Name: -------------------------------------------Bayesian Statistics, 22S:138Midterm 2, 2010Eight college statistics majors are playing basketball together for the first time. Just forthe fun of it, they decide to use Bayesian methods to estimate each individual student’ssuccess probability of making a basket from 10 feet away, as well as the overall averagesuccess probability of the group. They gather data in the f ollowing way:One at a time, each student stands at a position 10 feet from the basket and keeps shootinguntil he finally makes a basket. Another student records xi– how many failures shooter ihad before successfully making his first basket.The geometric probability mass function, which we met in midterm 1, is app ropriate formod eling each student’s number of failures before the first success. The students have neverplayed together before, so they don’t have any knowledge about who the go od shootersand poor sho oters might be. Thus, in their Bayesian model, they consider the successprobabilities pi, i = 1, . . . , 8 to be random draws from a common Beta d en sity. Theycomplete their Bayesian model by specifying priors on the parameters of th e Beta density.OpenBUGS co de and output for fitting the students’ model to their data are attached.Note that the geometric distribution is a special case of the negative binomial distribution,namely negative binomial with the second parameter equal to 1. Thus,x[i] ~ dnegbin(p[i], 1)says that xiis drawn from a geometric distrib ution with parameter pi.1. On the OpenBUGS code, indicate which line or lines represent the second stage ofthe model.2. What quantity in the OpenBUGS model should be monitored to get samples from theposterior den sity of the overall average success probability? If it’s already named inthe model code, just write the n ame here. If not, in the OpenBUGS code itself, writethe line(s) that should b e added to define the quantity, and then write the name here.3. If the students had used a Gamma(1,2) prior on the parameter beta instead ofGamma(1,1), would that have been likely to make any difference in the resultingposterior means of the individual pis? Explain briefly.14. Three plots are included in the OpenBUGS output provided. Refer to them in an-swering these questions.(a) The autocorrelation between values of the beta parameter drawn 50 iterationsapart is closest to (circle one):i. 1.0ii. 0.5iii. 0.0iv. -0.5v. -1.0vi. Plots give no information on this.(b) How many iterations would you discard as burn-in? Exp lain h ow you decided.5. What is the 95% credible set for p2? (Give numeric values from OpenBUGS ou tp ut).6. Student 2 and Student 8 have exactly the same data values: x2= x8= 0. But theestimated posterior m eans shown in the OpenBUGS output for p2and p8are notequal. Why might that be the case?7. If the data for student number 2 was analyzed separately, the frequentist mle wouldbe ˆp = 1. However, the Bayesian posterior mean for p2from this hierarchical modelis only about 0.55. This is an example of a phenomenon in hierarchical models called(circle one):(a) exchangeability(b) invariance to transf ormations(c) shrinkage(d) multiple stages(e) none of the above28. In specifying their mo del, the students treated all the pis as random draws from thesame Beta density. This shows that they considered the pis to be (circle one):(a) exchangeable(b) invariant to transformations(c) marginally indepen dent(d) nuisance parameters(e) none of the above9. Draw a directed graph of the students’
View Full Document
Unlocking...