Please use D2L to turn in both the PDF output and your R Markdown file.

Q1. (25 pts)

For this question, consider modeling bat counts using a Poisson distribution as a sampling model.

a. Prior (5 pts)

Select and justify a prior for this setting.

b. Posterior (10 pts)

Using nightly bat counts recorded over a two week period (below), compute a posterior distribution for the mean term in the Poisson distribution.

bat.counts <- c(3,3,92,101,54,5,28,2,0,0,6,0)

Please formally write out this distribution and create a figure to visualize the distribution.

c. Posterior Predictive Distribution (10 pts)

Create and plot your posterior predictive distribution. Use this to compute:

Now do a posterior predictive check by comparing your posterior predictive distribution with the observed data. Please indicate and justify whether you feel your model is adequate.

Q2.

After enjoying numerous Bridger Bowl powder days during your time at MSU, you have received a job offer in New Mexico. You decision hinges on the quality of snow at your potential new home mountain, Taos.

According to https://www.onthesnow.com/new-mexico/taos-ski-valley/historical-snowfall.html annual snowfall totals can be obtained.

Below are the snowfall totals for Taos and Bridger Bowl for the last nine years.

taos <- c(78,192,169,179,191,204,197,116,195)
bridger <- c(271,209,228,166,316,254,344,319,247) 

Specifically the BSF is interested in computing three probabilistic statements.

  1. \(Pr[\theta_{bridger} > \theta_{taos}]\) where \(\theta_{bridger}\) is the mean annual snow fall at Bridger Bowl and \(\theta_{taos}\) is the mean annual snow fall at Taos.
  2. \(Pr[\theta_{bridger} > 250]\)
  3. \(Pr[\theta_{taos} > 200]\)

a. (5 pts)

How would you go about addressing the questions in a classical framework? Would you be able to compute these probabilities?

b. (20 pts)

Using the prior structure where \(p(\sigma_{bridger}^2,\theta_{bridger}) = p(\theta_{bridger}|\sigma_{bridger}^2) p(\sigma_{bridger}^2)\) and \(p(\sigma_{taos}^2,\theta_{taos}) = p(\theta_{taos}|\sigma_{taos}^2) p(\sigma_{taos}^2)\) compute the marginal posterior distributions \(p(\theta_{bridger}|y_{{bridger},1}, \dots, y_{{bridger},15})\) and \\(p(\theta_{taos}|y_{{taos},1}, \dots, y_{{taos},9})\), where \(\sigma_1^2\) and \(\sigma_{taos}^2\) are the variances for snowfall (in inches) a, and \(y_{i,j}\) is the observed snowfall at location \(i\) for reading \(j\). Then using posterior samples from each distribution compute the three values specified above.