2. [Jan. 13 Choose a conjugate prior setting and investigate the performance of credible intervals (let's say at the 80% level) in this setting by doing the following. (a) Demonstrate that for some combination of little information in the data (i.e., a small sample size) and non-negligible information in the prior (e.g., not a uniform distribution as the prior), the frequentist coverage will be wrong. More specifically, repeatedly generate datasets for a true parameter value in the tail of the prior, and verify that the proportion of 80% credible intervals containing this true value is less than the nominal 80%. Similarly, show that higher-than-nominal coverage obtains for a true value 'in the middle' of the prior. (b) Now take the setting you chose in part (a), and investigate a different kind of coverage. Repeatedly sample a (parameter,dataset) pair by sampling a parameter value from the prior distribution and then sampling a dataset from the model using this parameter value. What do you find about the proportion of 80% credible intervals containing the parameter value in this kind of repeated sampling? (You could regard this kind of repeated sampling as corresponding to a series of studies of different phenomena, so that the true parameter value is different for each study.) NOTE: When using simulation, please be mindful to control and/or report "simulation error.' or "simulation variability' if you prefer. Sometimes statisticians are quick to tell other kinds of scientists to control/report variability, but slow to do this in their own domain