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<section class="slide level2">

<h1>
<p>An Intro to <code>Stan</code></p>
</h1>
<hr>
<h3>
<p>Sean Pinkney, Group Director TV Analytics</p>
</h3>
<h3>
<p>2022-10-20</p>
</h3>
<p><br></p>
<h3>
<p><svg aria-hidden="true" role="img" viewbox="0 0 496 512" style="height:1em;width:0.97em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:black;overflow:visible;position:relative;"><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3 .3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5 .3-6.2 2.3zm44.2-1.7c-2.9 .7-4.9 2.6-4.6 4.9 .3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3 .7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3 .3 2.9 2.3 3.9 1.6 1 3.6 .7 4.3-.7 .7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3 .7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3 .7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"></path></svg> &nbsp; <a href="https://github.com/spinkney/intro-to-stan" class="uri">https://github.com/spinkney/intro-to-stan</a></p>
</h3></section>
<section id="who-am-i" class="slide level2">
<h2>Who am I?</h2>
<ul>
<li><p>Group Director of TV Analytics at OMG since April 2022</p></li>
<li><p>Stan Developer (for a few years now)</p></li>
<li><p>Stan Governance Board Member 2022</p></li>
</ul>
</section>
<section id="probabilistic-programming-languages" class="slide level2">
<h2>Probabilistic Programming Languages</h2>
<p><a href="https://en.wikipedia.org/wiki/Probabilistic_programming" class="uri">https://en.wikipedia.org/wiki/Probabilistic_programming</a></p>
</section>
<section id="probabilistic-programming-the-what-why-and-how" class="slide level2">
<h2><a href="https://www.youtube.com/watch?v=cvD9DnTDxmY&amp;ab_channel=ACMSIGPLAN">Probabilistic Programming: The What, Why and How</a></h2>
<p><span class="math display">\[
y = f(x)
\]</span></p>
<ul>
<li><p>Conventional programming</p>
<ul>
<li><p>Given: input <span class="math inline">\(x\)</span>, function <span class="math inline">\(f\)</span></p></li>
<li><p>Want: output <span class="math inline">\(y\)</span></p></li>
</ul></li>
<li><p>Probabilistic programming</p>
<ul>
<li>Given: output <span class="math inline">\(y\)</span>, function <span class="math inline">\(f\)</span></li>
<li>Want: probability distribution on the input <span class="math inline">\(x\)</span></li>
</ul></li>
<li><p>Deep learning (supervised)</p>
<ul>
<li>Given: input x, output y</li>
<li>Want: function <span class="math inline">\(f\)</span></li>
</ul></li>
</ul>
</section>
<section id="what-is-stan" class="slide level2">
<h2>What is Stan?</h2>
<p>Stan is a state-of-the-art platform for statistical modeling and high-performance statistical computation.</p>

<img data-src="https://mc-stan.org/images/stan_logo.png" class="r-stretch quarto-figure-center"><p>To learn more about Stan see <a href="https://mc-stan.org/" class="uri">https://mc-stan.org/</a>.</p>
</section>
<section id="why-use-stan" class="slide level2">
<h2>Why use Stan?</h2>
<h4 id="a-biased-perspective">A biased perspective</h4>
<ul>
<li>It’s fun!</li>
<li>Excellent documentation - <a href="https://mc-stan.org/docs/stan-users-guide/index.html">Stan User Guide</a></li>
<li>Supportive community - <a href="https://discourse.mc-stan.org/" class="uri">https://discourse.mc-stan.org/</a></li>
</ul>
<h4 id="a-mainstream-perspective">A mainstream perspective</h4>
<ul>
<li>Encode your assumptions directly into the model</li>
<li>Model what you want</li>
<li>Really understand the problems/limitations with your assumptions or model</li>
