25 06 2024 E TS

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BAYESIAN STATISTICS A. Guglielmi & M. Gianella 25.06.2024 Properly justify all your answers. Use the indicator function to denote the support of a distr. . Exercise 1 We consider the conditional density fX(x|β) = βe−βx1(0,∞)(x), for β > 0, (1) and let (X1, . . . , Xn) be a random sample from (1). Denote by ( x1, . . . , xn) the corresponding observed sample. 1. Compute θ = P(Xi ≤ 1|β) for any i = 1, . . . , n. Instead of observing the whole sample ( x1, . . . , xn), we only know that observations are less than or equal to 1 (≤ 1), or larger than 1. 2. Introduce r.v.s ( Y1, . . . , Yn) as a transformation of ( X1, . . . , Xn), representing the only information that we have. 3. Derive the likelihood, i.e. the conditional density of ( Y1, . . . , Yn), given β, under the sole information that (exactly) k observations in (x1, . . . , xn) are less than or equal to 1, where k is an integer number between 0 and n. 4. Use reparameterization θ = g(β), derived at point 1., and find a conjugate prior πθ for the new parameter θ, under the conditional distribution of ( Y1, . . . , Yn), given θ. Transform back πθ, and assume this prior πβ as the prior for the likelihood derived at point 3. 5. Write the expression of the posterior distribution of β up to a constant, under the sole information that k observations in (x1, . . . , xn) are less than or equal to 1. How would you simulate from this posterior? 6. Set the hyperparameters so that, a priori, the mean and the variance of θ, introduced at point 4., are 1/2 and 1/12, respectively. Then compute Eπβ(β). Solution of Exercise 1 1. It is straightforward to compute θ = P(Xi ≤ 1|β) = R 1 0 fX(x; β)dx = 1 − e−β×1 = 1 − e−β. 2. We define Yi = 1 if Xi ≤ 1 and Yi = 0 if Xi > 1. This means that, conditionally to β, (Y1, . . . , Yn) is a…