Question

Tumor counts: 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. Type A mice have been well studied, and information from other laboratories suggest that Type A mice have tumor counts that are approximately Poisson-distributed with a mean of 12. Tumor count rates for Type B mice are unknown, but type B mice are related to type A mice. The observed tumor counts 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). 1). (b) Compute and plot the posterior expectation of θB under the prior distribution θB ∼ gamma(12 × n0, n0) for each value of n0 = 1, 2, . . . , 50. Describe what sort of prior beliefs about θB would be necessary in order for the posterior expectation of θB to be close to that of θA. (c) Should knowledge about population A tell us anything about population B? Discuss whether or not it makes senes to have independent priors p(θA, θB) = p(θA) × p(θB).

Answer 1

We Know that , The proor distribution on the parameter    ,  .

   

Assume that the prior is ;

The posterior is

I simulate the result choosing

The extimate of the probability using the Monte carlo sampling is 0.604. The program is attached:

R program:

rm(list = ls())
r_a = 1; r_b = 1;
nu_a = 1.5; nu_b = 1.5
y_a = c(12, 9, 12, 14, 13, 13, 15, 8, 15, 6);
y_b = c(11, 11, 10, 9, 9, 8, 7, 10, 6, 8,8, 9, 7);
n = 1000;
z1 = rgamma(n = 1000, shape = r_a + sum(y_a), scale = nu_a + 1)
z2 = rgamma(n = 1000, shape = r_b + sum(y_b), scale = nu_b + 1);

s =length(which((z1 - z2)<0)); est = s/n; est; 1 - est

b) The following table gives the estimate of the event for different values of

n_0 Estimate of probability

10 0.579

10.1 0.512

10.2 0.436

10.3 0.373

10.4 0.374

10.5 0.268

10.6 0.216

10.7 0.171

10.8 0.139

12 0.001

So the probability is a decreasing function of  .