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UI STAT 4520 - Bayesian Statistics

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Name: ______________________________________Bayesian Statistics, 22S:138Fall 2007, Instructor: CowlesFinal examPRACTICE PROBLEMS for 2008 FINAL EXAMShow any computations that you carry out.1. The Humane Society of the United States reports the following statistics r egarding dogownership in the U.S.:• Four in ten (or 40,000,000) hous eh olds own at least one dog.• 63% of households with dogs have only one dog.• 24% of households with dogs have two dogs.• 13% of households with dogs have three or more dogs.(a) What is the probability that a household selected at random from among all U.S.households will own two or more dogs?2. For each of the following cases, name the family of distributions (e.g. beta, binomial,gamma, normal, poisson) that is most likely to b e appropriate for the first stage of aBayesian model (if you do not transform the variable in any way). Briefly justify yourchoice.(a) A rand om sample of 100 households in Iowa is selected, and the variable measuredon each household is how many people live in the household.(b) A random sample of 25 Illinois counties is selected, and the var iable measured on eachcounty is what proportion of children under age 18 are covered by health insurance.(c) A random sample of registered voters in Iowa is selected, and the survey questionasked of each voter is whether or not he/she plans to attend his/her precinct caucus.3. Consider a model in which the likelihood isy |β ∼ Exponential(β)1and the prior isβ ∼ Gam ma(2, 5)(a) Find the posterior d istribution of β if the data is a single observation, y = 8. (Use theparameterization of the Exponential distribution given in the table of distributionsfrom the textbook.)(b) Is the Gamma the conjugate prior family for the Exponential likelihood? Justifyyour answer in one sentence.(c) Find the Jeffreys prior for the Exponential likelihood. Write an expression to whichit is proportional, and, if possible, identify it as a member of a parametric family.4. A sample of size 20 was drawn from a population of adult men from a particular de-mographic group, and systolic bloo d pressure was measured on each man in the sample.Suppose it was kn own that the typical systolic blood pressure for all adult males is 125mmHg, and the researchers wish to determine whether the population mean for the par-ticular demographic group is above or below that value. That is, the researchers wish totest the hypotheses:H0: µ >= 125H1: µ < 125See the WinBUGS code and output on the last two pages.(a) The prior specified on µ (circle one):i. implies that the prior odds in favor of H0= 0ii. implies that the prior odds in favor of H0= 0.52iii. implies that the prior odds in favor of H0= 1iv. contains no information about the prior odds in favor of H0(b) Based on the WinBUGS output, what is the estimated posterior probability of H0(numeric value)?(c) What is the Bayes factor in favor of H0vs. H1(numeric value)?(d) In a sampler iteration in which the value drawn for µ was 123.8, the value of p.abovewould be (circle one):i. 0ii. 0.5iii. 0.8271iv. 1v. impossible to determine(e) Explain the meaning of th e 95% posterior credible set f or σ.(f) The posterior mean of mu in the WinBUGS output is slightly different from that ofy.new. Would this difference be expected from the theory behind the model? If not,how could it happen? Briefly explain.(g) The MC error for µ is (circle one):i. the probability that the WinBUGS software has made an errorii. the standard deviation of the posterior distribution of µiii. a measur e of the uncertainty in estimating the mean of the posterior distributionof µiv. none of the above(h) The values produced by WinBUGS for th e node y.new over all sampler iterationsare draws from (circle one):i. a degenerate distributionii. a posterior distribution3iii. a posterior predictive distributioniv. a prior distributionv. none of the above(i) In this model, the 20 data valuesi. are treated as exchangeableii. are not treated as exchangeableiii. whether exchangeability was assumed cannot be determined from the informa-tion given(j) Explain the meaning of the quantity pD at the end of the WinBUGS output.4Systolic blood pressureProgram 2.1 from Congdon, much modifiedmodel{# likelihoodfor (i in 1:N) {y[i] ~ dnorm(mu,tau)}# priorsmu ~ dnorm(125, 0.0001)tau ~ dgamma(0.001, 0.001)# samples where H0 holdsp.above <- step(mu-125)# prediction for ’new’ persony.new ~ dnorm(mu,tau)sigma <- 1/sqrt(tau)}Datalist(N=20, y=c(98,160,136,128,130,114,123,134,128,107,123,125,129,132,154,115,126,132,136,130))Initslist(tau = 0.01, mu = 100)list(tau = 0.1, mu = 150)list(tau = 0.001, mu = 200)5Node statisticsnode mean sd MC error 2.5% median 97.5% start samplemu 128.0 3.284 0.01307 121.5 128.0 134.5 501 60000p.above 0.8271 0.3782 0.00159 0.0 1.0 1.0 501 60000sigma 14.49 2.498 0.01036 10.56 14.18 20.25 501 60000y.new 127.9 15.09 0.06281 98.02 127.9 157.7 501 60000Dbar = post.mean of -2logL; Dhat = -2LogL at post.mean of stochastic nodesDbar Dhat pD DICy 163.107 161.048 2.059


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UI STAT 4520 - Bayesian Statistics

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