5102-Solution-Set-03

5102-Solution-Set-03 - SOLUTIONS FOR PROBLEM SET 3 STAT...

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SOLUTIONS FOR PROBLEM SET 3 STAT 5102-004 Since the first three problems deal with binomial sampling and beta priors, a few preliminary remarks will help to set the context. Let Y be distributed binomially over { 0 , . . . , n } with success probability θ . The likelihood function for θ given y satisfies lik( θ | y ) θ y (1 - θ ) n - y . A Beta( α, β ) prior for θ has mean μ 0 = α α + β . It’s density satisfies π ( θ ) θ α - 1 (1 - θ ) β - 1 . The posterior density is proportional to the prior density times the likelihood π ( θ | y ) θ α + y - 1 (1 - θ ) β + n - y - 1 . Since the right hand side of this proportionality is the the kernel of a beta density, the posterior distribution is a Beta( α + y, β + n - y ). (1) Let Y Bin( n, θ ). Assume a Beta(5 , 10) prior. The posterior is Beta(5+ y, 10+ n - y ). For n = 20 and y = 1, the posterior mean 5 + y 5 + 10 + n = 6 35 is the Bayes estimate of θ . (2) Let Y = n ¯ X n The Bayes estimator under squared error loss is the posterior mean ˆ θ n = α + n ¯ X n α + β + n = n ¯ X n α + β + n + α α + β + n = n α + β + n ¯ X n + α + β α + β + n μ 0 . Let γ n = n α + β + n . The posterior mean ˆ θ n = γ n ¯ X n + (1 - γ n ) μ 0 is an average of the prior mean and the sample mean. Since α + β is fixed, the ratio of n to α + β + n goes to unity as n increases. Thus γ n -→ 1 and (1 - γ n ) μ 0 -→ 0 as n -→ ∞ .
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