A beta prior for the binomial parameter ???? has ????1 = ????2 = ????.

a. When y = 0, for what values of ???? is the posterior density of ???? monotone decreasing and hence the HPD posterior interval of the form (0, U)? For such ????, report U as the quantile of a beta distribution.

b. Show that the improper case ???? = 0 corresponds to an improper uniform prior for log[????∕(1 − ????)]. With it, when y = 0 or 1, show that the posterior distribution of ???? is also improper.

c. Show that the uniform prior for ???? corresponds to a logistic distribution prior for the logit.