wip

2022-10-01 00:25:34 -07:00
parent 47caee3ddd
commit 8e18e3ddc1
24 changed files with 3225 additions and 5 deletions
--- a/R/chapter-3-beta-binomial/ch3-beta-binomial.html
+++ b/R/chapter-3-beta-binomial/ch3-beta-binomial.html
@@ -148,7 +148,8 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni

 <div class="cell">
 <div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(bayesrules)</span>
-<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span>
+<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(patchwork)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </div>
 <p>The chapter is set up with an example of polling results. We are put into the scenario where we are managig the campaing for a candidate. We know that on average her support based on recent polls is around 45%. In the next few sections we’ll work through our Bayesian framework and incorporate a new tool the <strong>Beta-Binomial</strong> model. This model will take develop a continuous prior, as opposed to the discrete one’s we’ve been working with so far.</p>
 <section id="the-beta-prior" class="level2">
@@ -229,7 +230,7 @@ The Standard Uniform
 <div class="callout-body-container callout-body">
 <p>When <span class="math inline">\pi</span> can take equally take on any value between 0 and 1, we can model <span class="math inline">\pi</span> using the standard uniform model.</p>
 <p><span class="math display">\pi \sim Unif(0, 1)</span></p>
-<p>the pdf of <span class="math inline">Unif(0, 1)</span> is <span class="math inline">\f(\pi) = 1</span></p>
+<p>the pdf of <span class="math inline">Unif(0, 1)</span> is <span class="math inline">f(\pi) = 1</span></p>
 <p>Note that <span class="math inline">Unif(0, 1)</span> is just a special case of the Beta with hyperparameters <span class="math inline">\alpha = \beta = 1</span>, see <a href="#fig-std-unif-as-beta">Figure&nbsp;2</a></p>
 </div>
 </div>
@@ -282,6 +283,136 @@ The Standard Uniform
 <p>it follows that</p>
 <p><span class="math display">SD = \sqrt{Var(\pi)}</span></p>
 </section>
+<section id="tuning-the-beta-prior" class="level3">
+<h3 class="anchored" data-anchor-id="tuning-the-beta-prior">Tuning the beta prior</h3>
+<p>Now that we know we can use the beta model to represent <span class="math inline">\pi</span> for <span class="math inline">\pi \in [0, 1]</span>, what we need to try and do is tune the model so that it best represents our prior. Our options of course are chaning values of <span class="math inline">\alpha</span> and <span class="math inline">\beta</span>. We can make use of the fact that we know that on average the candidate is polling at around 45%. That is we know that</p>
+<p><span class="math display">E(\pi) \approx .45 \approx \frac{\alpha}{\alpha + \beta}</span></p>
+<p><em>so I tried to work through the algebra myself, but was unable to do so, the book gets to the following conclusion</em></p>
+<p><span class="math display">\alpha = \frac{9}{11}\beta</span></p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>p <span class="ot">&lt;-</span> <span class="fu">plot_beta</span>(<span class="dv">9</span>, <span class="dv">11</span>) <span class="sc">+</span> </span>
+<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>    <span class="fu">labs</span>(<span class="at">title=</span><span class="st">"Beta(9,11)"</span>) <span class="sc">+</span> </span>
+<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a>    <span class="fu">scale_x_continuous</span>(<span class="at">breaks=</span><span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="at">by=</span>.<span class="dv">1</span>))</span>
+<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a>p2 <span class="ot">&lt;-</span> <span class="fu">plot_beta</span>(<span class="dv">27</span>, <span class="dv">33</span>) <span class="sc">+</span> </span>
+<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a>    <span class="fu">labs</span>(<span class="at">title=</span><span class="st">"Beta(27,33)"</span>) <span class="sc">+</span> </span>
+<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a>    <span class="fu">scale_x_continuous</span>(<span class="at">breaks=</span><span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="at">by=</span>.<span class="dv">1</span>))</span>
+<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a>p3 <span class="ot">&lt;-</span> <span class="fu">plot_beta</span>(<span class="dv">45</span>, <span class="dv">55</span>) <span class="sc">+</span> </span>
+<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a>    <span class="fu">labs</span>(<span class="at">title=</span><span class="st">"Beta(45,55)"</span>) <span class="sc">+</span> </span>
+<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a>    <span class="fu">scale_x_continuous</span>(<span class="at">breaks=</span><span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="at">by=</span>.<span class="dv">1</span>))</span>
+<span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a>p <span class="sc">/</span> p2 <span class="sc">/</span> p3</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<p><img src="ch3-beta-binomial_files/figure-html/unnamed-chunk-6-1.png" class="img-fluid" width="672"></p>
