24 01 2023 E TS

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BAYESIAN STATISTICS A. Guglielmi & M. Beraha 24.01.2023 Properly justify all your answers. Use the indicator function to denote the support of a distr. . Exercise 1 Let X1, X2, . . . , Xn be (conditionally) independent and identically distributed (iid) absolutely con- tinuous random variables with density fX(x; α, β) = βαα xα+1 I[β,+∞)(x), α, β > 0 . (1) For the moment, consider β = 3 fixed. 1. Compute the likelihood L(α; x1, . . . , xn) and show that the gamma distribution is a conjugate prior for α. Derive the hyperparameters of the associated posterior distribution. (Hint: denote by (a, b) the hyperparameters of the prior gamma density π, with prior variance for α given by the ratio a/(b2)) 2. Compute the marginal distribution of X1. 3. For the prior π as at point 1, consider prior hyperparameters (1 , b) (i.e., a = 1) and fix b such that Eπ[α] = 5. Propose a two-step method to sample from the marginal distribution of X1 using the inverse- CDF method. You sample two iid random variables from the Uniform distribution U([0, 1]) and observe U1 = 0.235 and U2 = 0.901. Which value of X1 do you get? We observe data (x1, . . . , x7) such that P i log(xi) = 9.73. 4. Test the hypotheses H0 : α = 1.5 vs H1 : α ̸= 1.5 using the Bayes factor, making explicit your conclusion under available data and the prior at points 1 and 3. For any integer n, if X1, X2, . . . , Xn are (conditionally) iid with density (1), assume that both α and β are random and a priori independent. For α assume the marginal prior derived at point 1. 5. Propose a prior for β and find the associated full conditional distribution, in such a way that you can easily sample from it. 6. Describe a Gibbs sampling algorithm to sample from the posterior distribution of ( α, β), with any dataset (x1, . . . , xn).…