>15</td>
</tr>
<tr>
<td class="progress-cell">
<progress max="2000" value="2000">
</progress>
</td>
<td>2000</td>
<td>5</td>
<td>0.53</td>
<td>7</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</section>
<section id="sampler-statistics-overview" class="level2">
<h2 class="anchored" data-anchor-id="sampler-statistics-overview">Sampler Statistics Overview</h2>
<p>The sampler statistics can be grouped into several categories:</p>
<section id="basic-hmc-statistics" class="level3">
<h3 class="anchored" data-anchor-id="basic-hmc-statistics">Basic HMC Statistics</h3>
<p>These statistics are always collected and are essential for basic diagnostics:</p>
<div id="ffa51d43" class="cell" data-execution_count="2">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Access through trace.sample_stats</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>basic_stats <span class="op">=</span> [</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a> <span class="st">'depth'</span>, <span class="co"># Tree depth for current draw</span></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a> <span class="st">'maxdepth_reached'</span>, <span class="co"># Whether max tree depth was hit</span></span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a> <span class="st">'logp'</span>, <span class="co"># Log probability of current position</span></span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a> <span class="st">'energy'</span>, <span class="co"># Hamiltonian energy</span></span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a> <span class="st">'diverging'</span>, <span class="co"># Whether the transition diverged</span></span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a> <span class="st">'step_size'</span>, <span class="co"># Current step size</span></span>
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a> <span class="st">'step_size_bar'</span>, <span class="co"># Current estimate of an ideal step size</span></span>
<span id="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a> <span class="st">'n_steps'</span> <span class="co"># Number of leapfrog steps</span></span>
<span id="cb2-11"><a href="#cb2-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-12"><a href="#cb2-12" aria-hidden="true" tabindex="-1"></a>]</span>
<span id="cb2-13"><a href="#cb2-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-14"><a href="#cb2-14" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot step size evolution during warmup</span></span>
<span id="cb2-15"><a href="#cb2-15" aria-hidden="true" tabindex="-1"></a>trace.warmup_sample_stats.step_size_bar.plot.line(x<span class="op">=</span><span class="st">"draw"</span>, yscale<span class="op">=</span><span class="st">"log"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display">
<div>
<figure class="figure">
<p><img src="sample-stats_files/figure-html/cell-3-output-1.png" class="img-fluid figure-img"></p>
</figure>
</div>
</div>
</div>
</section>
<section id="mass-matrix-adaptation" class="level3">
<h3 class="anchored" data-anchor-id="mass-matrix-adaptation">Mass Matrix Adaptation</h3>
<p>These statistics track how the mass matrix evolves:</p>
<div id="146dc128" class="cell" data-execution_count="3">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>(</span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a> trace</span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a> .warmup_sample_stats</span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a> .mass_matrix_inv</span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a> .plot</span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a> .line(</span>
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a> x<span class="op">=</span><span class="st">"draw"</span>,</span>
<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a> yscale<span class="op">=</span><span class="st">"log"</span>,</span>
<span id="cb3-9"><a href="#cb3-9" aria-hidden="true" tabindex="-1"></a> col<span class="op">=</span><span class="st">"chain"</span>,</span>
<span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a> col_wrap<span class="op">=</span><span class="dv">2</span>,</span>
<span id="cb3-11"><a href="#cb3-11" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb3-12"><a href="#cb3-12" aria-hidden="true" tabindex="-1"></a>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display">
<div>
<figure class="figure">
