Stat 544 Final Exam. May 2, I have neither given nor received unauthorized assistance on this examination.

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1 Stat 544 Final Exam May, 006 I have neither given nor received unauthorized assistance on this examination. signature date 1

2 1. Below is a directed acyclic graph that represents a joint distribution for the variables α, β, p, p, p, X, X, and X Some information about this joint distribution is that: α and β are independent, α Exp 1, β Exp 1 ( ) ( ) ( αβ) p p p ( αβ) conditioned on,,,, are iid Beta, 1 3 conditioned on p, p, p, the variables X, X, X are independent, with Bi n ( 100, ), Bi n ( 100,min (, )), Bin ( 100, ) X p X p p X p a) In general (for all distributions represented by this DAG) are X1 and p 3 independent? Argue carefully that your answer is correct. b) Are X1 and p 3 conditionally independent given p 1? Argue carefully that your answer is correct.

3 c) Write out a joint "density" for these 8 random variables, h(,, p, p, p, x, x, x ) αβ d) Suppose that X1 = 10, X = 15, and X3 = 70 and that one wishes to sample from the conditional distribution of the other variables given these values using a Gibbs sampler. Write out a function of p 1 proportional to the pdf on ( 0,1 ) used in a " * * * * * 1 α, β, p1, p, p3. p update" of the vector ( ) 3

4 e) Finish the WinBUGS code below for implementing the Gibbs sampling alluded to in part d). (The WinBUGS User Manual says step(e) is 1 if e>0; 0 otherwise.) model { alpha ~ beta ~ for ( ) { p[i] ~ p<-( )*step( )+p[1] x1 ~ dbin ( ) x ~ dbin (p,100) x3 ~ dbin ( ) #data are next list( ). Suppose that conditional on p1 and p, 1 and X i Bin 10, prior distributions for ( p1, p ) are Prior/Model 1: p1, p iid U( 0,1) Prior/Model : p1 = p, p1 U( 0,1) Find the Bayes Factor for comparing these two models (likelihood plus prior) based on X = x and X = x. 1 1 X X are independent with ( p ) i. Two 4

5 3. Two versions of an industrial process are run with the intention of comparing effectiveness. (There is an "old/#1" and a "new/#" process.) Six different batches of raw material are used in the study. For a response variable yij = the yield of the jth run made using raw material batch i, we'll consider an analysis based on the model y = μ + β + ε ij process( ij) i ij where process( ij ) takes the value either 1 or depending upon which version of the process is used, μ1 and μ are mean yields for the two versions of the process, the i are iid N( 0, β ) β σ independent of the ij iid N 0,σ, and the parameters of the model are μ1, μ, σβ, and σ. Attached to this exam is a summary of a WinBUGS session run to analyze the total of n = 5 runs made in the study. Use it to answer the following questions. ε which are ( ) a) Describe in standard mathematical (Stat 54) notation (NOT in WinBUGS terminology/notation) μ, μ, σ, σ in the analysis. (What was the form of what prior distribution was used for ( 1 β ) h( μ 1, μ, σβ, σ )?) b) As it turns out, the 95% Bayes credible intervals for the parameters provided on the WinBUGS output are very much like the corresponding approximate 95% confidence intervals one obtains from an R "lme" mixed effects model analysis of the data. Why is this not surprising? Would you expect this to have been true if a substantially different prior had been used? Explain. c) Is there a clear difference between the mean yields under the two different versions of the process? Explain. 5

6 d) What is a 95% credible interval for y new, a yield for a future run of the production process made using a new batch of raw material and the "new/#" version of the process? 4. A response, y, whose mean is known to increase with a covariate/predictor, x, is investigated in a study where (coded) values x = 1,,3, 4,5,6 are used and there are 3 observations for each level of the predictor. Some summary statistics from that study are (in the obvious notation) y1. =.51, y. =.15, y3. =.95, y4. = 5.88, y5. = 6.65, y6. = 7.59, and MSE =.99 Attached to this exam is summary of a Bayes analysis of the data based on the model y = μ + ε (*) xj x xj where μ1 μ μ3 μ4 μ5 μ6 are unknown means and the xj N 0,σ. a) What is a 95% credible interval for μ3 μ under the prior used in the Bayes analysis? ε are iid ( ) b) The prior used in the Bayes analysis for model (*) is improper. Exactly how would you modify the prior to create one that is proper, but should produce results close to those provided here? (Fully specify your alternative prior.) c) One alternative to the model (*) with the order restriction on the 6 means is a model y = β + β x+ ε (**) xj 0 1 where β0 and β1 0 are unknown parameters and the xj would you use with model (**)? (Fully specify this prior.) xj ε are iid N( 0,σ ). What prior distribution 6

7 WinBUGS Session Summary for Problem #3 model { for (i in 1:) { process[i] ~ dnorm (0,.0001) diff<-process[]-process[1] logsigb ~ dflat() taubatch<- exp(-*logsigb) sigmabatch<- exp(logsigb) for (j in 1:7) { batch[j] ~ dnorm (0,taubatch) logsig~dflat() tau<-exp(-*logsig) sigma<- exp(logsig) for (l in 1:7) { mu[l]<-process[p[l]]+batch[b[l]] for (l in 1:7) { y[l]~ dnorm (mu[l],tau) list(b=c(1,1,1,1,,,3,3,3,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7), p=c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,,,,,,,,,1,), y=c(8.7, 78.31, 8.0, 81.18, 80.06, 81.09, 78.71, 77.48, 76.06, 87.77, 84.4, 84.8, 78.61, 77.47, 77.80, 81.58, 77.50, 78.73, 78.3, 76.40, 81.64, 83.04, 8.40, 81.93, 8.96,NA,NA)) node mean sd MC error.5% median 97.5% start sample batch[1] batch[] batch[3] batch[4] batch[5] batch[6] batch[7] diff process[1] process[] sigma sigmabatch y[6] y[7]

8 WinBUGS Session Summary for Problem #4 model { mu1 ~ dnorm(0,.0001) mu[1] <- mu1 for (i in :6) { delta[i] ~ dexp(.4) for (i in :6) { mu[i] <- mu[i-1]+delta[i] logsigma ~ dflat() sigma <- exp(logsigma) tau <- exp(-*logsigma) 8

9 for (j in 1:N) { y[j] ~ dnorm(mu[group[j]],tau) list(n=18,group=c(1,1,1,,,,3,3,3,4,4,4,5,5,5,6,6,6), y=c( , , , , , , , , , , , , , , , , , )) list(mu1=0,logsigma=0) node mean sd MC error.5% median 97.5% start sample delta[] delta[3] delta[4] delta[5] delta[6] mu[1] mu[] mu[3] mu[4] mu[5] mu[6] sigma E

Stat 544 Exam 2. May 5, 2008

Stat 544 Exam 2. May 5, 2008 Stat 544 Exam 2 May 5, 2008 I have neither given nor received unauthorized assistance on this examination. signature name printed There are 10parts on this exam. I will score every part out of 10 points.