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<h1 class="title">Chapter 3 Beta-Binomial Bayesian Model Notes</h1>
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<div class="quarto-title-meta">
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<div class="quarto-title-meta-heading">Author</div>
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<p>Emanuel Rodriguez </p>
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<div class="cell">
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<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>
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<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>
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<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>
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<section id="the-beta-prior" class="level2">
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<h2 class="anchored" data-anchor-id="the-beta-prior">The Beta prior</h2>
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Probability Density Function
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</div>
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</div>
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<p>Let <span class="math inline">\pi</span> be a continuous random variable with probability density function (pdf) <span class="math inline">f(\pi)</span>. Then <span class="math inline">f(\pi)</span> has the following properties:</p>
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<ol type="1">
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<li><span class="math inline">f(\pi) \geq 0</span></li>
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<li><span class="math inline">\int_{\pi}f(\pi)d\pi = 1</span> (this is analogous to <span class="math inline">\sum</span> in the case of pmfs)</li>
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<li><span class="math inline">P(a < \pi < b) = \int_a^bf(\pi)d\pi</span> when <span class="math inline">a\leq b</span></li>
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Tip
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<div class="callout-body-container callout-body">
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<p>a quick note on (1) above. Note that it does not place a restriction on <span class="math inline">f(\pi)</span> being less than 1. This means that we can’t interpret values of <span class="math inline">f</span> as probabilities, we can however use to interpret plausability of two different events, the greater the value of <span class="math inline">f</span> the more plausible. To calculate probabilities using <span class="math inline">f</span> we must determine the area under the curve it defines, as shown in (3).</p>
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