Suppose $y|\theta \sim$ Exponential$(\theta)$, and the prior distribution of $\theta$ is Gamma$(α,β)$.
(a) Suppose we observe that $y \geq y'$, but do not observe the exact value of $y$. What is the posterior distribution, $p(\theta|y≥y')$, as a function of $\alpha$ and $\beta$? What are the posterior mean and variance of $\theta$?
(b) Suppose that we are now told that y is exactly 100. Now what are the posterior mean and variance of θ?
It seems clear that this distribution will exploit the memoryless property of the prior exponential distribution. However, I am not exactly clear on how to prove this; in particular I am not sure how to properly incorporate the information that $y \geq y'$ into the likelihood function in order to derive the posterior distribution for the censored data. (Note: assume $y'$ is a known constant.)
This question is a generalized version of a question from Gelman, Bayesian Data Analysis, 3rd. ed., ch. 2 exercise 20.