<p><img src="sample-stats_files/figure-html/cell-4-output-1.png" class="img-fluid figure-img"></p>
</figure>
</div>
</div>
</div>
<p>Variables that are a source of convergence issues, will often show high variance in the final mass matrix estimate across chains.</p>
<p>The mass matrix will always be fixed for 10% of draws at the end, because we only run final step size adaptation during that time, but high variance in the mass matrix before this final window and indicate that more tuning steps might be needed.</p>
</section>
<section id="detailed-diagnostics" class="level3">
<h3 class="anchored" data-anchor-id="detailed-diagnostics">Detailed Diagnostics</h3>
<p>These are only available when explicitly requested:</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>detailed_stats <span class="op">=</span> [</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a> <span class="st">'gradient'</span>, <span class="co"># Gradient at current position</span></span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a> <span class="st">'unconstrained_draw'</span>, <span class="co"># Parameters in unconstrained space</span></span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a> <span class="st">'divergence_start'</span>, <span class="co"># Position where divergence started</span></span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a> <span class="st">'divergence_end'</span>, <span class="co"># Position where divergence ended</span></span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a> <span class="st">'divergence_momentum'</span>, <span class="co"># Momentum at divergence</span></span>
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a> <span class="st">'divergence_message'</span> <span class="co"># Description of divergence</span></span>
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<section id="idintify-divergences" class="level4">
<h4 class="anchored" data-anchor-id="idintify-divergences">Idintify Divergences</h4>
<p>We can for instance use this to identify the sources of divergences:</p>
<div id="44f63008" class="cell" data-execution_count="4">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> xarray <span class="im">as</span> xr</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>draws <span class="op">=</span> (</span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a> trace</span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a> .sample_stats</span>
<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a> .unconstrained_draw</span>
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a> .assign_coords(kind<span class="op">=</span><span class="st">"draw"</span>)</span>
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a>divergence_locations <span class="op">=</span> (</span>
<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a> trace</span>
<span id="cb5-11"><a href="#cb5-11" aria-hidden="true" tabindex="-1"></a> .sample_stats</span>
<span id="cb5-12"><a href="#cb5-12" aria-hidden="true" tabindex="-1"></a> .divergence_start</span>
<span id="cb5-13"><a href="#cb5-13" aria-hidden="true" tabindex="-1"></a> .assign_coords(kind<span class="op">=</span><span class="st">"divergence"</span>)</span>
<span id="cb5-14"><a href="#cb5-14" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb5-15"><a href="#cb5-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-16"><a href="#cb5-16" aria-hidden="true" tabindex="-1"></a>points <span class="op">=</span> xr.concat([draws, divergence_locations], dim<span class="op">=</span><span class="st">"kind"</span>)</span>
<span id="cb5-17"><a href="#cb5-17" aria-hidden="true" tabindex="-1"></a>points.to_dataset(<span class="st">"unconstrained_parameter"</span>).plot.scatter(x<span class="op">=</span><span class="st">"log_sigma"</span>, y<span class="op">=</span><span class="st">"x_0"</span>, hue<span class="op">=</span><span class="st">"kind"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display">
<div>
<figure class="figure">
<p><img src="sample-stats_files/figure-html/cell-5-output-1.png" class="img-fluid figure-img"></p>
</figure>
</div>
</div>
</div>
</section>
<section id="covariance-of-gradients-and-draws" class="level4">
<h4 class="anchored" data-anchor-id="covariance-of-gradients-and-draws">Covariance of gradients and draws</h4>
