Answer the following short questions and provide a brief justification for each question.

(a) Assume y binomial(, n = 10). Suppose that y = 1 was observed. Consider the following priors for : (i) U (0, 1); (ii) beta(1, 5); and (iii) beta(10, 10).

(a1) Are these priors conjugate? Are these priors proper?

(a2) What are the corresponding posteriors?

(a3) Which prior is the most informative and which prior is the least informa- tive? In order to answer this question, you need to set up a reasonable criterion.