For fixed n and r, consider the following two experiments: E1: X Bi(n, ) and E2: Y NBin(r, ), using a negative binomial

For fixed n and r, consider the following two experiments:

E1: X ∼ Bi(n, θ) and E2: Y ∼ NBin(r, θ), using a negative binomial with f(y | θ) = C y+n−1 y θ

n

(1 − θ)

y where C n

k is the binomial coefficient and θ is a Bernoulli success probability.

a. Find Jeffreys’ prior h(θ) in both experiments.

Hint: E(Y) = r(1 − θ)/θ for the negative binomial distribution in E2 (in contrast to the parametrization used in Appendix A).

b. Use n = 2, r = 1. Using the priors found in part (a), compute p(θ > 0.5

|

X = 1) in E1; and p(θ > 0.5

| Y = 1) in E2.

c. The two probabilities computed in part

(b) will differ. Based on parts

(a) and (b), argue that inference based on Jeffreys’ prior can violate the Likelihood Principle (stated below for reference).

Likelihood Principle: Consider any two experiements with observed data x1 and x2 such that f(x1 | θ) = c(x1, x2) f(x2 | θ), i.e., the likelihood functions are proportional as a function of θ. The two experiments bring the same information about θ and must lead to identical inference (Robert, 1994, ch. 1).2