09 06 2022 E TS

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BAYESIAN STATISTICS A. Guglielmi & M. Beraha 09.06.2022 Properly justify all your answers . Exercise 1 Consider the following model for count data ( y1, y2, . . . , yn): Y1, Y2, . . . , YA1|λ1, A1 iid∼ P oisson(λ1), Y A1+1, . . . , Yn|λ2, A1 iid∼ P oisson(λ2) where A1 ∈ {1, 2, . . . , n− 1} is called change point and the two samples are independent, conditional on A1, λ1, λ2. Denote by A2 = n − A1 the size of the second sample YA1+1, . . . , Yn. First, consider A1 fixed and equal to a1 ∈ {1, 2, . . . , n− 1}. 1. Verify that the moment generating function M(t; λ) of X ∼ P oisson(λ) with density fX(k; λ) = e−λ λk k! 10,1,2,...(k) for λ > 0 is M(t; λ) = E etX
= exp λ(et − 1)
t ∈ R. 2. Write the likelihood of ( y1, y2, . . . , ya1, ya1+1, . . . , yn). 3. Find the conjugate prior for ( λ1, λ2) such that π(λ1, λ2) = π1(λ1) × π2(λ2) and derive the hyperparameters of the posterior distribution. (Hint: the marginal priors π1 and π2 can be parameterized by parameters (αj, βj), j = 1 , 2, respectively. Denote the posterior hyperparameters as (α1a1, β1a1, α2a2, β2a2).) 4. In order to set the values for ( αj, βj), j = 1 , 2, we ask to an expert: she thinks that, before the change point, marginal prior values are such that E ( Y1) = 4, Var( Y1) = 8, and, after the change point, E ( Yn) = 8, Var( Yn) = 16. Compute the prior hyperparameters and derive the corresponding posterior hyperparameters. Now, suppose that A1 is random: in particular, consider A1 uniformly distributed on {1, 2, . . . , n− 1} and independent on ( λ1, λ2). Here ( λ1, λ2) has the prior derived at points 3 and 4. 5. Check that, in this case, the likelihood has the same analytic form as at point 2. Then determine the marginal posterior of A1, up to a normalizing constant, denoted by K. (Hint: first derive the…