26 Homework Alternative Priors - M4L11HW1

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--- title: "Homework Alternative Priors - M4L11HW1" subtitle: "Bayesian Statistics: From Concept to Data Analysis" categories: - Bayesian Statistics keywords: - Exponential Distribution - Homework --- ::::: {.content-visible unless-profile="HC"} ::: {.callout-caution} Section omitted to comply with the Honor Code ::: ::::: ::::: {.content-hidden unless-profile="HC"} ::: {#exr-alt-priors-1} Suppose we flip a coin five times to estimate $\theta$, the probability of obtaining heads. We use a Bernoulli likelihood for the data and a non-informative (and improper) $\mathrm{Beta}(0,0)$ prior for $\theta$. We observe the following sequence: $(H, H, H, T, H)$. Because we observed at least one H and at least one T, the posterior is proper. What is the posterior distribution for $\theta$ ::: ::: {.solution .column-screen-inset-right .callout-tip collapse="true"} #### Solution: Beta(4,1) We observed four "successes" and one "failure," and these counts are the parameters of the posterior beta distribution. ::: ::: {#exr-alt-priors-2} Continuing the previous question, what is the posterior mean for $\theta$ ? Round your answer to one decimal place. ::: ::: {.solution .column-screen-inset-right .callout-tip collapse="true"} #### Solution: $\bar{y} = 0.8 $ ::: ::: {#exr-alt-priors-3} Consider again the thermometer calibration problem from Lesson 10. Assume a normal likelihood with unknown mean $\theta$ and known variance $\sigma^2=0.25$. Now use the non-informative (and improper) flat prior for $\theta$ across all real numbers. This is equivalent to a conjugate normal prior with variance equal to $\infty$. You collect the following $n=5$ measurements: (94.6, 95.4, 96.2, 94.9, 95.9). What is the posterior distribution for $\theta$? ::: ::: {.solution .column-screen-inset-right .callout-tip collapse="true"} #### Solution: Recall from the lesson that with a flat prior on $\theta$ , the posterior distribution is $$ \mathcal{N}(\bar{y},\frac{\sigma^2}{n})=\mathcal{N}(95.4,0.05) $$ ::: ::: {#exr-alt-priors-4} Which of the following graphs shows the Jeffreys prior for a Bernoulli/binomial success probability p? Hint: The Jeffreys prior in this case is $\mathrm{Beta}(1/2, 1/2)$. ::: ::: {.solution .column-screen-inset-right .callout-tip collapse="true"} #### Solution: Beta(1/2, 1/2). ::: ::: {#exr-alt-priors-5} Scientist A studies the probability of a certain outcome of an experiment and calls it $\theta$ . To be non-informative, he assumes a $Uniform(0,1)$ prior for $\theta$ . Scientist B studies the same outcome of the same experiment using the same data, but wishes to model the odds $\phi= \frac{\theta}{1−\theta}$. Scientist B places a uniform distribution on $\phi$. If she reports her inferences in terms of the probability $\theta$, will they be equivalent to the inferences made by Scientist A? ::: ::: {.solution .column-screen-inset-right .callout-tip collapse="true"} #### Solution: No, they did not use the Jeffreys prior. The uniform prior on $\theta$ implies the following prior PDF for $$ f(\phi)= \frac{1}{(1+\phi)^2} I_{\{\phi≥0\}} $$ which is not the uniform prior used by Scientist B. They would obtain equivalent inferences if they both use the Jeffrey's prior. ::: :::::