# Bayesian posterior of censored exponential distribution

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.

• Do I understand correctly that $y'$ is a (known) constant, so that the aswer to (a) would depend on $\alpha,\beta, y'$? Have you tried a "brute force" approach of writing the posterior as proportional to prior * likelihood? – Juho Kokkala Sep 25 '17 at 15:38
• Yes, you are correct; question updated to reflect this. I have indeed tried the "brute force" approach but the solution is not obvious from this; in particular, as I ask, the question that remains to me is how the truncation affects the posterior (otherwise, the distribution seems to be the same; a truncated exponential distribution is just another exponential). – jpgard Sep 25 '17 at 20:19
• (NeilG seems to have given a full answer, but in case you did not look at that yet and want a hint) By "brute force" I meant: don't try to reason about "truncation", just plug in $p(\theta \mid \textrm{Data}) \propto p(\textrm{Data} \mid \theta)\,p(\theta)$ – Juho Kokkala Sep 26 '17 at 6:04
• Just a note to future readers, I found the example (without truncation) useful here, in the slide titled "the exponential-gamma system" halweb.uc3m.es/esp/Personal/personas/mwiper/docencia/English/… – jpgard Sep 26 '17 at 15:49

1. Calculate the likelihood, which is the probability than an exponential distribution with rate $\theta$ has a sample greater than $y'$: $L(\theta \mid y>y') = e^{-y'{\theta}}$. This is an exponential distribution with rate $y'$, i.e., a gamma with shape 1 and rate $y'$. The posterior is therefore $\alpha$ and $\beta + y'$.
2. This is just the Bayesian update of $\alpha$ and $\beta$, which gives a gamma distribution with parameters $\alpha+1$ and $\beta + 100$.
• @jpgard write out the density $f(x) \propto e^{-\frac{x}{\theta}}$, after truncation, it is the same density over a different support. – Neil G Sep 23 '17 at 15:34