Monte Carlo methods

A cancer laboratory is estimating the rate of tumorigenesis in two strains of mice, A and B. They have tumor count data for 10 mice in strain A and 13 mice in strain B. The observed tumor counts for the two populations are ya = {12, 9, 12, 14, 13, 13, 15, 8, 15, 6} YB = {11, 11, 10, 9, 9, 8, 7, 10,6,8,8,9,7} 2 > = 7 > Consider Yai ~ Pois(@A) for i = 1,...,na and Y Bi Pois(OB) for = 1, ..., nB as the correspoding random variables for the counts of each pop- ulation (A and B). = 2 (5 marks] Reparametrize the model using @ A = 0 and A = y. Consider O~ Ga(2, 1) and y~ Ga(8,8). Derive the full conditionals for 017, YA, YB and y|0, ya, yb and implement in R the relative Gibbs-sampling using 5000 simulations. For 7, obtain an acceptance rate around 23%. Plot the obtained chains; after choosing an appropriate burnin, estimate the posterior mean and variance of 0 and y. > A cancer laboratory is estimating the rate of tumorigenesis in two strains of mice, A and B. They have tumor count data for 10 mice in strain A and 13 mice in strain B. The observed tumor counts for the two populations are ya = {12, 9, 12, 14, 13, 13, 15, 8, 15, 6} YB = {11, 11, 10, 9, 9, 8, 7, 10,6,8,8,9,7} 2 > = 7 > Consider Yai ~ Pois(@A) for i = 1,...,na and Y Bi Pois(OB) for = 1, ..., nB as the correspoding random variables for the counts of each pop- ulation (A and B). = 2 (5 marks] Reparametrize the model using @ A = 0 and A = y. Consider O~ Ga(2, 1) and y~ Ga(8,8). Derive the full conditionals for 017, YA, YB and y|0, ya, yb and implement in R the relative Gibbs-sampling using 5000 simulations. For 7, obtain an acceptance rate around 23%. Plot the obtained chains; after choosing an appropriate burnin, estimate the posterior mean and variance of 0 and y. >