</ul>
</section>
<section id="how-does-it-work" class="slide level2">
<h2>How does it work?</h2>
<p>You specify a log-density up to a constant. That’s the “loss” function. The sampler runs around the geometry of that space finding regions of density.</p>
<ul>
<li>Uses a variant of Hamiltonian Monte Carlo (HMC)</li>
</ul>
<p><a href="https://www.youtube.com/watch?v=Vv3f0QNWvWQ&amp;t=1s&amp;ab_channel=DavidDuvenaud" class="uri">https://www.youtube.com/watch?v=Vv3f0QNWvWQ&amp;t=1s&amp;ab_channel=DavidDuvenaud</a></p>
<p><a href="https://chi-feng.github.io/mcmc-demo/app.html#HamiltonianMC,banana" class="uri">https://chi-feng.github.io/mcmc-demo/app.html#HamiltonianMC,banana</a></p>
</section>
<section id="how-does-it-work-1" class="slide level2">
<h2>How does it work?</h2>
<p>It is gradient based.</p>
<ul>
<li><p>No discrete parameters</p></li>
<li><p>Multimodal posteriors are really difficult for the sampler</p></li>
<li><p>The goal is to explore the posterior space and get a great approximation to the full posterior</p></li>
</ul>
</section>
<section id="anatomy-of-a-stan-program" class="slide level2">
<h2>Anatomy of a Stan Program</h2>
<div class="sourceCode" id="cb1"><pre class="sourceCode numberSource stan number-lines code-with-copy"><code class="sourceCode stan"><span id="cb1-1"><a href="#cb1-1"></a><span class="kw">functions</span> {</span>
<span id="cb1-2"><a href="#cb1-2"></a>}</span>
<span id="cb1-3"><a href="#cb1-3"></a><span class="kw">data</span>{ </span>
<span id="cb1-4"><a href="#cb1-4"></a>}</span>
<span id="cb1-5"><a href="#cb1-5"></a><span class="kw">transformed data</span>{</span>
<span id="cb1-6"><a href="#cb1-6"></a>}</span>
<span id="cb1-7"><a href="#cb1-7"></a><span class="kw">parameters</span>{</span>
<span id="cb1-8"><a href="#cb1-8"></a>}</span>
<span id="cb1-9"><a href="#cb1-9"></a><span class="kw">transformed parameters</span>{</span>
<span id="cb1-10"><a href="#cb1-10"></a>}</span>
<span id="cb1-11"><a href="#cb1-11"></a><span class="kw">model</span>{ </span>
<span id="cb1-12"><a href="#cb1-12"></a>}</span>
<span id="cb1-13"><a href="#cb1-13"></a><span class="kw">generated quantities</span> {</span>
<span id="cb1-14"><a href="#cb1-14"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</section>
<section id="simple-stan" class="slide level2">
<h2>Simple Stan</h2>
<div class="sourceCode" id="cb2"><pre class="sourceCode numberSource stan number-lines code-with-copy"><code class="sourceCode stan"><span id="cb2-1"><a href="#cb2-1"></a><span class="kw">data</span> {</span>
<span id="cb2-2"><a href="#cb2-2"></a> <span class="dt">int</span>&lt;<span class="kw">lower</span>=<span class="dv">0</span>&gt; N;</span>
<span id="cb2-3"><a href="#cb2-3"></a> <span class="dt">array</span>[<span class="dv">10</span>] <span class="dt">int</span> counts;</span>
<span id="cb2-4"><a href="#cb2-4"></a>}</span>
<span id="cb2-5"><a href="#cb2-5"></a><span class="kw">parameters</span> {</span>
<span id="cb2-6"><a href="#cb2-6"></a> <span class="dt">simplex</span>[<span class="dv">10</span>] theta;</span>
<span id="cb2-7"><a href="#cb2-7"></a>}</span>
<span id="cb2-8"><a href="#cb2-8"></a><span class="kw">model</span> {</span>
<span id="cb2-9"><a href="#cb2-9"></a>  counts ~ multinomial(theta);</span>
<span id="cb2-10"><a href="#cb2-10"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</section>
<section id="run-it" class="slide level2">
<h2>Run it</h2>
<div class="cell">
<div class="cell-output cell-output-stdout">
<pre><code>Running MCMC with 4 parallel chains...