+</div>
+</div>
+<p>when compared to the original polling values, we can see that the closest of these is <span class="math inline">Beta(45, 55)</span></p>
+<p>we can get the pdf, and then mean, mode and variance of the pdf.</p>
+<p><span class="math display">f(\pi) = \frac{\Gamma(100)}{\Gamma(45)\Gamma(55)}\pi^{44}(1-\pi)^{54}</span></p>
+<p><span class="math display">E(\pi) = \frac{45}{45 + 55} = .4500</span> <span class="math display">\text{Mode}(\pi) = \frac{45 - 1}{45 + 55 - 2} = .449</span> <span class="math display"> SD(\pi) = .05</span></p>
+<div class="cell" data-fig.caption="the prior distrubtions with Y = 30 hihglighted">
+<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a>rb1 <span class="ot">&lt;-</span> <span class="fu">rbinom</span>(<span class="dv">10000</span>, <span class="dv">50</span>, .<span class="dv">2</span>)</span>
+<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a>rb2 <span class="ot">&lt;-</span> <span class="fu">rbinom</span>(<span class="dv">10000</span>, <span class="dv">50</span>, .<span class="dv">3</span>)</span>
+<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a>rb3 <span class="ot">&lt;-</span> <span class="fu">rbinom</span>(<span class="dv">10000</span>, <span class="dv">50</span>, .<span class="dv">5</span>)</span>
+<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a>rb4 <span class="ot">&lt;-</span> <span class="fu">rbinom</span>(<span class="dv">10000</span>, <span class="dv">50</span>, .<span class="dv">6</span>)</span>
+<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a>rb5 <span class="ot">&lt;-</span> <span class="fu">rbinom</span>(<span class="dv">10000</span>, <span class="dv">50</span>, .<span class="dv">7</span>)</span>
+<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a>rb6 <span class="ot">&lt;-</span> <span class="fu">rbinom</span>(<span class="dv">10000</span>, <span class="dv">50</span>, .<span class="dv">8</span>)</span>
+<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a>results <span class="ot">&lt;-</span> <span class="fu">tibble</span>(</span>
+<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">Bin(50, .2)</span><span class="st">`</span> <span class="ot">=</span> rb1,</span>
+<span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">Bin(50, .3)</span><span class="st">`</span> <span class="ot">=</span> rb2,</span>
+<span id="cb7-11"><a href="#cb7-11" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">Bin(50, .5)</span><span class="st">`</span> <span class="ot">=</span> rb3,</span>
+<span id="cb7-12"><a href="#cb7-12" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">Bin(50, .6)</span><span class="st">`</span> <span class="ot">=</span> rb4,</span>
+<span id="cb7-13"><a href="#cb7-13" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">Bin(50, .7)</span><span class="st">`</span> <span class="ot">=</span> rb5,</span>
+<span id="cb7-14"><a href="#cb7-14" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">Bin(50, .8)</span><span class="st">`</span> <span class="ot">=</span> rb6,</span>
+<span id="cb7-15"><a href="#cb7-15" aria-hidden="true" tabindex="-1"></a>) <span class="sc">|&gt;</span></span>
+<span id="cb7-16"><a href="#cb7-16" aria-hidden="true" tabindex="-1"></a>    <span class="fu">pivot_longer</span>(<span class="at">names_to =</span> <span class="st">"bin_model"</span>, <span class="at">values_to =</span> <span class="st">"value"</span>, </span>
+<span id="cb7-17"><a href="#cb7-17" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">Bin(50, .2)</span><span class="st">`</span><span class="sc">:</span><span class="st">`</span><span class="at">Bin(50, .8)</span><span class="st">`</span>) <span class="sc">|&gt;</span></span>
+<span id="cb7-18"><a href="#cb7-18" aria-hidden="true" tabindex="-1"></a>    <span class="fu">mutate</span>(<span class="at">is_30 =</span> value <span class="sc">==</span> <span class="dv">30</span>)</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>results <span class="sc">|&gt;</span></span>
+<span id="cb7-21"><a href="#cb7-21" aria-hidden="true" tabindex="-1"></a>    <span class="fu">ggplot</span>(<span class="fu">aes</span>(value, <span class="at">fill=</span>is_30)) <span class="sc">+</span> <span class="fu">geom_bar</span>(<span class="at">stat =</span> <span class="st">"count"</span>) <span class="sc">+</span></span>
+<span id="cb7-22"><a href="#cb7-22" aria-hidden="true" tabindex="-1"></a>    <span class="fu">scale_fill_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="st">"grey30"</span>, <span class="st">"darkblue"</span>), <span class="at">guide=</span><span class="st">"none"</span>) <span class="sc">+</span> </span>
+<span id="cb7-23"><a href="#cb7-23" aria-hidden="true" tabindex="-1"></a>    <span class="fu">facet_wrap</span>(<span class="fu">vars</span>(bin_model))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div id="fig-betas-for-poll" class="quarto-figure quarto-figure-center anchored">
+<figure class="figure">
+<p><img src="ch3-beta-binomial_files/figure-html/fig-betas-for-poll-1.png" class="img-fluid figure-img" width="672"></p>
