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<h2 class="hd hd-2 unit-title">Bayesian updating with discrete priors</h2>
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<h2 class="hd hd-2 unit-title">Class 11 reading questions: Bayesian updating</h2>
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<p><b class="bf">1.</b> Recall the coin example from the reading: there are three types of coins which are indistinguishable apart from their probability of landing heads when tossed. </p>
<ul class="itemize">
<li>
<p>
Type [mathjaxinline]A[/mathjaxinline] coins are fair, with probability 0.5 of heads </p>
</li>
<li>
<p>
Type [mathjaxinline]B[/mathjaxinline] coins have probability 0.6 of heads </p>
</li>
<li>
<p>
Type [mathjaxinline]C[/mathjaxinline] coins have probability 0.9 of heads </p>
</li>
</ul>
<p>
You have a drawer containing 4 coins: 2 of type [mathjaxinline]A[/mathjaxinline], 1 of type [mathjaxinline]B[/mathjaxinline], and 1 of type [mathjaxinline]C[/mathjaxinline]. You reach into the drawer and pick a coin at random. </p>
<p>
Suppose you toss the coin 5 times and it lands tails all 5 times. What are the posterior probabilities? </p>
<p>
Hint: do the updating in one go, i.e. the data is 5 tails and the likelihood is [mathjaxinline]P(5 \text { tails } | \text { type of coin})[/mathjaxinline]. </p>
<p>
We label the data [mathjaxinline]{\cal D}[/mathjaxinline]. Give your answer to 3 significant digits. </p>
<p>
[mathjaxinline]P(A|{\cal D})[/mathjaxinline] <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_c11-rq11_2_1" class=" capa_inputtype textline">
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[mathjaxinline]P(B|{\cal D})[/mathjaxinline] <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_c11-rq11_3_1" class=" capa_inputtype textline">
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<p><b class="bf">2. Stubborn priors.</b> Suppose you run an experiment to test 3 different hypotheses: [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline]. You are so convinced that [mathjaxinline]B[/mathjaxinline] is the correct hypothesis that your prior is </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]P(A) = 0 \qquad P(B) = 1 \qquad P(C) = 0.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
You collect data [mathjaxinline]{\cal D}[/mathjaxinline] with likelihoods </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]P({\cal D}|A) = 0.8 \qquad P({\cal D}|B) = 0.01 \qquad P({\cal D}|C) = 0.7[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p><b class="bf">2(a)</b> Compute the posterior probabilities: </p>
<p>
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<p><b class="bf">2(b)</b> Will any amount of data change your belief that [mathjaxinline]B[/mathjaxinline] is the correct hypothesis? </p>
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