Question 4 4. [21 marks] Let X = {XhX9,...,Xn) be i.i.d. observations from a density that for a given A 3.:- U is believed to he Poisson, i.e. you) = armi) - won, a: = o,1,e...., i s o. The prior on A is believed to be Gamma (1,4). [Note that, for any values of o :e i}, :3 :a- U, the density of the Gamma(o,,8] distribution is given by 1 _ cll _ T r{.i}={ gnurm 3' esp{ M13} ;ie}+ and Mo) = I: e"s\"'1d:r is the Gamma function with the property l"{o:+1] = oI'(o).] i} [ID marks] Find the posterior density MAM) of A given the sample X = (X11X3,...,Xn}. Show that, like the prior, the posterior density is also a member of the Gamma family. ii} [5 marks] Find the Bayesian estimator of .1 for quadratic error-loss with respect to the prior 11).}. iii} [it marks] Seven observations 4, 4* 3, 1'1", 12, 15* 3 are taken Determine the Insulting Bayesian estimate of A. 7. [14 POINTS] ACME Pharmaceuticals has developed a new arthritis medication. Based on prelimi- mary experiments, the statisticians believe the proportion @ of arthritis sufferers who will experience relief with this medication is a random variable with the following distribution: .50 .65 .60 65 P(0 = 0) .10 40 40 .10 The medication was administered to twenty individuals, fifteen of whom reported feeling relief. The likelihood function is binomial: P(Y - 15/0 - 9) = (75)915(1 -8). Complete the following table for Bayesian inference. Suppose it is assumed in this simple model that & can only take values from {0.50, 0.55, 0.60, 0.65). Note you must compute the normalization constant . P(Y = 1510 - 0) x P(0 - 8). P(0 = 0) P(Y - 1510 = 0) P(Y = 1510 - 0) x P(0 - 0) P(0 = 0|Y = 15) 0.50 0.10 0.014786 0.0014786 0.55 0.036471 0.60 0.40 0.65 0.10 0.0127199 Normalization constant = > P(Y = 15/0 = 0) xP(0 = 0) =(a) (b) (C) The Pareto distribution (written 9 ~ Pareto(a, #3)) has density a (a) - (0)={'0 +1 \"25 0 otherwise, where a, ,6 > 0. Let the prior for 9 be taken to be Pareto(a, ,6) and derive the posterior distribution (9L0)- In an experiment conducted in the Antarctic in the 19805 to study a particular species of fossil ammonite, the following was a linearly rescaled version of the data obtained, in ascending order: y = (0.4, 1.0, 1.5, 1.7, 2.0, 2.1, 3.1, 3.7, 4.3, 4.9). Prior information equivalent to a Pareto prior with (a, ,6) = (2.5, 4) was available. Plot the prior, likelihood, and posterior distributions arising from this data set on the same graph, and briey discuss what this picture implies about the updating of information from prior to posterior in this case. Make a table summarizing the mean and standard deviation for the prior, likeli- hood and posterior distributions, using the (0,13) choices and the data in part (b) above. In Bayesian updating the posterior mean is often a weighted average of the prior mean and the likelihood mean (with positive weights), and the posterior standard deviation is typically smaller than either the prior or likelihood standard deviations. Are each of these behaviors true in this case? Explain briey. Why confidence intervals for unknown parameters are more meaningful in Bayesian inference compared to confidence intervals in traditional inference? Let: X=c(2, 2, 3, 1, 3, 3,4, 2, 3, 1, 3, 5, 2, 2,4,4, 2, 2, 2,4, 3,4, 3, 4, 2: 1, 4, 5, 3, 2, 3, 5, 2: 5, 4, 5, 1, 3, 8, 1, 2: 5, 4, 5, 3, 3, 4, 5, 2, 2, 2, 1, 2, 2, 3, 2, 1, 2, 3, 2, 3, 3, 5, 4, 3, 2, 3, 3, 2, 4, 2, 2, 3: 7, 4, 3, 5, 4, 5, 1, 3: 3, 6, 7, '3, 3, 4, 2, 2: 4, 5, 2, 2, 3, 5, 2, 3, 1, 9, 4) Assuming that x are our data, we will fit a statistical model to x. Will a normal model with mean u and standard deviation 0' be appropriate for x? Why?|