Chapter 16.2, Problem 1E

Refer to the results of Example 16.2 given in Table 16.1.

1. a Which of the two priors has the smaller variance?
2. b Compare the means and variances of the two posteriors associated with the beta (1.3) prior. Which of the posteriors has mean and variance that differ more from the mean and variance of the beta (1.3) prior?
3. c Answer the questions in parts (a) and (b) for the beta (10. 30) prior.
4. d Are your answers to parts (a)–(c) supported by the graphs presented in Figure 16.1(a) and (b)?
5. e Compare the posteriors based on n = 5 for the two priors. Which of the two posteriors has mean and variance that differs more from the mean and variance of the corresponding priors?

EXAMPLE 16.2 Consider the virulent-disease scenario and the results of Example 16.1. Compare the prior and posterior distributions of the Bernoulli parameter p (the proportion of responders to the new therapy) if we chose the values for α and β and observed the hypothetical data given below:

1. a α = 1, β = 3, n = 5, Σy_i = 2.
2. b α = 1, β = 3, n = 25, Σy_i = 10.
3. c α = 10, β = 30, n = 5, Σy_i = 2.
4. d α = 10, β = 30, n = 25, Σy_i = 10.