Consider independent binary observations from two groups with ????i =

P(yi = 1) = 1 − P(yi = 0), and a binary predictor x. For the 2×2 contingency table summarizing the two binomials, let logit(????i) = ????0 + ????1xi, where xi = 1 or xi = 0 according to a subject’s group classification. Also, express logit(????i) = ????∗

0 + ????∗

1 x∗

i , where x∗

i = 1 when xi = 0 and x∗

i = 0 when xi = 1

(i.e., the classification when the group labels are reversed). Let ???? = exp(????1)

and ????∗ = exp(????∗

1 ) denote the corresponding odds ratios. The ML estimates satisfy ????̂∗

1 = −????̂

1 and ????̂∗ = 1∕????̂. For a Bayesian solution, denote the means of the posterior distributions of ????1 and ????∗

1 by ????̃

1 and ????̃∗

1 and the means of the posterior distributions of ???? and ????∗ by ????̃ and ????̃∗.

a. Explain why ????̃∗

1 = −????̃

1 but ????̃∗ ≠ 1∕????̃.

b. Let (L, U) denote the 95% HPD interval from the posterior distribution of ????. Explain why the 95% HPD interval from the posterior distribution of ????∗ is not (1∕U, 1∕L). Explain why such invariance does occur for the equal-tail interval and for frequentist inference.