Cancer deaths: Suppose for a set of counties i ∈ {1,..., n} we have infor­mation on the population size X_i = number of people in 10,000s, and Y_i = number of cancer fatalities. One model for the distribution of cancer fatalities is that, given the cancer rate 0, they are independently distributed with Y_{i ˜} Poisson(θX_i).

a) Identify the posterior distribution of θ given data (Y₁,.X₁),...., (Y_n, X_n) and a gamma(a, b) prior distribution.

The file cancer_react.dat contains 1990 population sizes (in 10,000s) and number of cancer fatalities for 10 counties in a Midwestern state that are near nuclear reactors. The file cancer_noreact.dat contains the same data on counties in the same state that are not near nuclear reactors. Consider these data as samples from two populations of counties: one is the population of counties with no neighboring reactors and a fatality rate of θ₁ deaths per 10,000, and the other is a population of counties having nearby reactors and a fatality rate of θ₂. In this exercise we will model beliefs about the rates as independent and such that θ_{1 ˜} gamma(a₁, b₁) and θ_{2 ˜} gamma(a₂, b₂).

b) Using the numerical values of the data, identify the posterior distributions for θ₁ and θ₂ for any values of (a_l, b₁, a₂, b₂).

c) Suppose cancer rates from previous years have been roughly θ˜ = 2.2 per 10,000 (and note that most counties are not near reactors). For each of the following three prior opinions, compute E[θ₁| data], E[θ₂|data], 95% quantile-based posterior intervals for θ₁ and θ₂, and Pr(θ₂ > θ₁| data). Also plot the posterior densities (try to put p(θ₁| data) and p(θ2|data) on the same plot). Comment on the differences across posterior opinions.

i. Opinion 1: (a_l = a₂ = 2.2 x 100, b₁ = b₂ = 100). Cancer rates for both types of counties are similar to the average rates across all counties from previous years.

ii. Opinion 2: (a_l = 2.2 x 100, b_l = 100, a₂ = 2.2, b₂ = 1). Cancer rates in this year for nonreactor counties are similar to rates in previous years in nonreactor counties. We don't have much information on reactor counties, but perhaps the rates are close to those observed previously in nonreactor counties.

iii. Opinion 3: (a_l = a₂ = 2.2, b₁ = b₂ = 1). Cancer rates in this year could be different from rates in previous years, for both reactor and nonreactor counties.