+<p></p><figcaption class="figure-caption">Figure&nbsp;4: <strong>?(caption)</strong></figcaption><p></p>
+</figure>
+</div>
+</div>
+</div>
+<p>After the finish the poll, we see that the actual value of <span class="math inline">Y</span> is 30. The vertical line in <a href="#fig-betas-for-poll">Figure&nbsp;4</a></p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a>results <span class="ot">&lt;-</span> <span class="fu">tibble</span>(</span>
+<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>    <span class="at">ys =</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">50</span>, </span>
+<span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">beta(50, .1)</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">dbinom</span>(<span class="at">x =</span> ys, <span class="at">size =</span> <span class="dv">50</span>, <span class="at">prob =</span> .<span class="dv">1</span>),</span>
+<span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">beta(50, .2)</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">dbinom</span>(<span class="at">x =</span> ys, <span class="at">size =</span> <span class="dv">50</span>, <span class="at">prob =</span> .<span class="dv">2</span>),</span>
+<span id="cb8-5"><a href="#cb8-5" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">beta(50, .3)</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">dbinom</span>(<span class="at">x =</span> ys, <span class="at">size =</span> <span class="dv">50</span>, <span class="at">prob =</span> .<span class="dv">3</span>),</span>
+<span id="cb8-6"><a href="#cb8-6" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">beta(50, .4)</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">dbinom</span>(<span class="at">x =</span> ys, <span class="at">size =</span> <span class="dv">50</span>, <span class="at">prob =</span> .<span class="dv">4</span>),</span>
+<span id="cb8-7"><a href="#cb8-7" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">beta(50, .5)</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">dbinom</span>(<span class="at">x =</span> ys, <span class="at">size =</span> <span class="dv">50</span>, <span class="at">prob =</span> .<span class="dv">5</span>),</span>
+<span id="cb8-8"><a href="#cb8-8" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">beta(50, .6)</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">dbinom</span>(<span class="at">x =</span> ys, <span class="at">size =</span> <span class="dv">50</span>, <span class="at">prob =</span> .<span class="dv">6</span>),</span>
+<span id="cb8-9"><a href="#cb8-9" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">beta(50, .7)</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">dbinom</span>(<span class="at">x =</span> ys, <span class="at">size =</span> <span class="dv">50</span>, <span class="at">prob =</span> .<span class="dv">7</span>),</span>
+<span id="cb8-10"><a href="#cb8-10" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">beta(50, .8)</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">dbinom</span>(<span class="at">x =</span> ys, <span class="at">size =</span> <span class="dv">50</span>, <span class="at">prob =</span> .<span class="dv">8</span>),</span>
+<span id="cb8-11"><a href="#cb8-11" aria-hidden="true" tabindex="-1"></a>    <span class="st">`</span><span class="at">beta(50, .9)</span><span class="st">`</span> <span class="ot">=</span> <span class="fu">dbinom</span>(<span class="at">x =</span> ys, <span class="at">size =</span> <span class="dv">50</span>, <span class="at">prob =</span> .<span class="dv">9</span>),</span>
+<span id="cb8-12"><a href="#cb8-12" aria-hidden="true" tabindex="-1"></a>) <span class="sc">|&gt;</span></span>
+<span id="cb8-13"><a href="#cb8-13" aria-hidden="true" tabindex="-1"></a>    <span class="fu">pivot_longer</span>(<span class="at">names_to =</span> <span class="st">"beta_model"</span>, <span class="at">values_to=</span><span class="st">"value"</span>, <span class="sc">-</span>ys) <span class="sc">|&gt;</span></span>
+<span id="cb8-14"><a href="#cb8-14" aria-hidden="true" tabindex="-1"></a>    <span class="fu">mutate</span>(</span>
+<span id="cb8-15"><a href="#cb8-15" aria-hidden="true" tabindex="-1"></a>        <span class="at">is_30 =</span> ys <span class="sc">==</span> <span class="dv">30</span>, </span>
+<span id="cb8-16"><a href="#cb8-16" aria-hidden="true" tabindex="-1"></a>        <span class="at">pie =</span> <span class="fu">rep</span>(<span class="fu">c</span>(.<span class="dv">1</span>, .<span class="dv">2</span>, .<span class="dv">3</span>, .<span class="dv">4</span>, .<span class="dv">5</span>, .<span class="dv">6</span>, .<span class="dv">7</span>, .<span class="dv">8</span>, .<span class="dv">9</span>), <span class="dv">50</span>))</span>
+<span id="cb8-17"><a href="#cb8-17" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb8-18"><a href="#cb8-18" aria-hidden="true" tabindex="-1"></a>results <span class="sc">|&gt;</span> <span class="fu">ggplot</span>(<span class="fu">aes</span>(ys, value, <span class="at">fill =</span> is_30)) <span class="sc">+</span> </span>
+<span id="cb8-19"><a href="#cb8-19" aria-hidden="true" tabindex="-1"></a>    <span class="fu">geom_col</span>() <span class="sc">+</span> </span>
