The observed data is y = ()1. . .. .yn), a sample from a negative binomial distribution with parameters q and r, where r is assumed to be known. The prior distribution for q is a beta distribution Beta(c.B). Suppose that y1 = . . . = yn = 0. Take n = 20 + A, where A is the third-to-last digit of your ID number; o = 10 + B.where B is the second-to-last digit of your ID number; r = 2; and B = 1. (a) What is the normalized posterior probability density function for q? [4] (b) Find an expression for the quantile function for this posterior distribution, and hence find the posterior median for q. [5] (c) Let x be a new data-point generated by the same negative binomial distribution with parameters q and r. Find P(x =0| y), the posterior predictive probability that x is 0. [6] Suppose now that we want to compare two models. Model M, is the model and prior distribution described above. Model M2 assumes that the data follow a negative binomial distribution with q known to be go = 0.9. (d) Find the Bayes factor B12 for comparing the two models. [6] (e) We assign prior probabilities of 1/2 that each model is the true model. Find the posterior probability that M2 is the true model. [3] We have observed data y = by:i= 1.....nj=1.....mil. Each y is the number of times a certain type of machine needs to be repaired during length of time Ti. where / = 1.....m; are the machines in factory i, for / = 1. ....n, with n 2 2. A hierarchical model is used to model the data. We assume that yy ~ Poisson(Tip;). ; is the repair rate for factory i, which varies between factories according to a gamma distribution Hi ~ Gamma(a.B). i = 1.....n. The parameters o and S are given prior distributions, p(o) and p(B). Suppose that we have generated a sample of size M from the joint posterior distribution P(#1.....Hm.(.Bly). (a) Explain how to obtain a sample from the marginal posterior distribution p(a.ly) using the joint posterior sample. (b) Given a sample from p(a.Bly). explain how to estimate the following: (i) The posterior median of a. (ii) The posterior median of v = (iii) A 95% equal tail credible interval for v. (c) Explain how to estimate the posterior probability P(# n). Each data-point y, is assumed to be generated by a distribution with the following probability density function: I (K)YETexp ( -"). MIZO. The unknown parameter is 0, with & assumed to be known, and 0. 0. I() is the gamma function. (a) Write down the likelihood for 0 given y. Find an expression for the maximum likelihood estimate (MLE) 0. [5] (b) A Gamma(a,B) distribution is chosen as the prior distribution for 0. Derive the resulting posterior distribution for & given y. [5] (c) We would like to choose the gamma prior distribution parameters so that o = 1, and P(0 > 100 + B) = 0.1. where B is the second-to-last digit of your ID number. Find the value of B that is needed. [4] (d) The data are y = (4.2.8.8.C+ 2). where C is the last digit of your ID number, with 1 = 5. Set * = 3. (i) What is the MLE A? [3] (ii) Using the prior distribution from part (c). what are the parameters of the posterior distribution for e? [3] (iii) What are the posterior mean and standard deviation for e? [2]
