24 01 2022 E TS

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BAYESIAN STATISTICS A. Guglielmi & M. Beraha 24.01.2022 Properly justify all your answers . Exercise 1 Consider the beta-binomial distribution, i.e. M | N, p ∼ Bin(N, p), where N is a fixed positive integer, p ∼ beta(α, β), α, β > 0. 1. Show that the prior marginal distribution of the r.v. M is the beta-binomial density, i.e., P (M = m) = N m B(α + m, β + N − m) B(α, β) , m = 0, 1, . . . , N (1) where B(α, β) = Γ(α)Γ(β)/Γ(α + β) is the beta function. 2. Compute E[ M] and Var[M]. Consider now an urn with N = 4 balls of which M are blue and the remaining 4 − M are red. Let X denote the number of blue balls obtained when sampling n = 2 balls without replacement from the urn. The r.v. X has the hypergeometric distribution, i.e. P (X = x | M) = M x
4−M 2−x
4 2
1SM (x), (2) where SM = {max{0, M − 2}, max{0, M − 2} + 1, . . . ,min{2, M}}. The statistical goal here is to make inference on M, with data X under likelihood (2). A priori, we assume that M (the only unknown parameter) is the beta-binomial r.v. with hyperparameters N = 4, α, β, i.e. its (discrete) density is (1). 3. Find the values of prior hyperparameters α and β such that E[ M] = N/2 = 2 and Var[ M] = N 2/10 = 16/10. Assume those values for the rest of the exercise. We observe data x = 1. 4. Derive the support of the posterior distribution of M, given X = 1. (Hint: check for which values of m = 0, 1, 2, 3, 4 the indicator function in (2) assumes value 1 ) 5. Compute the posterior probability P (M = m|X = 1), m = 0, 1, 2, 3, 4. (Hint: first ignore factors that do not depend on m in the expression and make this calculation for a general m in the support of the posterior ) 6. Test the hypotheses H0 : M = 2 vs H1 : M = 1 by computing the Bayes factor of the model (2)-(1) with available data. Write down your…