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Chain 4 Iteration: 1400 / 2000 [ 70%]  (Sampling) 
Chain 4 Iteration: 1500 / 2000 [ 75%]  (Sampling) 
Chain 4 Iteration: 1600 / 2000 [ 80%]  (Sampling) 
Chain 4 Iteration: 1700 / 2000 [ 85%]  (Sampling) 
Chain 4 Iteration: 1800 / 2000 [ 90%]  (Sampling) 
Chain 4 Iteration: 1900 / 2000 [ 95%]  (Sampling) 
Chain 4 Iteration: 2000 / 2000 [100%]  (Sampling) 
Chain 1 finished in 0.0 seconds.
Chain 2 finished in 0.0 seconds.
Chain 3 finished in 0.0 seconds.
Chain 4 finished in 0.0 seconds.

All 4 chains finished successfully.
Mean chain execution time: 0.0 seconds.
Total execution time: 0.2 seconds.</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code># A tibble: 10 × 10
   variable    mean median     sd    mad      q5   q95  rhat ess_bulk ess_tail
   &lt;chr&gt;      &lt;dbl&gt;  &lt;dbl&gt;  &lt;dbl&gt;  &lt;dbl&gt;   &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt;    &lt;dbl&gt;    &lt;dbl&gt;
 1 theta[1]  0.0998 0.0930 0.0462 0.0438 0.0357  0.185  1.00    7143.    2689.
 2 theta[2]  0.124  0.117  0.0506 0.0498 0.0525  0.218  1.00    6481.    2918.
 3 theta[3]  0.151  0.145  0.0563 0.0547 0.0701  0.251  1.00    7136.    2880.
 4 theta[4]  0.0752 0.0672 0.0411 0.0382 0.0214  0.154  1.00    6949.    2992.
 5 theta[5]  0.124  0.118  0.0520 0.0516 0.0522  0.218  1.00    7425.    3255.
 6 theta[6]  0.0504 0.0424 0.0343 0.0302 0.00942 0.120  1.00    6443.    2679.
 7 theta[7]  0.0495 0.0423 0.0343 0.0308 0.00890 0.116  1.00    5955.    2838.
 8 theta[8]  0.125  0.119  0.0510 0.0487 0.0536  0.218  1.00    6710.    3203.
 9 theta[9]  0.126  0.119  0.0517 0.0502 0.0529  0.218  1.00    7396.    3129.
10 theta[10] 0.0749 0.0674 0.0417 0.0384 0.0212  0.153  1.00    6335.    2598.</code></pre>
</div>
</div>
</section>
<section id="a-less-contrived-example" class="slide level2">
<h2>A less contrived example</h2>
<div class="columns">
<div class="column" style="width:60%;">
<ul>
<li><p>A small panel with full measurements, <span class="math inline">\(s\)</span></p></li>
<li><p>A larger panel with missing measurement, <span class="math inline">\(b\)</span></p></li>
<li><p>We have two measurements from each</p>
<ul>
<li><p>Number of households watching channel <span class="math inline">\(c\)</span> as <span class="math inline">\(y_c\)</span></p></li>
<li><p>Number of hours the households are watching channel <span class="math inline">\(c\)</span> as <span class="math inline">\(x_c\)</span></p></li>
</ul></li>
</ul>
</div><div class="column" style="width:40%;">
<div class="quarto-figure quarto-figure-right">
<figure>
<p><img data-src="images/paste-E808A6D6.png" width="350"></p>
</figure>
</div>
</div>
</div>
</section>
<section id="a-less-contrived-example-1" class="slide level2">
<h2>A less contrived example</h2>
<div class="cell">
<div class="cell-output cell-output-stdout">
<pre><code>[1] "Correlation of small: 0.754319141286783"</code></pre>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] "Correlation of big: 0.661268004354324"</code></pre>
</div>
<div class="cell-output-display">
<table>
<thead>
<tr class="header">
<th style="text-align: right;">n_big</th>
<th style="text-align: right;">n_small</th>
<th style="text-align: right;">hours_big</th>
<th style="text-align: right;">hours_small</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">2862</td>
