1. Laplace claimed that the probability that an event which has occurred n times, and has not hitherto failed, will occur again is (n+1)/(n+2) [see Laplace (1774)], which is sometimes known as Laplace’s rule of succession. Suggest grounds for this assertion.
2. Find a suitable interval of 90% posterior probability to quote in a case when your posterior distribution for an unknown parameter π is Be(20, 12), and compare this interval with similar intervals for the cases of Be(20.5, 12.5) and Be(21, 13) posteriors. Comment on the relevance of the results to the choice of a reference prior for the binomial distribution.
3. Suppose that your prior beliefs about the probability π of success in Bernoulli trials have mean 1/3 and variance 1/32. Give a 95% posterior HDR for π given that you have observed 8 successes in 20 trials.
4. Suppose that you have a prior distribution for the probability π of success in a certain kind of gambling game which has mean 0.4, and that you regard your prior information as equivalent to 12 trials. You then play the game 25 times and win 12 times. What is your posterior distribution for π?
5. Suppose that you are interested in the proportion of females in a certain organization and that as a first step in your investigation you intend to find out the sex of the first 11 members on the membership list. Before doing so, you have prior beliefs which you regard as equivalent to 25% of this data, and your prior beliefs suggest ...