Suppose that X is a discrete random variable with P(X=1) = theta and P(X=2) = (1-theta). Three independent observations are made: x_1 = 1, x_2 = 2, and x_3 = 2. Assume theta has a prior distribution that is uniform on [0, 1].

I was required to find the posterior dist'n, the Baye's estimate, and a 95% C.I. for theta. I found the posterior dist'n and the Baye's estimate based on the loss function, but I'm not sure how to construct a confidence interval for theta. Any ideas?

Thanks.