18 07 2022 E TS

Page 1

BAYESIAN STATISTICS A. Guglielmi & M. Beraha 18.07.2022 Properly justify all your answers . Use the indicator function to denote the support of a distr.. Exercise 1 Assume the following conditional density (w.r.t. the Lebesgue measure on R) for the data: X1, . . . , Xn|θ i.i.d.∼ fθ(x) = 3x2 θ3 1[0, θ](x), θ > 0. 1. Write down the likelihood L(θ; x1, . . . , xn), properly verifying that the indicator function in its expression is equal to 1[maxi xi, +∞)(θ). 2. Show that a conjugate prior density w.r.t. the Lebesgue measure on (0 , +∞) for this model is such that π(θ; α, β) ∝ 1 θα+1 1[β, +∞)(θ), where α, β > 0. Moreover, first derive the normalizing constant for π(θ; α, β). Then, denoting by αn, βn the posterior hyperparameters, find the analytic expression of the posterior density π(θ; αn, βn), deriving explicitly its normalizing constant. (Hint: remember that 1[a, +∞)(θ) × 1[b, +∞)(θ) = 1[c, +∞)(θ), where c = . . . (function of a and b)) 3. Find the posterior mean E π (θ|x1, . . . , xn) of θ, when the prior is the density π found at point 2. Compare it to the maximum likelihood estimate and make a comment. 4. By closely looking at the expression of the posterior mean of θ found at point 3., fix the prior hyperparameters (α, β) in π(θ; α, β) using an old dataset of size 7 and range equal to [2 .3, 15]. 5. Derive the hyperparameters of the posterior (using prior hyperparameters at point 4.), when the current available dataset contains 10 observations and its range is [1 .43, 5] (i.e. 1 .43 = min i xi and 5 = . . .). Moreover, compute the 95% HPD credible interval of θ. 6. By using the posterior odds, test the hypotheses H0 : θ ≤ θ0 vs H1 : θ > θ 0, for any θ0 in R+, using the posterior derived so far. Write down your conclusion when θ0 = 16. Solution of Exercize 1 . 1.…