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daslu committed Sep 21, 2023
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<!DOCTYPE html>
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<title>Probability basics for Bayesian analysis</title>
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<span id="cb5-2"><a href="#cb5-2"></a><span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x=</span>X, <span class="at">y=</span>Y)) <span class="sc">+</span> <span class="fu">geom_point</span>(<span class="at">size=</span><span class="dv">10</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>

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<section id="random-vectors" class="slide level2">
<h2>Random vectors</h2>
<p><span class="math inline">\((X,Y)\)</span> may be considered a random vector, viewed as a function <span class="math inline">\(\Omega \to \mathbb{R}^2\)</span> <span class="math display">\[\omega \mapsto (X(\omega), Y(\omega))\]</span></p>
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<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1"></a>Omega_XY_ <span class="sc">%&gt;%</span> <span class="fu">helper_plot_X_probs</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>

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<section id="distribution-examples---cont." class="slide level2">
<h2>Distribution examples - cont.</h2>
<div class="cell">
<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1"></a>Omega_XY_ <span class="sc">%&gt;%</span> <span class="fu">helper_plot_Y_probs</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>

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<section id="joint-distribution" class="slide level2">
<h2>Joint distribution</h2>
<p>Similarily, the distribution of a random vector <span class="math inline">\((X,Y)\)</span> is a probability measure over the plane <span class="math inline">\(\mathbb{R^2}\)</span>.</p>
Expand All @@ -1520,7 +1520,7 @@ <h2>Joint distribution</h2>
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1"></a>Omega_XY_ <span class="sc">%&gt;%</span> <span class="fu">helper_plot_XY_probs</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>

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<section id="types-of-distribution" class="slide level2">
<h2>Types of distribution</h2>
<p>A probability distribution <span class="math inline">\(P\)</span> is</p>
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<span id="cb14-7"><a href="#cb14-7"></a> <span class="fu">xlim</span>(<span class="dv">0</span>,<span class="dv">5</span>) <span class="sc">+</span> <span class="fu">ylim</span>(<span class="dv">0</span>,<span class="dv">10</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>

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<section>
<section id="expectation" class="title-slide slide level1 center">
<h1>Expectation</h1>
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<span id="cb15-3"><a href="#cb15-3"></a>helper_plot_X_probs</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>

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<section id="conditional-expectation-given-an-event" class="slide level2">
<h2>Conditional expectation given an event</h2>
<p>Given a random variable <span class="math inline">\(X:\Omega \to \mathbb{R}\)</span> and an event <span class="math inline">\(B\)</span> such that <span class="math inline">\(\mathbb{P}(B)&gt;0\)</span>, the conditional expectation <span class="math inline">\(\mathbb{E}(X \vert B)\)</span> is defined as the expectation of the conditional distribution.</p>
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