<td style="text-align: right;">55</td>
<td style="text-align: right;">28620</td>
<td style="text-align: right;">165</td>
</tr>
<tr class="even">
<td style="text-align: right;">2288</td>
<td style="text-align: right;">63</td>
<td style="text-align: right;">22440</td>
<td style="text-align: right;">315</td>
</tr>
<tr class="odd">
<td style="text-align: right;">3119</td>
<td style="text-align: right;">22</td>
<td style="text-align: right;">31190</td>
<td style="text-align: right;">66</td>
</tr>
<tr class="even">
<td style="text-align: right;">NA</td>
<td style="text-align: right;">99</td>
<td style="text-align: right;">NA</td>
<td style="text-align: right;">396</td>
</tr>
<tr class="odd">
<td style="text-align: right;">4072</td>
<td style="text-align: right;">16</td>
<td style="text-align: right;">24432</td>
<td style="text-align: right;">80</td>
</tr>
<tr class="even">
<td style="text-align: right;">NA</td>
<td style="text-align: right;">13</td>
<td style="text-align: right;">NA</td>
<td style="text-align: right;">65</td>
</tr>
<tr class="odd">
<td style="text-align: right;">2638</td>
<td style="text-align: right;">87</td>
<td style="text-align: right;">21104</td>
<td style="text-align: right;">435</td>
</tr>
<tr class="even">
<td style="text-align: right;">3981</td>
<td style="text-align: right;">84</td>
<td style="text-align: right;">27867</td>
<td style="text-align: right;">84</td>
</tr>
<tr class="odd">
<td style="text-align: right;">2773</td>
<td style="text-align: right;">49</td>
<td style="text-align: right;">22184</td>
<td style="text-align: right;">196</td>
</tr>
<tr class="even">
<td style="text-align: right;">1827</td>
<td style="text-align: right;">62</td>
<td style="text-align: right;">9135</td>
<td style="text-align: right;">286</td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
<section id="a-less-contrived-example-2" class="slide level2">
<h2>A less contrived example</h2>
<p>We’ll assume</p>
<p><span class="math display">\[
\begin{aligned}
\begin{bmatrix}
y_b \\
y_s
\end{bmatrix} \sim \mathcal{N}
\begin{bmatrix}
  \begin{bmatrix}
   \mu_b \\
   \mu_s
  \end{bmatrix}, \,
   \begin{bmatrix}
   \sigma_b^2 &amp; \rho \sigma_b \sigma_s \\
   \rho \sigma_b \sigma_s &amp; \sigma_s^2
  \end{bmatrix}
\end{bmatrix}
\end{aligned}
\]</span></p>
<p><span class="math display">\[
\begin{aligned}
\begin{bmatrix}
x_b \\
x_s
\end{bmatrix} \sim \mathcal{N}
\begin{bmatrix}
  \begin{bmatrix}
   \mu_{xb} \\
   \mu_{xs}
  \end{bmatrix}, \,
   \begin{bmatrix}
   \sigma_{xb}^2 &amp; \rho \sigma_{xb} \sigma_{xs} \\
   \rho \sigma_{xb} \sigma_{xs} &amp; \sigma_{xs}^2
  \end{bmatrix}
\end{bmatrix}
\end{aligned}
\]</span></p>
<p>where</p>
<p><span class="math display">\[
\begin{aligned}
\mu_y &amp;= \alpha_y + \beta_y x \\
\mu_x &amp;= \alpha_x + \beta_x y
\end{aligned}
\]</span></p>
</section>
<section id="code" class="slide level2">
<h2>Code</h2>
</section>
<section id="modifications" class="slide level2">
<h2>Modifications</h2>
<ul>
<li><p>Student t?</p></li>
<li><p>Use small panel data in regression for large panel imputation?</p></li>
<li><p>Estimation correlation between <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span>?</p></li>
</ul>

<img src="omg_logo.png" class="slide-logo r-stretch"><div class="footer footer-default">

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