In the recent literature, the Poisson distribution has been used to model COVID-19 death counts. Valid estimation of the death rate relies on the availability of accurate data on COVID-19 death counts. In some regions in the world, the accuracy of reported COVID-19 death counts is questionable. We are going to build a Bayesian model to estimate the death rate parameter due to COVID-19 using a Poisson sampling model but also allow for possible under-reporting of death counts. Let y, yn be the reported daily death counts for Region A for n different days. Let Z, Zn be the true (unreported) death counts, the sampling model assumption is z; id Pois (A). (Note, in this question we assume there is no over-reporting of death counts, just under-reporting so yi .) (a) [2 marks] Assume yi = zi (i 1,..., n). So there is no under-reporting of death counts. Define a prior distribution for A and derive the posterior distribution of A with your chosen prior. (Note, you do not need to provide specific hyperparameter values in your chosen prior assumption). = (b) [3 marks] Now we want to adjust our model for the possibility of under-reporting. The true rate of under-reporting is not known but it is assumed that the reported death count y is at least 80% of the true death count z; (for i = 1,..., n). That is, the under-reporting rate is at most 20%. Give the posterior distribution of up to a proportionality constant (that is, ignoring any normalising constant), under this revised model (c) [2 marks] Describe in words how we could use the posterior distribution in (b) to predict the unobserved true death counts zi.