<p>TODO this section should really use the transformed gradients and draws, not the unconstrained ones, as that avoids the manual mass matrix correction. This is only available for the normalizing flow adaptation at the moment though.</p>
<p>In models with problematic posterior correlations, the singular value decomposition of gradients and draws can often point us to the source of the issue.</p>
<p>Let’s build a little model with correlations between parameters:</p>
<div id="32313889" class="cell" data-execution_count="5">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="cf">with</span> pm.Model() <span class="im">as</span> model:</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a> x <span class="op">=</span> pm.Normal(<span class="st">'x'</span>)</span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a> y <span class="op">=</span> pm.Normal(<span class="st">"y"</span>, mu<span class="op">=</span>x, sigma<span class="op">=</span><span class="fl">0.01</span>)</span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a> z <span class="op">=</span> pm.Normal(<span class="st">"z"</span>, mu<span class="op">=</span>y, shape<span class="op">=</span><span class="dv">100</span>)</span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a>compiled <span class="op">=</span> nutpie.compile_pymc_model(model)</span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a>trace <span class="op">=</span> nutpie.sample(</span>
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a> compiled,</span>
<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a> tune<span class="op">=</span><span class="dv">1000</span>,</span>
<span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a> store_gradient<span class="op">=</span><span class="va">True</span>,</span>
<span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a> store_unconstrained<span class="op">=</span><span class="va">True</span>,</span>
<span id="cb6-12"><a href="#cb6-12" aria-hidden="true" tabindex="-1"></a> store_mass_matrix<span class="op">=</span><span class="va">True</span>,</span>
<span id="cb6-13"><a href="#cb6-13" aria-hidden="true" tabindex="-1"></a> seed<span class="op">=</span><span class="dv">42</span>,</span>
<span id="cb6-14"><a href="#cb6-14" aria-hidden="true" tabindex="-1"></a>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display">
<style>
:root {
--column-width-1: 40%; /* Progress column width */
--column-width-2: 15%; /* Chain column width */
--column-width-3: 15%; /* Divergences column width */
--column-width-4: 15%; /* Step Size column width */
--column-width-5: 15%; /* Gradients/Draw column width */
}
.nutpie {
max-width: 800px;
margin: 10px auto;
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
//color: #333;
//background-color: #fff;
padding: 10px;
box-shadow: 0 4px 6px rgba(0,0,0,0.1);
border-radius: 8px;
font-size: 14px; /* Smaller font size for a more compact look */
}
.nutpie table {
width: 100%;
border-collapse: collapse; /* Remove any extra space between borders */
}
.nutpie th, .nutpie td {
padding: 8px 10px; /* Reduce padding to make table more compact */
text-align: left;
border-bottom: 1px solid #888;
}
.nutpie th {
//background-color: #f0f0f0;
}
.nutpie th:nth-child(1) { width: var(--column-width-1); }
.nutpie th:nth-child(2) { width: var(--column-width-2); }
.nutpie th:nth-child(3) { width: var(--column-width-3); }
.nutpie th:nth-child(4) { width: var(--column-width-4); }
.nutpie th:nth-child(5) { width: var(--column-width-5); }
.nutpie progress {
width: 100%;
height: 15px; /* Smaller progress bars */
border-radius: 5px;
}
progress::-webkit-progress-bar {
background-color: #eee;
border-radius: 5px;
}
progress::-webkit-progress-value {
background-color: #5cb85c;
border-radius: 5px;
}
progress::-moz-progress-bar {
background-color: #5cb85c;
border-radius: 5px;
}
.nutpie .progress-cell {
width: 100%;
}
.nutpie p strong { font-size: 16px; font-weight: bold; }
@media (prefers-color-scheme: dark) {
.nutpie {
//color: #ddd;
//background-color: #1e1e1e;
box-shadow: 0 4px 6px rgba(0,0,0,0.2);
}
.nutpie table, .nutpie th, .nutpie td {
border-color: #555;
color: #ccc;
}
.nutpie th {
background-color: #2a2a2a;
}
.nutpie progress::-webkit-progress-bar {
background-color: #444;
}
.nutpie progress::-webkit-progress-value {
background-color: #3178c6;
}
.nutpie progress::-moz-progress-bar {
background-color: #3178c6;
}
}
</style>
</div>
<div class="cell-output cell-output-display">
<div class="nutpie">
