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Q4

Assume that the surveys were independent simple random samples from the populationof registered voters. Model the data as two different multinomial distributions, anduse independent Dirichlet priors, with all the hyperparameters equal to 2, for thepopulation proportions of these multinomial distributions. Forj= 1,2, letγjbe theproportion of voters who preferred Bush, out of those two had a preference for eitherBush or Dukakis at the time of surveyj.(a) Derive the posterior distributions ofγ1andγ2, respectively.(b) Draw 1000 samples from the posterior distribution ofγ1−γ2and draw the his-togram.(c) Based on the samples drawn in part (b), estimate the posterior probability thatthere was a shift towards Bush after the presidential debate.(d)∗Let,θ(j)BDdenote proportion of registered voters who had preference for either Bushor Dukakis at the time of surveyj. What are the posterior mean and posteriorvariance ofθ(1)BD−θ(2)BD?(e)∗Compute a 95% posterior credible interval forθ(1)BD−θ(2)BDby drawing 1000 samplesfrom the posterior distribution ofθ(1)BD−θ(2)BD. Does this result indicate any shift inthe combined support for either Bush or Dukakis, before and after the presidentialdebate ? Explain.

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