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<h2 class="hd hd-2 unit-title">Conjugate priors: beta, normal</h2>
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<h2 class="hd hd-2 unit-title">Class 15 reading questions: conjugate priors; beta</h2>
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<p><b class="bf">1.</b> Data [mathjaxinline]x[/mathjaxinline] follows a binomial(12,&#8201;[mathjaxinline]\theta[/mathjaxinline]) distribution. To begin with you have no idea about [mathjaxinline]\theta[/mathjaxinline] so you choose a flat prior. You run and experiment and find [mathjaxinline]x=3[/mathjaxinline]. What is the posterior distribution for [mathjaxinline]\theta[/mathjaxinline]? </p>
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<text> binomial(12,3)</text>
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<text> binomial(15,[mathjaxinline]\theta[/mathjaxinline])</text>
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<text> Beta(3,12)</text>
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<text> Beta(4,10)</text>
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<p><b class="bf">2.</b> True or false: Setting the prior probability of a hypothesis to 0 means that no amount of data will make the posterior probability of that hypothesis the maximum over all hypotheses. </p>
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Problem 3
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<p><b class="bf">3.</b> True or false: It is okay to have a prior that depends on more than one unknown parameter. </p>
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<h2 class="hd hd-2 unit-title">Class 15 reading questions: conjugate priors; choosing priors</h2>
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Problem 1
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<p><b class="bf">1.</b> Suppose our data [mathjaxinline]y \sim \text {N}(\theta ,6)[/mathjaxinline] where [mathjaxinline]\theta[/mathjaxinline] is unknown. (Remember, in our notation 6 is the variance.) </p>
<p>
Also suppose that our prior for [mathjaxinline]\theta[/mathjaxinline] is [mathjaxinline]\text {N}(2,10)[/mathjaxinline]. </p>
<p>
You run a five of experiments and collect the following data [mathjaxinline]2, 1, 4, -1, -1[/mathjaxinline]. </p>
<p>
Your posterior pdf for [mathjaxinline]\theta[/mathjaxinline] is [mathjaxinline]\text {N}(\mu _{\text {post}},\sigma ^2_{\text {post}})[/mathjaxinline]. What are [mathjaxinline]\mu _{\text {post}}[/mathjaxinline] and [mathjaxinline]\sigma ^2_{\text {post}}[/mathjaxinline]? </p>
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[mathjaxinline]\mu _{\text {post}}[/mathjaxinline]: <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_c15-rq21_2_1" class=" capa_inputtype textline">
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<p><b class="bf">2.</b> Suppose we have a measurement [mathjaxinline]x \sim N(\theta , \sigma ^2)[/mathjaxinline] where [mathjaxinline]\theta[/mathjaxinline] is our parameter of interest and the variance [mathjaxinline]\sigma ^2[/mathjaxinline] is known. Suppose also that the prior pdf on [mathjaxinline]\theta[/mathjaxinline] is normal. If the prior pdf is shown in blue and the single data point is at the red line then what is the posterior pdf? </p>
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