<p><strong>Sampler Progress</strong></p>
<p>Total Chains: <span id="total-chains">6</span></p>
<p>Active Chains: <span id="active-chains">0</span></p>
<p>
Finished Chains:
<span id="active-chains">6</span>
</p>
<p>Sampling for now</p>
<p>
Estimated Time to Completion:
<span id="eta">now</span>
</p>
<progress id="total-progress-bar" max="12000" value="12000">
</progress>
<table>
<thead>
<tr>
<th>Progress</th>
<th>Draws</th>
<th>Divergences</th>
<th>Step Size</th>
<th>Gradients/Draw</th>
</tr>
</thead>
<tbody id="chain-details">
<tr>
<td class="progress-cell">
<progress max="2000" value="2000">
</progress>
</td>
<td>2000</td>
<td>0</td>
<td>0.13</td>
<td>31</td>
</tr>
<tr>
<td class="progress-cell">
<progress max="2000" value="2000">
</progress>
</td>
<td>2000</td>
<td>0</td>
<td>0.16</td>
<td>31</td>
</tr>
<tr>
<td class="progress-cell">
<progress max="2000" value="2000">
</progress>
</td>
<td>2000</td>
<td>0</td>
<td>0.22</td>
<td>15</td>
</tr>
<tr>
<td class="progress-cell">
<progress max="2000" value="2000">
</progress>
</td>
<td>2000</td>
<td>0</td>
<td>0.14</td>
<td>31</td>
</tr>
<tr>
<td class="progress-cell">
<progress max="2000" value="2000">
</progress>
</td>
<td>2000</td>
<td>0</td>
<td>0.15</td>
<td>31</td>
</tr>
<tr>
<td class="progress-cell">
<progress max="2000" value="2000">
</progress>
</td>
<td>2000</td>
<td>0</td>
<td>0.15</td>
<td>31</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<p>Now we can compute eigenvalues of the covariance matrix of the gradient and draws (using the singular value decomposition to avoid quadratic cost):</p>
<div id="be02c1fa" class="cell" data-execution_count="6">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> covariance_eigenvalues(x, mass_matrix):</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">assert</span> x.dims <span class="op">==</span> (<span class="st">"chain"</span>, <span class="st">"draw"</span>, <span class="st">"unconstrained_parameter"</span>)</span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a> x <span class="op">=</span> x.stack(sample<span class="op">=</span>[<span class="st">"draw"</span>, <span class="st">"chain"</span>])</span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a> x <span class="op">=</span> (x <span class="op">-</span> x.mean(<span class="st">"sample"</span>)) <span class="op">/</span> np.sqrt(mass_matrix)</span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a> u, s, v <span class="op">=</span> np.linalg.svd(x.T <span class="op">/</span> np.sqrt(x.shape[<span class="dv">1</span>]), full_matrices<span class="op">=</span><span class="va">False</span>)</span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a> <span class="bu">print</span>(u.shape, s.shape, v.shape)</span>
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a> s <span class="op">=</span> xr.DataArray(</span>
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a> s,</span>
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a> dims<span class="op">=</span>[<span class="st">"eigenvalue"</span>],</span>
<span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a> coords<span class="op">=</span>{<span class="st">"eigenvalue"</span>: <span class="bu">range</span>(s.size)},</span>
<span id="cb7-11"><a href="#cb7-11" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb7-12"><a href="#cb7-12" aria-hidden="true" tabindex="-1"></a> v <span class="op">=</span> xr.DataArray(</span>
<span id="cb7-13"><a href="#cb7-13" aria-hidden="true" tabindex="-1"></a> v,</span>
<span id="cb7-14"><a href="#cb7-14" aria-hidden="true" tabindex="-1"></a> dims<span class="op">=</span>[<span class="st">"eigenvalue"</span>, <span class="st">"unconstrained_parameter"</span>],</span>
<span id="cb7-15"><a href="#cb7-15" aria-hidden="true" tabindex="-1"></a> coords<span class="op">=</span>{</span>
<span id="cb7-16"><a href="#cb7-16" aria-hidden="true" tabindex="-1"></a> <span class="st">"eigenvalue"</span>: s.eigenvalue,</span>
<span id="cb7-17"><a href="#cb7-17" aria-hidden="true" tabindex="-1"></a> <span class="st">"unconstrained_parameter"</span>: x.unconstrained_parameter,</span>
<span id="cb7-18"><a href="#cb7-18" aria-hidden="true" tabindex="-1"></a> },</span>
<span id="cb7-19"><a href="#cb7-19" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb7-20"><a href="#cb7-20" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> s <span class="op">**</span> <span class="dv">2</span>, v</span>