+<span id="cb8-20"><a href="#cb8-20" aria-hidden="true" tabindex="-1"></a>    <span class="fu">scale_fill_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="st">"darkgray"</span>, <span class="st">"darkblue"</span>), <span class="at">guide=</span><span class="st">"none"</span>) <span class="sc">+</span> </span>
+<span id="cb8-21"><a href="#cb8-21" aria-hidden="true" tabindex="-1"></a>    <span class="fu">facet_wrap</span>(<span class="fu">vars</span>(beta_model))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<p><img src="ch3-beta-binomial_files/figure-html/unnamed-chunk-8-1.png" class="img-fluid" width="672"></p>
+</div>
+</div>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>results <span class="sc">|&gt;</span></span>
+<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a>    <span class="fu">filter</span>(is_30) <span class="sc">|&gt;</span></span>
+<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a>    <span class="fu">ggplot</span>(<span class="fu">aes</span>(pie, value)) <span class="sc">+</span> <span class="fu">geom_col</span>() <span class="sc">+</span> </span>
+<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a>    <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="fu">seq</span>(.<span class="dv">1</span>, .<span class="dv">9</span>, <span class="at">by =</span> .<span class="dv">1</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<p><img src="ch3-beta-binomial_files/figure-html/unnamed-chunk-9-1.png" class="img-fluid" width="672"></p>
+</div>
+</div>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summarize_beta_binomial</span>(<span class="at">alpha=</span><span class="dv">45</span>, <span class="at">beta=</span><span class="dv">55</span>, <span class="at">n=</span><span class="dv">50</span>, <span class="at">y=</span><span class="dv">30</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>      model alpha beta mean      mode         var         sd
+1     prior    45   55 0.45 0.4489796 0.002450495 0.04950248
+2 posterior    75   75 0.50 0.5000000 0.001655629 0.04068942</code></pre>
+</div>
+</div>
+</section>
+</section>
+<section id="simulation" class="level2">
+<h2 class="anchored" data-anchor-id="simulation">Simulation</h2>
+<p>In this simulation we first generate 10,000 random values from a beta distribution with parameter <span class="math inline">\alpha = 45, \beta = 55</span>. Recall, that this is what we believe to be support based on recent polls for the candidate, and therefore our prior. We use each one of these are the <code>prob</code> value (the <span class="math inline">\pi</span> value) into 10,000 more random this time from the binomial distribution with <span class="math inline">n = 50</span>. This binomial distribution is the model for <span class="math inline">Y</span> conditioned on <span class="math inline">\pi</span>.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>sim <span class="ot">&lt;-</span> <span class="fu">tibble</span>(</span>
+<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a>    <span class="at">pi =</span> <span class="fu">rbeta</span>(<span class="dv">10000</span>, <span class="dv">45</span>, <span class="dv">55</span>)</span>
+<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a>) <span class="sc">|&gt;</span></span>
+<span id="cb12-4"><a href="#cb12-4" aria-hidden="true" tabindex="-1"></a>    <span class="fu">mutate</span>(<span class="at">y =</span> <span class="fu">rbinom</span>(<span class="dv">10000</span>, <span class="at">size =</span> <span class="dv">50</span>, <span class="at">prob =</span> pi))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a>sim <span class="sc">|&gt;</span></span>
+<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a>    <span class="fu">ggplot</span>(<span class="fu">aes</span>(pi, y)) <span class="sc">+</span> <span class="fu">geom_point</span>(<span class="fu">aes</span>(<span class="at">color =</span> (y <span class="sc">==</span> <span class="dv">30</span>))) <span class="sc">+</span> </span>
+<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a>    <span class="fu">scale_color_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="st">"darkgrey"</span>, <span class="st">"navyblue"</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<p><img src="ch3-beta-binomial_files/figure-html/unnamed-chunk-12-1.png" class="img-fluid" width="672"></p>
+</div>
+</div>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>sim <span class="sc">|&gt;</span></span>
+<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a>    <span class="fu">filter</span>(y <span class="sc">==</span> <span class="dv">30</span>) <span class="sc">|&gt;</span> <span class="co"># the posterior</span></span>
+<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a>    <span class="fu">ggplot</span>(<span class="fu">aes</span>(pi)) <span class="sc">+</span> <span class="fu">geom_density</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<p><img src="ch3-beta-binomial_files/figure-html/unnamed-chunk-13-1.png" class="img-fluid" width="672"></p>
+</div>
+</div>
 </section>

 </main>