Exercise 5.13 Show that Jeffreys prior distribution for 2 of a normal distribution with given mean, results

Exercise 5.13 Show that Jeffreys prior distribution for σ2 of a normal distribution with given mean, results in a proper posterior distribution when there are at least two observations.

Exercise 5.14 Berger (2006) reported on a Bayesian analysis to estimate the positive predictive value θ of a diagnostic test for a disease from

(a) the prevalence of the disease (p0

(b) the sensitivity of the diagnostic test (p1) and

(c) 1-specificity of the diagnostic test (p2). Bayes theorem states that θ = p0 p1 p0 p1 + (1 − p0 ) p2 . Suppose that data xi (i = 0, 1, 2) are available with binomial distributions Bin(ni, pi). Show that an equal tail interval CI for θ based on Jeffreys priors π (pi) ∝ p −1/2 i (1 − pi)−1/2 enjoys good coverage properties. Employ sampling for this and compare the 95% equal-tail CI with a classical 95% CI.