08 02 2022 E T

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BAYESIAN STATISTICS A. Guglielmi & M. Beraha 08.02.2022 Properly justify all your answers . Exercise 1 Mario buys useless stuff every Sunday. Whenever he comes home, he tells Francesca the total amount of money (in euros) he has spent. If she guesses the correct number of items that Mario has bought, he will have to take her out for a fancy dinner. After several unsuccessful attempts, Francesca decides to seek the help of an expert Bayesian statistician. Let Y be the total amount of money spent on a given Sunday. We model Y through the following distribution Y := NX j=1 Xj, X j | α, β iid∼ gamma(α, β) j = 1, 2, 3, . . . (1) N |λ ∼ Poisson(λ), i.e. P (N = n) = e−λ λn n! if n = 0, 1, 2, . . . , (2) where N is the number of items purchased and the Xj’s represent individual prices of each item. For the moment, assume α, β, λ fixed (and positive) . Note that, when N = 0, the sum above is equal to 0, i.e. Y = 0 a.s.. 1. Show that if Z1 ∼ gamma(α1, β) and Z2 ∼ gamma(α2, β) are independent random variables, then V := Z1 + Z2 ∼ gamma(α1 + α2, β). (Hint: you can either show the result using moment generating or characteristic functions, e.g. m(s) = E(e sZi); alternatively, find the probability density function of V , writing P (V ∈ dv) =R ? ? P (V ∈ dv|Z1 = z)P (Z1 ∈ dz)dz. Substitution z = vt might become useful to solve the integral.) 2. Use 1. to derive the distribution of Y given N (and given α and β). (Hint: it is useful to assume Z ∼ gamma(0, β) ⇔ Z = 0 a.s.) 3. Compute E[ Y ] and Var[Y ]. (Hint: If N ∼ Poisson(λ), remember that E(N) = Var(N).) Now, and for the rest of the exercise, as before consider both α, λ > 0 fixed, but assume β random. 4. As a prior for β, consider a semi-conjugate prior, i.e. a prior that is conjugate with respect to the conditional distribution of…