2. A Mixture Prior for Heart Transplant Surgeries A hospital in the United States wants to evaluate their success rate of heart transplant surgeries. We observe the number of deaths, y, in a number of heart transplant surgeries. Let y ~ Pois(v)) where is the rate of deaths/patient and r is the exposure (total number of heart transplant patients). When measuring rare events with low rates, maximum likelihood estimation can be notoriously bad. We'll tak a Bayesian approach. To construct your prior distribution you talk to two experts. The first expert thinks that p (A) with a gamma(3, 2000)$ density is a reasonable prior. The second expert thinks that pz(X) with a gamma(7, 1000) density is a reasonable prior distribution. You decide that each expert is equally credible so you combine their prior distributions into a mixture prior with equal weights: p()) = 0.5*pi (X) +0.5# pz(X) 2a. What does each expert think the mean rate is, a priori? Which expert is more confident about the value of A a priori (i.e. before seeing any data)? Type your answer here, replacing this text. 2b. Plot the mixture prior distribution. # YOUR CODE HERE 2c. Suppose the hospital has y = 8 deaths with an exposure of v = 1767 surgeries performed. Write the posterior distribution up to a proportionality constant by multiplying the likelihood and the prior density. Warning: be very careful about what constitutes a proportionality constant in this example. Type your answer here, replacing this text. 2d. Let K = [ L(X; y)p(A)dA be the integral of the proportional posterior. Then the proper posterior density, i.e. a true density integrates to 1, can be expressed as p() | y) = Helped. Compute this posterior density and clearly express the density as a mixture of two gamma distributions. Type your answer here, replacing this text. 2e. Plot the posterior distribution. Add vertical lines clearly indicating the prior means from each expert. Also add a vertical line for the maximum likelihood estimate. # YOUR CODE HERE