<span id="cb7-21"><a href="#cb7-21" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-22"><a href="#cb7-22" aria-hidden="true" tabindex="-1"></a>mass_matrix <span class="op">=</span> trace.sample_stats.mass_matrix_inv.isel(draw<span class="op">=-</span><span class="dv">1</span>, chain<span class="op">=</span><span class="dv">0</span>)</span>
<span id="cb7-23"><a href="#cb7-23" aria-hidden="true" tabindex="-1"></a>draws_eigs, draws_eigv <span class="op">=</span> covariance_eigenvalues(trace.sample_stats.unconstrained_draw, mass_matrix)</span>
<span id="cb7-24"><a href="#cb7-24" aria-hidden="true" tabindex="-1"></a>grads_eigs, grads_eigv <span class="op">=</span> covariance_eigenvalues(trace.sample_stats.gradient, <span class="dv">1</span> <span class="op">/</span> mass_matrix)</span>
<span id="cb7-25"><a href="#cb7-25" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-26"><a href="#cb7-26" aria-hidden="true" tabindex="-1"></a>draws_eigs.plot.line(x<span class="op">=</span><span class="st">"eigenvalue"</span>, yscale<span class="op">=</span><span class="st">"log"</span>)</span>
<span id="cb7-27"><a href="#cb7-27" aria-hidden="true" tabindex="-1"></a>grads_eigs.plot.line(x<span class="op">=</span><span class="st">"eigenvalue"</span>, yscale<span class="op">=</span><span class="st">"log"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>(6000, 102) (102,) (102, 102)
(6000, 102) (102,) (102, 102)</code></pre>
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<p><img src="sample-stats_files/figure-html/cell-7-output-2.png" class="img-fluid figure-img"></p>
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<p>We can see one very large and one very small eigenvalue in both covariances. Large eigenvalues for the draws, and small eigenvalues for the gradients prevent the sampler from taking larger steps. Small eigenvalues in the draws, and large eigenvalues in the grads mean, that the sampler has to move far in parameter space to get independent draws. So both lead to problems during sampling. For models with many parameters, typically only the large eigenvalues of each are meaningful, because of estimation issues with the small eigenvalues.</p>
<p>We can also look at the eigenvectors to see which parameters are responsible for the correlations:</p>
<div id="202af166" class="cell" data-execution_count="7">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>(</span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a> draws_eigv</span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a> .sel(eigenvalue<span class="op">=</span><span class="dv">0</span>)</span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a> .to_pandas()</span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a> .sort_values(key<span class="op">=</span><span class="bu">abs</span>)</span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a> .tail(<span class="dv">10</span>)</span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a> .plot.bar(x<span class="op">=</span><span class="st">"unconstrained_parameter"</span>)</span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p><img src="sample-stats_files/figure-html/cell-8-output-1.png" class="img-fluid figure-img"></p>
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<div id="52404958" class="cell" data-execution_count="8">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>(</span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a> grads_eigv</span>
<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a> .sel(eigenvalue<span class="op">=</span><span class="dv">0</span>)</span>
<span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a> .to_pandas()</span>
<span id="cb10-5"><a href="#cb10-5" aria-hidden="true" tabindex="-1"></a> .sort_values(key<span class="op">=</span><span class="bu">abs</span>)</span>
<span id="cb10-6"><a href="#cb10-6" aria-hidden="true" tabindex="-1"></a> .tail(<span class="dv">10</span>)</span>
<span id="cb10-7"><a href="#cb10-7" aria-hidden="true" tabindex="-1"></a> .plot.bar(x<span class="op">=</span><span class="st">"unconstrained_parameter"</span>)</span>
<span id="cb10-8"><a href="#cb10-8" aria-hidden="true" tabindex="-1"></a>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p><img src="sample-stats_files/figure-html/cell-9-output-1.png" class="img-fluid figure-img"></p>
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